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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
Restart when you are ready to check your answers.
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
Write your answer in interval notation.
|||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.
Restart when you are ready to check your answer.
Objective 3: Quadratic Inequalities
Example: Solve the inequality 2 12 35x x
Write your answer in interval notation.
Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 5 of 12
Example: Solve the inequality 220 3 2 0x x
Write your answer in interval notation.
Example: Solve the inequality 2 5x
Write your answer in interval notation.
Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 6 of 12
Pause the video and try these problems.
Solve the inequality. Write your answer in interval notation.
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
Restart when you are ready to check your answers.
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
Write your answer in interval notation.
Example: Solve the inequality 3 2
4 1 5 0x x x
Write your answer in interval notation.
Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 9 of 12
Pause the video and try these problems.
Solve the inequality. Write your answer in interval notation.
1. 3 25 6x x x
2. 2
3 8 0x x
Restart when you are ready to check your answers.
Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 10 of 12
Objective 5: Rational Inequalities
Example: Solve the inequality 3
07
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
22x
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
27 3
01
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.
Solve the inequality. Write your answer in interval notation.
1. 2 3
01
x
x
Cypress College Math Department – CCMR Notes Solving Nonlinear Inequalities, Page 12 of 12
2. 7
73x
3.
22 1 4
07
x x
x
Restart when you are ready to check your answers.
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