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What Are Quadratic Inequalities?

Quadratic inequalities are polynomial inequalities of degree 2. You solve them using sign charts. This is how to do it:

Rule

Solving Quadratic Inequalities

1.: Rearrange the inequality to get 0 is on the right-hand side.
2.: Rearrange the expression on the left-hand side in order of descending degree.
3.: Factorize the expression.
4.: Make a sign chart.
5.: Read off the solution.

Example 1

Solve the inequality $x^{2} - 4 \geq 3 x$

\begin{array}{l} x^{2} - 4 & \geq 3 x \\ x^{2} - 3 x - 4 & \geq 0 \end{array}

You can factorize this by using the quadratic formula:

\begin{array}{l} (x - 4) (x + 1) & \geq 0 \end{array}

Next, you make a sign chart:

The sign charts of x-4 and x+1 are combined to create the sign chart of (x-4)(x+1)

You can read off that $x^{2} - 4 \geq 3 x$ when $x \in (- \infty, - 1] \cup [4, \infty)$ . Because we had a greater than or equal sign ( $\geq$ ) in the original inequality, the intervals are closed on the side that does not tend to infinity.

How to Solve Inequalities of Degree 3 or More

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