>Logarithmic Equations>How to Solve Exponential Inequalities

How to Solve Exponential Inequalities

When solving exponential inequalities it is very important that you look at the value of the argument $a$ in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by $\ln a$ or $\log a$ .

Rule

Solving exponential inequalities

When

a > 1

\ln (a) > 0

and you can solve the inequality as usual.

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & > \frac{\ln b}{\ln a} & x & > \frac{\log b}{\log a} \end{array}

When $0 < a < 1$ , $\ln (a) < 0$ , you will have to turn the inequality sign as you end up dividing or multiplying by a negative number!

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & < \frac{\ln b}{\ln a} & x & < \frac{\log b}{\log a} \end{array}

Example 1

Solve the inequality ${3.5}^{x} > 439$

\begin{array}{l} 3.5 x > 439 \ln {3.5}^{x} & > \ln 439 x \ln 3.5 & > \ln 439 & | \div \ln 3.5 x & > \frac{\ln 439}{\ln 3.5} \approx 4.9Example 2 Solve the inequality 50 \cdot {1.05}^{x} > 300 \begin{array}{l} 50 \cdot {1.05}^{x} > 300 | \div 50 {1.05}^{x} & > 6 \ln {1.05}^{x} & > \ln 6 x \ln 1.05 & > \ln 6 & | \div \ln 1.05 x & > \frac{\ln 6}{\ln 1.05} \approx 36.7Example 3 Solve the inequality 3 \cdot {0.25}^{x} > 27 \begin{array}{l} 3 \cdot {0.25}^{x} > 27 | \div 3 {0.25}^{x} & > 9 \lg {0.25}^{x} & > \lg 9 x \lg 0.25 & > \lg 9 & | \div \log 0.25 x & < \frac{\lg 9}{\log 0.25} \approx - 1.6Previous entry What Are Logarithmic Equations? Next entry How to Solve Logarithmic InequalitiesGo backHome Curriculum Stats LeagueAbout Us About House of Math Employees Career Media Lectures Contact support@houseofmath.com Address House of Math 470 Ramona Street Palo Alto, CA 94301 USA FAQ Links WeTakeAction Junior Math Pen & Paper exercises Encyclopedia Terms Privacy Cookies Download our app{"props":{"pageProps":{"campaign":false,"articleBody":" When solving exponential inequalities it is very important that you look at the value of the argument a in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by ln ⁡ a or log ⁡ a. Rule Solving exponential inequalities When a \u003e 1, ln ⁡ (a) \u003e 0 and you can solve the inequality as usual. ax \u003e b ax \u003e b ln ⁡ ax \u003e ln ⁡ b log ⁡ ax \u003e log ⁡ b x ln ⁡ a \u003e ln ⁡ b x log ⁡ a \u003e log ⁡ b x \u003e ln ⁡ b ln ⁡ a x \u003e log ⁡ b log ⁡ a When 0 \u003c a \u003c 1, ln ⁡ (a) \u003c 0, you will have to turn the inequality sign as you end up dividing or multiplying by a negative number! ax \u003e b ax \u003e b ln ⁡ ax \u003e ln ⁡ b log ⁡ ax \u003e log ⁡ b x ln ⁡ a \u003e ln ⁡ b x log ⁡ a \u003e log ⁡ b x \u003c ln ⁡ b ln ⁡ a x \u003c log ⁡ b log ⁡ a Example 1 Solve the inequality 3.5x \u003e 439 3.5x \u003e 439 ln ⁡ 3.5x \u003e ln ⁡ 439 x ln ⁡ 3.5 \u003e ln ⁡ 439 | ÷ ln ⁡ 3.5 x \u003e ln ⁡ 439 ln ⁡ 3.5 ≈4.9 Example 2 Solve the inequality 50 ⋅1.05x \u003e 300 50 ⋅1.05x \u003e 300 | ÷ 50 1.05x \u003e 6 ln ⁡ 1.05x \u003e ln ⁡ 6 x ln ⁡ 1.05 \u003e ln ⁡ 6 | ÷ ln ⁡ 1.05 x \u003e ln ⁡ 6 ln ⁡ 1.05 ≈36.7 Example 3 Solve the inequality 3 ⋅0.25x \u003e 27 3 ⋅0.25x \u003e 27 | ÷3 0.25x \u003e 9 lg ⁡ 0.25x \u003e lg ⁡ 9 xlg ⁡ 0.25 \u003e lg ⁡ 9 | ÷log ⁡ 0.25 x \u003c lg ⁡ 9 log ⁡ 0.25 ≈−1.6 ","topic":{"breadcrumbs":[{"_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","h1":"Encyclopedia","path":"/encyclopedia"},{"_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21","h1":"Algebra","path":"/encyclopedia/algebra"},{"_key":"5746e0ab-24ae-5841-af0f-f1b35cfab6c9","h1":"Logarithms","path":"/encyclopedia/algebra/logarithms"},{"_key":"db5be093-ee70-5f75-bdd2-f3d4e77d9fda","h1":"Logarithmic Equations","path":"/encyclopedia/algebra/logarithms/logarithmic-equations"},{"_key":"22deaf56-6168-575e-8a39-4a3d360193ac","h1":"How to Solve Exponential Inequalities","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-exponential-inequalities"}],"cssUrl":"/8c2ce5fa-9e80-4618-888c-fb8e70afc5d1-en-1.css","h1":"How to Solve Exponential Inequalities","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     When solving exponential inequalities it is very important that you look at the value of the argument \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e or \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-701\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     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form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e and you can solve the inequality as usual. \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                         \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace 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mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/math\u003e, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e, you will \u003cspan  class=\"ec-lmri-12x-x-144\"\u003ehave to \u003c/span\u003eturn the inequality sign as you end up dividing or multiplying by a negative number! \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                         \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                              \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-702\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                          \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                                          \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-703\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                        \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e                             \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e  \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                      \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-704\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                          \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003elg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" 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personalize your experience.","student":{"better-grades":{"title":"Better grades"},"prepare-for-test":{"title":"Prepare for a test or exam"},"catch-up-basics":{"title":"Catch up on the basics"},"get-ahead":{"title":"Get ahead / extra challenge"},"build-confidence":{"title":"Build confidence in math"},"improving-math-reasoning":{"title":"Get better at problem-solving"},"homework-support":{"title":"Homework support"},"school-choice-future-studies":{"title":"School choice / future studies"},"math-for-work":{"title":"Math for work (or career goals)"},"real-life-math":{"title":"Real-life math (money, measurement, everyday problems)"}},"parent":{"better-grades":{"title":"Improve their grades"},"prepare-for-test":{"title":"To prepare for upcoming tests or exams"},"catch-up-basics":{"title":"To prevent them from falling behind"},"get-ahead":{"title":"To build a strong foundation for future courses"},"build-confidence":{"title":"To reduce stress and frustration at 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if you have the right tool?","other":"Do you sometimes feel like: “I am not a math person”?"}},"lack-inner-power":{"title":{"student":"Are you happy with your math grade?","parent":"Do new math methods make it harder to help at home?","teacher":"Reaching every student so they all learn in each math class is challenging","other":"Are you happy with your math grade?"}},"like-math":{"title":{"student":"Do you like math?","parent":"How are your own skills in math?","teacher":"Is it challenging to teach students with very different math abilities?","other":"Do you like math?"},"subtitle":"We have your back no matter what you answer."},"like-to-be-encouraged":{"title.student":"Choose ways you like to be encouraged","title.teacher":"Choose ways you like to encourage your students","title.other":"Choose ways you like to be encouraged","title.parent":"Choose ways you like to encourage your children","subtitle.student":"You may choose more than one option.","subtitle.teacher":"You may choose more than one option.","subtitle.other":"You may choose more than one option.","subtitle.parent":"You may choose more than one option.","student":{"direct-honest":{"title":"Motivational push","subtitle":"challenge me, raise the bar"},"supportive-reassuring":{"title":"Supportive \u0026 reassuring","subtitle":"steady confidence and calm"},"celebrate-progress":{"title":"Celebrate progress","subtitle":"notice small wins along the way"},"high-expectations":{"title":"High-expectations","subtitle":"“I know you can do better” energy"},"low-pressure":{"title":"Low-pressure","subtitle":"gentle nudges, no urgency unless needed"}},"parent":{"direct-honest":{"title":"Motivational push","subtitle":"challenge them, raise the bar"},"supportive-reassuring":{"title":"Supportive \u0026 reassuring","subtitle":"steady confidence and calm"},"celebrate-progress":{"title":"Celebrate progress","subtitle":"notice small wins along the way"},"high-expectations":{"title":"High-expectations","subtitle":"“I know you can do better” energy"},"low-pressure":{"title":"Low-pressure","subtitle":"gentle nudges, no urgency unless needed"}},"teacher":{"direct-honest":{"title":"Motivational push","subtitle":"challenge me, raise the bar"},"supportive-reassuring":{"title":"Supportive \u0026 reassuring","subtitle":"steady confidence and calm"},"celebrate-progress":{"title":"Celebrate progress","subtitle":"notice small wins along the way"},"high-expectations":{"title":"High-expectations","subtitle":"“I know you can do better” energy"},"low-pressure":{"title":"Low-pressure","subtitle":"gentle nudges, no urgency unless needed"}}},"math-confident":{"title":{"student":"How confident do you feel in math right now?","parent":"It is easy to fit math practice into our weekly schedule","teacher":"I have tools that continuously assess understanding and guide my teaching","other":"How confident do you feel in math right now?"}},"math-plan-ready":{"title.student":"Your personal learning analysis is ready","title.other":"Your personal learning analysis is ready","title.parent":"We are ready to map your children's math skills","title.teacher":"You are now ready to check out the system where all students can learn math every time.","subtitle.student":"You are now ready to map your first math skills in 90 seconds without doing math! You heard right - without doing math!","subtitle.other":"You are now ready to map your first math skills in 90 seconds without doing math! You heard right - without doing math!","subtitle.parent":"You are now ready to map your first math skills in 90 seconds without doing math! You heard right - without doing math!","subtitle.teacher":"It is built to optimize student learning, so the teacher panel is not complete yet."},"math-practice-challenges":{"title":{"student":"My math practice could be more challenging","parent":"I see untapped potential in my children","teacher":"I see untapped potential in many of my students","other":"My math practice could be more challenging"}},"math-routine":{"title":{"student":"Mastering math makes me feel smart","parent":"My support makes a real difference in my children's math progress","teacher":"I have time to make all my students feel seen","other":"Mastering math makes me feel smart"}},"succeed-in-something":{"title.student":"When you get stuck, what do you do first?","title.teacher":"How do you usually celebrate student accomplishments?","title.other":"When you succeed in something, you'd rather...","title.parent":"How do you celebrate your children's accomplishments?","subtitle":"","student":{"think-of-what-could-be-done-better":{"title":"Give up pretty fast"},"celebrate-the-win":{"title":"Search for help"}},"parent":{"think-of-what-could-be-done-better":{"title":"Think about the next step"},"celebrate-the-win":{"title":"Celebrate the win"}},"teacher":{"think-of-what-could-be-done-better":{"title":"Think about the next step"},"celebrate-the-win":{"title":"Celebrate the win"}}},"time-you-commit":{"title.student":"How much time can you commit?","title.teacher":"Do you feel you catch learning gaps in time?","title.other":"How much time can you commit?","title.parent":"Rewards help motivate my children to practice math","subtitle":"","student":{"5-min":{"title":"5 min/day","subtitle":"quick daily practice"},"15-min":{"title":"15 min/day","subtitle":"steady progress"},"30-min":{"title":"30 min/day","subtitle":"Fast improvement"},"60-min":{"title":"60+ min/day","subtitle":"Maximum results"}},"parent":{"5-min":{"title":"Works like a charm","subtitle":"Finally self-going"},"15-min":{"title":"Increases stamina","subtitle":"They do more math"},"30-min":{"title":"Not sure","subtitle":"Does it have an effect?"},"60-min":{"title":"It is noise","subtitle":"They lose focus"}},"teacher":{"5-min":{"title":"Yes, full control","subtitle":"Impressive!"},"15-min":{"title":"Most of the time","subtitle":"We'll take you the last stretch"},"30-min":{"title":"Now and then","subtitle":"We'll help you see all"},"60-min":{"title":"Not really","subtitle":"No worries, we'll fix that"}}},"find-challenging-dragon":{"title":{"parent":{"1":"Helping with math feels difficult and uncertain","2":"You can help sometimes, but it doesn't always feel easy","3":"You generally feel comfortable helping your children with math","4":"Your confidence is a real advantage for your children"},"student":{"1":"Your future isn't decided — still much time to shape it into what you want","2":"Growth often starts with small steps and support","3":"Progress comes from showing up and trying – skills for life!","4":"Effort opens doors to math and a bright future"},"teacher":{"1":"One-size-fits-all homework often falls short","2":"Designing homework for mixed levels is hard","3":"Homework mostly works, but edge cases remain","4":"You already differentiate effectively"},"other":{"1":"Your future isn't decided — still much time to shape it into what you want","2":"Growth often starts with small steps and support","3":"Progress comes from showing up and trying – skills for life!","4":"Effort opens doors to math and a bright future"}},"subtitle":{"parent":{"1":"We'll guide you step by step so you know exactly how to support your children — even if math wasn't your strong subject.","2":"We'll strengthen your understanding and give you clear guidance so you feel more confident when your children need support.","3":"We'll help you stay aligned with what they need most to keep their progress — and grades — moving up.","4":"We'll help you use that strength strategically to support top performance and strong test results."},"student":{"1":"Our founder flunked math in high school and became a mathematician now helping millions master math. We know how to help you. Let's go!","2":"Even a little effort can unlock understanding and new possibilities over time. We are here to get you going and guide you on your way.","3":"Steady effort leads to improvement and prepares you for bigger challenges ahead. We'll take you even one step further.","4":"Working hard builds skills, confidence, and habits that create opportunities in school and in life. We are here to make your journey fast, easy and fun!"},"teacher":{"1":"We'll map individual gaps and generate personalized practice that supports every learner so they progress between every class","2":"We'll identify where students diverge and create targeted paths that meet them where they are. That way all students progress between classes.","3":"We'll help you spot where adjustments can unlock better outcomes for more students. That way all students progress between classes.","4":"You stay in control, with clear insight into how to support each student and maintain progress between classes."},"other":{"1":"Our founder flunked math in high school and became a mathematician now helping millions master math. We know how to help you. Let's go!","2":"Even a little effort can unlock understanding and new possibilities over time. We are her to get you going and guide you on your way.","3":"Steady effort leads to improvement and prepares you for bigger challenges ahead. We'll take you even one step further.","4":"Working hard builds skills, confidence, and habits that create opportunities in school and in life. We are here to make your journey fast, easy and fun!"}}},"personal-math-recipe":{"title.student":{"1":"Your personal learning recipe is 72% ready!","2":"Your personal learning recipe is 72% ready!","3":"Your personal learning recipe is 72% ready!","4":"Your personal learning recipe is 72% ready!"},"title.other":{"1":"Your personal learning recipe is 72% ready!","2":"Your personal learning recipe is 72% ready!","3":"Your personal learning recipe is 72% ready!","4":"Your personal learning recipe is 72% ready!"},"title.parent":{"1":"Your learning analysis is 72% ready!","2":"Your learning analysis is 72% ready!","3":"Your learning analysis is 72% ready!","4":"Your learning analysis is 72% ready!"},"title.teacher":{"1":"Your learning analysis is 72% ready!","2":"Your learning analysis is 72% ready!","3":"Your learning analysis is 72% ready!","4":"Your learning analysis is 72% ready!"},"subtitle.parent":{"1":"Just a little more, and you are ready to give your children a new math start","2":"Just a little more, and you are ready to give your children a new math start","3":"Just a little more, and you are ready to give your children a new math start","4":"Just a little more, and you are ready to give your children a new math start"},"subtitle.teacher":{"1":"Just a little more, and you are ready to give your students a new math start","2":"Just a little more, and you are ready to give your students a new math start","3":"Just a little more, and you are ready to give your students a new math start","4":"Just a little more, and you are ready to give your students a new math start"},"subtitle.other":{"1":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","2":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","3":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","4":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe."},"subtitle.student":{"1":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","2":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","3":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","4":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe."}},"friendly-push-dragon":{"title":{"student":{"1":"You rarely need external encouragement","2":"An occasional nudge can be helpful","3":"A friendly push is a powerful motivator","4":"Support and encouragement make a big difference for you"},"other":{"1":"You rarely need external encouragement","2":"An occasional nudge can be helpful","3":"A friendly push is a powerful motivator","4":"Support and encouragement make a big difference for you"},"parent":{"1":"Stressful and hard to manage","2":"Often challenging","3":"Mostly manageable","4":"Smooth and stress-free"},"teacher":{"1":"You're not alone! It can be hard to know if the basics are truly sticking","2":"Foundations can look solid until the content ramps up","3":"You're in a good place","4":"It sounds like your students are on a strong track"}},"subtitle":{"student":{"1":"You're highly self-driven. You're confident in staying motivated and moving forward on your own. Excellent trait to have to master math. But being seen is always nice, so we'll be here to cheer you on.","2":"You're generally self-motivated. You usually stay on track without much outside support. We'll be your little friendly reminder to keep going steadily.","3":"We'll tailor practice, explanations, and recommendations to help you keep progressing toward mastery.","4":"We'll proactively guide and motivate you with timely nudges to help you stay focused and make steady progress with your math."},"other":{"1":"You're highly self-driven. You're confident in staying motivated and moving forward on your own. Excellent trait to have to master math. But being seen is always nice, so we'll be here to cheer you on.","2":"You're generally self-motivated. You usually stay on track without much outside support. We'll be your little friendly reminder to keep going steadily.","3":"We'll tailor practice, explanations, and recommendations to help you keep progressing toward mastery.","4":"We'll proactively guide and motivate you with timely nudges to help you stay focused and make steady progress with your math."},"parent":{"1":"Homework at home is usually stressful and difficult for our family to handle.","2":"Homework time can be tough, and my children often get stuck or frustrated.","3":"Most days it goes fine, but sometimes my children need extra time or a bit of help from me.","4":"Homework time at home is usually calm, and my children can finish without much stress."},"teacher":{"1":"A structured baseline check can pinpoint missing prerequisites and give you a clear plan for strengthening their foundation step by step.","2":"A targeted diagnostic can quickly show which core skills need reinforcing so you can focus your time where it matters most","3":"A short skills check can validate the foundation and reveal any small gaps before they become barriers in higher-level topics","4":"A quick mastery snapshot can help confirm consistency across the class and highlight who's ready for stretch challenges"}}},"personal-math-recipe-2":{"title.student":{"1":"Your personal learning recipe is 94% ready!","2":"Your personal learning recipe is 94% ready!","3":"Your personal learning recipe is 94% ready!","4":"Your personal learning recipe is 94% ready!"},"title.other":{"1":"Your personal learning recipe is 94% ready!","2":"Your personal learning recipe is 94% ready!","3":"Your personal learning recipe is 94% ready!","4":"Your personal learning recipe is 94% ready!"},"title.parent":{"1":"Your learning analysis is 94% ready!","2":"Your learning analysis is 94% ready!","3":"Your learning analysis is 94% ready!","4":"Your learning analysis is 94% ready!"},"title.teacher":{"1":"Your learning analysis is 94% ready!","2":"Your learning analysis is 94% ready!","3":"Your learning analysis is 94% ready!","4":"Your learning analysis is 94% ready!"},"subtitle.student":{"1":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","2":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","3":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","4":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps."},"subtitle.other":{"1":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","2":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","3":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","4":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps."},"subtitle.parent":{"1":"You are nearly there now!","2":"You are nearly there now!","3":"You are nearly there now!","4":"You are nearly there now!"},"subtitle.teacher":{"1":"You are nearly there now!","2":"You are nearly there now!","3":"You are nearly there now!","4":"You are nearly there now!"}},"succeed-in-something-dragon":{"title":{"student":{"1":"Getting stuck can feel frustrating and overwhelming","2":"Not giving up is a winning strategy in math and life"},"other":{"1":"Getting stuck can feel frustrating and overwhelming","2":"Not giving up is a winning strategy in math and life"},"parent":{"1":"You're a math improver","2":"You balance progress and celebration"},"teacher":{"1":"You're forward-looking by nature","2":"You balance progress and celebration"}},"subtitle":{"parent":{"1":"You like refining your skills and spotting what to fix next. We'll use that mindset to provide targeted feedback and practice that tightens the small gaps fast.","2":"You notice what works and what needs improvement. We'll tailor the learning experience with steady challenges and clear milestones, so your children will keep moving forward and feel the wins."},"student":{"1":"It's normal to feel overwhelmed sometimes. We'll support you through tough moments to turn struggle into learning breakthroughs.","2":"Asking questions and using available resources builds understanding and confidence. We'll be your No. 1 resource in these moments."},"teacher":{"1":"You like refining their skills and spotting what they need fixing next. We'll use that mindset to provide targeted feedback and practice that tightens the small gaps fast.","2":"You notice what works and what needs improvement. 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We'll be your No. 1 resource in these moments."}}},"like-math-dragon":{"title":{"parent":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it ","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"},"student":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"},"teacher":{"1":"You handle mixed-ability classrooms with confidence.","2":"You've found ways to manage differences in ability","3":"Different ability levels is a challenge in some lessons","4":"Teaching wide ability ranges can feel overwhelming"},"other":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"}},"subtitle":{"parent":{"1":"Don't worry! We'll help you and your children to build confidence step by step, so math becomes more approachable over time.","2":"Don't worry! We'll help your children to strengthen the basics and discover strategies that make their math learning faster and easier.","3":"We'll support you with practice and guidance to help your children to achieve great performance.","4":"Together with House of Math, you can inspire your children to reach their full potential"},"student":{"1":"We'll focus on building confidence step by step, so math becomes more approachable over time. Our system is the fastest and easiest way to learn math.","2":"We'll help you strengthen the basics and discover strategies that make math learning faster and easier.","3":"We'll support you with practice and guidance to help you continue improving to achieve great performance.","4":"We help you find tons of tips on how to reach your full potential during practice and testing."},"teacher":{"1":"When differentiation already works well, smart insights can help you go even further without adding extra work.","2":"Extra insight helps you fine-tune support as gaps grow or shrink. We'll help you differentiate down to each and every student.","3":"Balancing pace and support takes effort. We'll be the flexible tool that helps you adapt without constantly re-planning.","4":"When students are far apart, it's hard to support everyone at once. 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We help you build real understanding to keep your momentum going."},"teacher":{"1":"We'll help you prioritize and use small moments more effectively so every student feels noticed and supported","2":"We'll help you spot where small adjustments can make a big difference","3":"We'll help you turn those moments into consistent, meaningful support for everyone","4":"We'll help you scale that care so no one slips through the cracks"},"other":{"1":"Progress isn't always loud, but the work you put in matters and deserves recognition.","2":"Even if it doesn't excite you yet, we'll help make progress feel clearer—and worth noticing.","3":"With the right guidance and practice, we help turn steady progress into moments where math really clicks.","4":"That feeling matters. We help you build real understanding to keep your momentum going."}}},"feel-math-skills-dragon":{"title":{"parent":{"1":"Motivation is a real challenge right now","2":"Motivation comes and goes","3":"Your children are generally motivated.","4":"Your children are highly motivated"},"student":{"1":"We'll make you feel less nervous and more confident before your math tests","2":"We'll make you feel less nervous before your math tests","3":"Math tests is a moment of  happiness for you","4":"You love taking math tests and we understand why!"},"teacher":{"1":"Motivation is often a challenge","2":"Motivation comes and goes","3":"Most students are generally motivated","4":"Your students are highly motivated to learn math"},"other":{"1":"We'll make you feel less nervous and more confident before your math tests","2":"We'll make you feel less nervous before your math tests","3":"Math tests is a moment of  happiness for you","4":"You love taking math tests and we understand why!"}},"subtitle":{"parent":{"1":"We'll help make progress visible and rewarding, so your children stay engaged and confident","2":"We'll support you with strategies that keep your children focused and moving forward — especially when things get hard.","3":"We'll help maintain that momentum with clear goals and progress tracking","4":"We'll help channel that drive into consistent improvement and strong test performance."},"student":{"1":"That's okay. With our support, practice, and feedback, we help reduce anxiety and build confidence over time.","2":"You're not alone. We help you prepare step by step so tests feel less stressful and more manageable.","3":"With clear goals and targeted practice, we help turn that happiness into even more confidence and fun.","4":"That confidence comes from preparation and skills. We help you practice in ways that make test day feel familiar and fun."},"teacher":{"1":"We'll help you identify what's holding students back and introduce strategies that spark engagement and confidence","2":"We'll help you create more consistent momentum and make math feel more rewarding for your students","3":"We'll help you strengthen what works and re-engage those who occasionally fall behind","4":"We'll help you sustain that energy and translate it into deep, lasting understanding"},"other":{"1":"That's okay. With our support, practice, and feedback, we help reduce anxiety and build confidence over time.","2":"You're not alone. We help you prepare step by step so tests feel less stressful and more manageable.","3":"With clear goals and targeted practice, we help turn that happiness into even more confidence and fun.","4":"That confidence comes from preparation and skills. 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Rows without a sis_id are new pupils every time, so their usernames must not be taken already.","importSubmit":"Import","importTargetClassTeachers":"Teachers → class","importTargetLabel":"What are you importing?","importTargetSchoolTeachers":"Teachers","importTargetStudents":"Students → class","importTeachersSummary":"{{ created }} of {{ rowsRead }} teachers created, {{ alreadyPresent }} already in the school.","importTitle":"Roster import","introEmpty":"Create a school, then add its classes and teachers.","inviteConflictOtherSchool":"{{ email }} already teaches at {{ school }}. A teacher can belong to only one school, so remove them there first, or invite a different address.","inviteConflictOtherSchoolUnnamed":"{{ email }} already belongs to another school. A teacher can belong to only one school, so remove them there first, or invite a different address.","inviteConflictRole":"{{ email }} is already at this school in another role. 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Refresh and try again.","removeFromClassLabel":"Remove {{ name }} from {{ class }}","removedFromClass":"{{ name }} no longer teaches {{ class }}.","retry":"Try again","save":"Save changes","saving":"Saving…","schoolCount_one":"{{ count }} school","schoolCount_other":"{{ count }} schools","schools":"Schools","searchPlaceholder":"Search schools by name","searchTeachers":"Search by name or email","sendInvite":"Send invite","sending":"Sending…","showingCount":"Showing {{ shown }} of {{ total }} schools","showingTeacherCount":"Showing {{ shown }} of {{ total }} teachers","sortBy":"Sort by","sortFewestTeachers":"Fewest teachers","sortName":"Name (A–Z)","sortNewest":"Newest first","status":"Status","statusActive":"Active","statusArchived":"Archived","statusSuspended":"Suspended","studentCount_one":"{{ count }} student","studentCount_other":"{{ count }} students","studentsLabel":"students","studentsUnit_one":"student","studentsUnit_other":"students","suspendWarning":"Suspending this school stops its teachers and students from signing in.","suspendedConfirm":"{{ name }} is suspended — its teachers and students can no longer sign in.","teacherActive":"Active","teacherCount_one":"{{ count }} teacher","teacherCount_other":"{{ count }} teachers","teacherEmail":"Teacher email","teacherEmailPlaceholder":"name@school.org","teacherFiltersUnavailable":"There are more than {{ count }} teachers, so search is unavailable and the list is paged instead.","teacherRemoved":"Removed","teachers":"Teachers","teachersLabel":"teachers","teachersUnit_one":"teacher","teachersUnit_other":"teachers","updateClassFailed":"Could not save the changes. Try again.","updateFailed":"Could not save the changes. Try again.","updated":"Changes saved","view":"View school","viewClass":"View class","viewClassrooms":"Classrooms","viewTeachers":"Teachers","districtSchoolId":"District school id","districtSchoolIdHint":"The school's id in the district's roster files, exactly as they write it. Optional: sign-in and the uploads work without it.","districtSchoolIdTaken":"Another school already has this district school id.","districtClassId":"District class id","districtClassIdHint":"The class's id in the district's roster files, exactly as they write it (5A is not 5a). Required for a class in the district's roster: the school-wide student and teacher uploads find the class by it. 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Please try again later","notFamilySubscription":"Last subscription is not a family plan","alreadyInThisFamily":"You are already in this family","invalidEmail":"Invalid email address","adminAlreadyHasLicense":"Admin already has math access","userAlreadyHasLicense":"User already has math access","noActiveSubscription":"You don’t have an active subscription","subscriptionNotActiveOrTrialing":"To upgrade, your subscription must be active or trialing. You can either reactivate it or wait until your current cancellation period ends before upgrading.","vippsNotSupportedForFamilyUpgrade":"Family upgrade isn’t supported with Vipps payment yet. We are working on it.","alreadyOnFamilyPlan":"You’re already on a family plan.","learnerCannotUpgrade":"You’re part of a family plan and can’t upgrade. The family admin has to remove you from the family before you can upgrade.","paymentFailed":"Your payment failed. Your card details may be missing or expired.","subscriptionScheduledForCancellation":"Can not perform the action as the subscription is scheduled for cancellation."},"logoutMessage":"This user need to login with their own username and password","removingLicense":"You are removing a math access.","removedAfterPeriod":"This math access will remain active until the end of your current billing period: {{date}}","willLoseAccess":"\u003citalic\u003e{{username}}\u003c/italic\u003e associated with this math access will lose access on that date.","willNotBeDeleted":"\u003citalic\u003e{{username}}\u003c/italic\u003e will not be deleted.","pendingRemoval":"Pending removal {{date}}","removingTheLast":"You are about to remove the last member of your family group.","toProceed":"To proceed, go to the subscription management page","toManageSubscription":"Go to mange subscription page","buyLicense":"Buy math access","assignMathAccess":"Assign Math access","userAccessSuccess":"{{user}} is now a math learner","createPin":"Create admin pin","pinProtection":"Protect the admin panel by\u003cbr /\u003e setting a PIN.","setPinDescription":"Use this pin to ensure no one else can access the admin panel","connectAccount":"Do you want to connect this profile to an existing House of Math account?","ConfirmPay":"Confirm and pay","toPayNow":"To pay now","yourPlan":"Your plan","acceptAs":"You will be accepting the invite as","useOther":"Want to use another account?","subscriptionWarning":"By accepting this invitation, your subscription will be transferred to the Family Account. 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wareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"notificationApi\"}},\"homApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":5,\"keepUnusedDataFor\":1,\"reducerPath\":\"homApi\"}},\"soundControl\":{\"muted\":false,\"volume\":1},\"blogApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"blogApi\"}},\"staticPageApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"staticPageApi\"}},\"activityApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"activityApi\"}},\"bootcampApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"bootcampApi\"}},\"craneGameApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"craneGameApi\"}},\"rewardApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"rewardApi\"}},\"dragonApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"dragonApi\"}},\"gemShopApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"gemShopApi\"}},\"usersCountApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"usersCountApi\"}},\"mathVaultActivityApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"mathVaultActivityApi\"}},\"mediaApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"mediaApi\"}},\"adminApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":true,\"keepUnusedDataFor\":1,\"reducerPath\":\"adminApi\"}},\"multipleChoiceTestApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"multipleChoiceTestApi\"}},\"questApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":false,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"questApi\"}},\"studytipsApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\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