Quadratic Inequality

A quadratic inequality asks for all values of $x$ that make a quadratic expression greater than, less than, at least, or at most another value. The standard tool is to rewrite it so one side is $0$, find the zeros, and use those zeros to decide which intervals satisfy the inequality.

A quadratic inequality involves a degree-$2$ expression and an inequality sign, such as

$$ax^2 + bx + c > 0$$

or

$$ax^2 + bx + c \le 0,$$

with $a \ne 0$. Here $a$, $b$, and $c$ are the coefficients of the quadratic. The key difference from a quadratic equation is the goal: an equation asks for the roots, while an inequality asks for the interval or intervals where the quadratic stays above or below $0$.

Why the zeros drive the method

The zeros matter because they are the only real points where the sign can change. Between consecutive zeros, a quadratic expression stays entirely positive or entirely negative, so testing a single value from each interval is enough to determine the sign there. That is what makes the sign-chart method reliable.

The graph makes this visible. The graph of a quadratic is a parabola. A solution to $ax^2 + bx + c > 0$ is any $x$-value where the parabola is above the $x$-axis; a solution to $ax^2 + bx + c < 0$ is any $x$-value where it is below. When the quadratic has two real roots, that gives a fast check:

• If the parabola opens upward, it is often positive outside the roots and negative between them.
• If the parabola opens downward, that pattern reverses.

This shortcut depends on the quadratic having real zeros. If there are no real zeros, the sign does not switch across the number line, so you must reason from the graph or the leading coefficient.

The method, step by step

1. Move everything to one side so the other side is $0$.
2. Find the zeros by factoring or another solving method.
3. Use the zeros to split the number line into intervals.
4. Test one value from each interval, or reason from the graph if the roots are clear.
5. Keep the intervals that make the inequality true.

If the inequality is strict, like $>$ or $<$, do not include the zeros. If it is inclusive, like $\ge$ or $\le$, include them.

Worked example: $x^2 - 5x + 6 > 0$

The quadratic is already compared with $0$, so start by factoring:

$$x^2 - 5x + 6 = (x - 2)(x - 3).$$

Now the zeros are $x = 2$ and $x = 3$. These split the number line into three intervals:

• $(-\infty, 2)$
• $(2, 3)$
• $(3, \infty)$

Test one value from each interval.

For $x = 0$:

$$(0 - 2)(0 - 3) = 6 > 0$$

So $(-\infty, 2)$ works.

For $x = 2.5$:

$$(2.5 - 2)(2.5 - 3) = (0.5)(-0.5) < 0$$

So $(2, 3)$ does not work.

For $x = 4$:

$$(4 - 2)(4 - 3) = 2 > 0$$

So $(3, \infty)$ works.

The solution is

$$x < 2 \text{ or } x > 3.$$

In interval notation, that is

$$(-\infty, 2) \cup (3, \infty).$$

Because the original inequality is $>$, the endpoints $2$ and $3$ are not included.

Practice it yourself

Solve $x^2 - 4x - 5 \le 0$. Factor first, mark the zeros, and decide whether the endpoints belong before you test the intervals. As a second check, compare your interval answer with the graph and confirm both methods agree.

Calculation traps to avoid

The most common trap is solving the related equation and stopping with the roots. The roots are usually the boundaries of the answer, not the full answer.

Another is including endpoints when the inequality is strict. In $x^2 - 5x + 6 > 0$, the values $x = 2$ and $x = 3$ make the expression equal to $0$, so they do not belong in the solution set.

A third is assuming the answer is always between the roots. That depends on the sign you want and on whether the parabola opens upward or downward.

When quadratic inequalities are used

Quadratic inequalities appear in algebra, graphing, optimization, and applied problems with limits. They are useful when you need a range of valid inputs rather than one exact answer, for example describing when a height stays above a threshold, when a profit model is positive, or when a formula stays within an allowed region.