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

**Related Pages**

Solving Quadratic Equations

Quadratic Inequalities 2

More Algebra Lessons

**Example:**

Solve the quadratic inequality x^{2} – 4x > –3

**Solution:**

Step 1: Make one side of the inequality zero

x^{2} – 4x > –3

x^{2} – 4x + 3 > 0

Step 2: Factor the quadratic expression

x^{2} – 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 x^{2} – 4x + 3 > 0 and x^{2} – 4x + 3 < 0.
Scroll down the page for more examples and solutions.

**Example:**

Solve 2x^{2} < 9x + 5

**Solution:**

Step 1: Make one side of the inequality zero

2x^{2} < 9x + 5

2x^{2} – 9x – 5 < 0

Step 2: Factor the quadratic expression

2x^{2} – 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:**

- Solve x
^{2}- 6x - 16 ≤ 0 - Solve 2x
^{2}- 11x + 12 > 0 - Solve x
^{2}+ 4 > 0

**Solve a Quadratic Inequality**

**Example:**

x^{2} - x - 12 ≤ 0

**How to solve a Quadratic Inequality?**

**Example:**

2x^{2} + 3x - 5 > 0

**Solving Quadratic Inequalities**

- Turn inequality into an equation.
- Find solutions.
- Make a number line, and check each solution and interval.

**Example:**

Solve x^{2} + 2x - 8 ≥ 0

**Solving Quadratic Inequalities - Step by step**

**Example:**

Find all the solutions to

2x^{2} + 5x - 12 ≥ 0

**Solving Quadratic Inequalities**

**Example:**

Find all the solutions to

2x^{2} < -4x + 6

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