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

In these lessons, we will look at solving quadratic inequalities.
*x*^{2} – 4*x* > –3

*x*^{2} – 4*x* + 3 > 0

*x*^{2} – 4*x* + 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

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

2*x*^{2} – 9*x* – 5 < 0

*x*^{2} – 9*x* – 5 < 0

(2*x* + 1)(*x* – 5) < 0

*x* + 1)(*x* – 5) < 0 (*y* is negative): we choose the interval for which the curve is below the *x*-axis.

**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**

1) Turn inequality into an equation.

2) Find solutions.

3) Make a number line, and check each solution and interval.

Examples:

Solve x^{2} + 2x - 8 ≥ 0

**Solving Quadratic Inequalities - Step by step**

Find all the solutions to

2x^{2} + 5x - 12 ≥ 0

**Solving Quadratic Inequalities**

Find all the solutions to

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

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

** Example:**

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

**Solution:**

**Step 1:** Make one side of the inequality zero

**Step 2:** Factor the quadratic expression

(

(

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

**Solution:**

**Step 1: **Make one side of the inequality zero

2

**Step 2: **Factor the quadratic expression

(2

**Step 3:** Find the range of values of *x* which satisfies the inequality.

**Note:** If the quadratic inequality was (2*x* + 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

1. Find all the zeros of the polynomial, and arrange the zeros in increasing order. The zeros are called its critical numbers.

2. Plot those numbers on the number line as open or closed points based upon the original inequality symbol.

3. 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.

4. Clearly graph your solution and state the solution using interval notation or inequalities.

Examples:

1. Solve x^{2} - 6x - 16 ≤ 0

2. Solve 2x^{2} - 11x + 12 > 0

3. Solve x^{2} + 4 > 0

Example:

x

Example:

2x

1) Turn inequality into an equation.

2) Find solutions.

3) Make a number line, and check each solution and interval.

Examples:

Solve x

Find all the solutions to

2x

Find all the solutions to

2x

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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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