Lesson 4: Quadratic Inequalities

New Concept

A quadratic inequality like $ax^2 + bx + c > 0$ asks for which $x$-values a parabola's graph is above the $x$-axis. This lesson teaches you how to find these solution intervals using both graphs and algebra.

What’s next

You'll start by solving inequalities visually by interpreting graphs. Then, you will apply your skills through a series of interactive examples and practice cards.

Property

To solve a quadratic inequality graphically, graph the corresponding quadratic function. The solution set consists of the $x$-values for which the graph lies above or below the $x$-axis (or a given horizontal line), as required by the inequality. For an inequality like $ax^2 + bx + c > k$, we find the $x$-values where the parabola $y = ax^2 + bx + c$ is above the line $y=k$.

Examples

• To solve $x^2 > 9$ graphically, graph $y=x^2$ and $y=9$. The parabola is above the line for $x < -3$ or $x > 3$.

• To solve $x^2 - 2x - 3 ext{≤} 0$, graph $y = x^2 - 2x - 3$. The parabola is below or on the $x$-axis for $x$-values between $-1$ and $3$, so the solution is $-1 ext{≤} x ext{≤} 3$.

Property

A solution set like $a < x < b$ is called a compound inequality because it involves more than one inequality symbol. It describes the set of all numbers that are simultaneously greater than $a$ and less than $b$. We read '$a < x < b$' as '$x$ is greater than $a$ and less than $b$' or '$x$ is between $a$ and $b$'.

Examples

• The statement 'a number $n$ is greater than 5 and less than 12' is written as the compound inequality $5 < n < 12$.

• If a thermostat must be kept between 68 and 75 degrees, the temperature $T$ can be described as $68 ext{≤} T ext{≤} 75$.

Property

An interval is a set that consists of all the real numbers between two numbers $a$ and $b$.

1. The closed interval $[a, b]$ is the set $a \le x \le b$.
2. The open interval $(a, b)$ is the set $a < x < b$.
3. Intervals may also be half-open or half-closed, like $[a, b)$ which is $a \le x < b$.
4. The infinite interval $[a, \infty)$ is the set $x \ge a$.
5. The infinite interval $(-\infty, a]$ is the set $x \le a$.

A union of intervals, denoted with $\cup$, combines two or more sets.

Examples

• The inequality $-4 \le x < 1$ is written in interval notation as $[-4, 1)$.

• The set of all numbers greater than 5 is written as the infinite interval $(5, \infty)$.

Property

To solve a quadratic inequality algebraically:

1. Write the inequality in standard form: One side is zero, and the other has the form $ax^2 + bx + c$.
2. Find the $x$-intercepts of the graph of $y = ax^2 + bx + c$ by setting $y = 0$ and solving for $x$.
3. Make a rough sketch of the graph, using the sign of $a$ to determine whether the parabola opens upward or downward.
4. Decide which intervals on the $x$-axis give the correct sign for $y$ based on the inequality.

Examples

• To solve $x^2 - 16 > 0$, find intercepts at $x = -4$ and $x = 4$. The parabola opens up, so it's positive (above the axis) on the outside: $x < -4$ or $x > 4$.

• To solve $x^2 - 7x + 10 ext{≤} 0$, factor to get $(x-2)(x-5) ext{≤} 0$. The intercepts are $2$ and $5$. The parabola opens up, so it's negative (below the axis) between them: $2 ext{≤} x ext{≤} 5$.