Quadratic Inequalities

These free video lessons with examples and solutions help Algebra students learn how to solve quadratic inequalities.

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Example:
Solve the quadratic inequality x² – 4x > –3

Solution:
Step 1: Make one side of the inequality zero
x² – 4x > –3
x² – 4x + 3 > 0

Step 2: Factor the quadratic expression
x² – 4x + 3 > 0
(x – 3)(x – 1) > 0

Step 3: Find the range of values of x which satisfies the inequality.
(x – 3)(x – 1) > 0 (y is positive): we choose the interval for which the curve is above the x-axis.
x < 1 or x > 3

Note: If the quadratic inequality was (x – 3)(x – 1) < 0 (y is negative), we would have chosen the interval for which the curve is below the x-axis i.e. 1 < x < 3

The following graphs show the solutions for x² – 4x + 3 > 0 and x² – 4x + 3 < 0. Scroll down the page for more examples and solutions.

Solve Quadratic Inequalities

Example:
Solve 2x² < 9x + 5

Solution:
Step 1: Make one side of the inequality zero
2x² < 9x + 5
2x² – 9x – 5 < 0

Step 2: Factor the quadratic expression
2x² – 9x – 5 < 0
(2x + 1)(x – 5) < 0

Step 3: Find the range of values of x which satisfies the inequality.
(2x + 1)(x – 5) < 0 (y is negative): we choose the interval for which the curve is below the x-axis.

Note: If the quadratic inequality was (2x + 1)(x – 5) > 0 (y is positive) we would have chosen the interval for which the curve is above the x-axis i.e. or x > 5

How to solve quadratic inequalities?

Guidelines for solving Quadratic Inequalities

Find all the zeros of the polynomial, and arrange the zeros in increasing order. The zeros are called its critical numbers.
Plot those numbers on the number line as open or closed points based upon the original inequality symbol.
Choose a test value in each interval to see if the interval satisfies the inequality or not. If the test value produces a true statement, the entire interval will be true. If the interval produces a false statement, the entire interval is false.
Clearly graph your solution and state the solution using interval notation or inequalities.

Examples: