1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 2.2: Solving Linear Inequalities

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Objectives o Solving linear inequalities. o Solving compound linear inequalities. o Solving absolute values inequalities.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Linear Inequalities o If the equality symbol in a linear equation is replaced with,,, or, the result is a linear inequality. o For example, a linear inequality in is of the form where and are real numbers and. o The solution of a linear inequality typically consists of some interval of real numbers described in set notation, graphically or with interval notation.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Linear Inequalities Cancellation Properties for Inequalities Throughout this table, and, represent algebraic expressions. These properties are true for all inequalities. Property Description Adding the same quantity to both sides of an inequality results in an equivalent inequality. If both sides of an inequality are multiplied by a positive quantity, the sign of the inequality is unchanged. If both sides of an inequality are multiplied by a negative quantity, the sign of the inequality is reversed. If

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 1: Linear Inequalities Solve the following linear inequality. Step 1: Distribute. Step 3: Divide by. Note the reversal of the inequality sign. Step 2: Combine like terms.

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 2: Linear Inequalities Solve the following linear inequality.

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Graphing a Solution o The solutions in the previous examples were described using interval notation, but graphing can also be used to describe solutions. o Like in interval notation, parentheses are used when endpoints are not included in the interval and brackets are used when the endpoints are included in the interval. o For example, is graphed as follows:

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Compound Linear Inequalities A compound inequality is a statement containing two inequality symbols, and can be interpreted as two distinct inequalities joined by the word “and”. For example, in a course where the grade depends solely on the grades of 5 exams, the following compound inequality could be used to determine the final exam grade needed to score a B in the course.

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: Solving Compound Inequalities Solve the compound inequality from the previous slide. Step 1: Multiply all sides by. Step 2: Subtract from all sides. Note: If this compound inequality relates to test scores, as indicated on the previous slide, the solution set is, assuming is the highest score possible.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: Solving Compound Inequalities Solve the compound inequality. Note: each inequality is reversed since we are dividing by a negative number!

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Absolute Value Inequalities An absolute value inequality is an inequality in which some variable expression appears inside absolute value symbols. can be interpreted as the distance between and zero on the real number line. This means that absolute value inequalities can be written without absolute values as follows, assuming is a positive real number: or and

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 5: Solving Absolute Value Inequalities Solve the following absolute value inequality. Step 1: Subtract. Step 2: Rewrite the inequality without absolute values. Step 3: Solve as compound inequality.

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 6: Solving Absolute Value Inequalities Solve the following absolute value inequality and graph the solution. or

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 7: Solving Absolute Value Inequalities Solve the following absolute value inequality The solution set is the empty set, as it is impossible for the absolute value of any expression to be negative.

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Interlude: Translating Inequality Phrases o Many real-world applications leading to inequalities involve notions such as “is not greater than”, “at least as great as”, “does not exceed”, and so on. o Phrases such as these all have precise mathematical translations that use one of the four inequality symbols,.

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Interlude: Translating Inequality Phrases SignPhraseExample “is greater than”The average temperature in New Mexico during the month of May is greater than 75. “at least as great as”The average price for a large screen digital TV is at least $9000. “is less than”The number of skiers on the mountain is less than 100. “is not greater than”In a quality test, the number of defective products cannot be greater than 5.

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 8: Translating Inequality Phrases The average daily high temperature in Phoenix, Arizona over the course of three days exceeded 80 degrees. Given that the high on the first day was 78 and the high on the third day was 85, what was the minimum high temperature on the second day? The high temperature on the second day was greater than 77.

18. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 9: Translating Inequality Phrases As a test for quality at a plant manufacturing silicon wafers, a random sample of 10 batches of 1000 wafers each must not detect more than 5 defective wafers per batch on average. In the first 9 batches tested, the average number of defective wafers per batch is found to be 4.89. What is the maximum number of defective wafers that can be found in the 10 th batch for the plant to pass the quality test?