# Solving Nonlinear Inequalities - Cypress · PDF file Objective 5: Rational Inequalities Example: Solve the inequality 3 0 7 x x Graph the solution to the inequality. Write the solution

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• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 1 of 12

Solving Nonlinear Inequalities

Objective 1: Interval Notation and Set Builder Notation Inequality Graph Set Builder Notation Interval Notation

Example: Graph the solution to the compound inequality. Write the solution in set builder notation and in interval

notation. 3 1 5 11x or x or x     

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 2 of 12

Pause the video and try these problems.

Graph the solution to the inequality. Write the solution to the inequality both in set builder notation and interval

notation.

1. 3 4x  

2. 9x 

3. 8 2x or x  

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 3 of 12

Objective 2: Use the Graph of a Function to Solve an Inequality

Example: Solve the inequality 2 5 6x x 

||| 106-3

0

0

is negative

is positive

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 4 of 12

Pause the video and try these problems.

1. Consider the graph of the function 212y x x   .

Use the graph to solve the inequality 212 0x x   .

Write the solution to the inequality both in set builder

notation and interval notation.

Example: Solve the inequality 2 12 35x x  

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 5 of 12

Example: Solve the inequality 220 3 2 0x x  

Example: Solve the inequality 2 5x 

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 6 of 12

Pause the video and try these problems.

1. 2 6x x

2. 26 5 0x x  

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 7 of 12

3. 2 7 0x  

Objective 4: Polynomial Inequalities

Example: Solve the inequality   4 2 0x x x   Write your answer in interval notation.

Find critical values:

1. Move all terms to one side.

2. Factor.

3. Set each factor equal to

zero and solve.

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 8 of 12

Example: Solve the inequality 3 16x x

Example: Solve the inequality     3 2

4 1 5 0x x x   

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 9 of 12

Pause the video and try these problems.

1. 3 25 6x x x  

2.     2

3 8 0x x  

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 10 of 12

Objective 5: Rational Inequalities

Example: Solve the inequality 3

0 7

x

x

 

Graph the solution to the inequality. Write the solution to the inequality both in set builder notation and interval

notation.

Example: Solve the inequality 3

2 2x 

Graph the solution to the inequality. Write the solution to the inequality both in set builder notation and interval

notation.

3 0 7 0

3 7

x or x

x or x

   

  

Find critical values:

1. Move all terms to one side.

2. Combine into one fraction.

3. Factor numerator and denominator.

4. Set each factor in both the numerator

and denominator equal to zero and solve.

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 11 of 12

Example: Solve the inequality   

2 7 3

0 1

x x

x

  

Graph the solution to the inequality. Write the solution to the inequality both in set builder notation and interval

notation.

Pause the video and try these problems.

1. 2 3

0 1

x

x

 

• Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 12 of 12

2. 7

7 3x 

3.   

2 2 1 4

0 7

x x

x

   

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