Holt Algebra 1

3-6 Solving Compound Inequalities

Solve compound inequalities with one variable. Graph solution sets of compound inequalities with one variable.

Objectives

Holt Algebra 1

3-6 Solving Compound Inequalities

Holt Algebra 1

3-6 Solving Compound Inequalities

You can graph the solutions of a compound inequality involving AND by using the idea of an overlapping region. The overlapping region is called the intersection and shows the numbers that are solutions of both inequalities.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 2A: Solving Compound Inequalities Involving

ANDSolve the compound inequality and graph the solutions.

–5 < x + 1 < 2 –5 < x + 1 < 2 –1 – 1 – 1–6 < x < 1

–10 –8 –6 –4 –2 0 2 4 6 8 10

Since 1 is added to x, subtract 1 from each part of the inequality.

Graph –6 < x.

Graph x < 1.Graph the intersection by

finding where the two graphs overlap.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 2B: Solving Compound Inequalities Involving

ANDSolve the compound inequality and graph the solutions.

8 < 3x – 1 ≤ 11

Holt Algebra 1

3-6 Solving Compound Inequalities

Solve the compound inequality and graph the solutions.

Check It Out! Example 2a

–9 < x – 10 < –5

Holt Algebra 1

3-6 Solving Compound Inequalities

Solve the compound inequality and graph the solutions.

Check It Out! Example 2b

–4 ≤ 3n + 5 < 11

Holt Algebra 1

3-6 Solving Compound Inequalities

You can graph the solutions of a compound inequality involving OR by using the idea of combining regions. The combine regions are called the union and show the numbers that are solutions of either inequality.

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Holt Algebra 1

3-6 Solving Compound InequalitiesExample 3A: Solving Compound Inequalities Involving OR

Solve the inequality and graph the solutions.8 + t ≥ 7 OR 8 + t < 28 + t ≥ 7 OR 8 + t < 2–8 –8 –8 −8

t ≥ –1 OR t < –6

Solve each simple inequality.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Graph t ≥ –1.

Graph t < –6.

Graph the union by combining the regions.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 3B: Solving Compound Inequalities Involving OR

Solve the inequality and graph the solutions.4x ≤ 20 OR 3x > 21

Holt Algebra 1

3-6 Solving Compound Inequalities

Solve the compound inequality and graph the solutions.

Check It Out! Example 3a

2 +r < 12 OR r + 5 > 19

Holt Algebra 1

3-6 Solving Compound Inequalities

Solve the compound inequality and graph the solutions.

Check It Out! Example 3b

Holt Algebra 1

3-6 Solving Compound Inequalities

Every solution of a compound inequality involving AND must be a solution of both parts of the compound inequality. If no numbers are solutions of both simple inequalities, then the compound inequality has no solutions.

The solutions of a compound inequality involving OR are not always two separate sets of numbers. There may be numbers that are solutions of both parts of the compound inequality.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 4A: Writing a Compound Inequality from a

GraphWrite the compound inequality shown by the graph.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 4B: Writing a Compound Inequality from a Graph

Write the compound inequality shown by the graph.

Holt Algebra 1

3-6 Solving Compound InequalitiesCheck It Out! Example 4a

Write the compound inequality shown by the graph.

Holt Algebra 1

3-6 Solving Compound InequalitiesCheck It Out! Example 4b

Write the compound inequality shown by the graph.

Holt Algebra 1

3-6 Solving Compound Inequalities

Lesson Quiz: Part I

Solve each compound inequality and graph the solutions.1. 2 ≤ 2w + 4 ≤ 12

2. 3 + r > −2 OR 3 + r < −7

Holt Algebra 1

3-6 Solving Compound Inequalities

Lesson Quiz: Part II

Write the compound inequality shown by each graph.

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