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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 3.4 Compound Inequalities.

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1 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 3.4 Compound Inequalities

2 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Objectives Basic Concepts Symbolic Solutions and Number Lines Numerical and Graphical Solutions Interval Notation

3 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Basic Concepts A compound inequality consists of two inequalities joined by the words and or or. 2x > –5 and 2x 8 x + 3 4 or x – 2 < –6

4 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Determine whether the given x-values are solutions to the compound inequalities. x + 2 3 x = 4, –4 Solution x + 2 3 Substitute 4 into the given compound inequality. 4 + 2 3 6 3 True and True Both inequalities are true, so 4 is a solution.

5 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Determine whether the given x-values are solutions to the compound inequalities. x + 2 3 x = 4, –4 Solution x + 2 3 Substitute –4 into the given compound inequality. –4 + 2 3 – 2 3 True and False To be a solution both inequalities must be true, so –4 is not a solution.

6 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Symbolic Solutions and Number Lines We can use a number line to graph solutions to compound inequalities, such as x –3. x < 7 x > –3 x –3 Note: A bracket, either [ or ] or a closed circle is used when an inequality contains or. A parenthesis, either ( or ), or an open circle is used when an inequality contains.

7 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve 3x + 6 > 12 and 5 – x < 11. Graph the solution. Solution 3x + 6 > 12 and 5 – x < 11

8 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve each inequality. Graph each solution set. Write the solution in set-builder notation. a.b.c. Solution a.b.

9 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) c.

10 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve x + 3 2 Solution x + 3 2 x –1

11 Copyright © 2013, 2009, 2005 Pearson Education, Inc.

12 Example Write each expression in interval notation. a. –3 x < 7 b. x 4 c. x < –3 or x 5 d. {x|x > 0 and x 5} e. {x|x 2 or x 5}

13 Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve 2x + 3 –3 or 2x + 3 5 Solution 2x + 3 –3 or 2x + 3 5 2x –6 or 2x 2 x –3 or x 1 The solution set may be written as (, 3] [1, )


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