CHAPTER 1 – EQUATIONS AND INEQUALITIES 1.6 – SOLVING COMPOUND AND ABSOLUTE VALUE INEQUALITIES Unit 1 – First-Degree Equations and Inequalities

1.6 – Solving Compound and Absolute Value Inequalities In this section we will review:  Solving compound inequalities  Solving absolute value inequalities

1.6 – Solving Compound and Absolute Value Inequalities Compound inequality – consists of two inequalities joined by the word and or the word or.  To solve a compound inequality, you must solve each part of the inequality The graph of a compound inequality containing and is the intersection of the solution sets of the two inequalities  Compound inequalities with and are called conjunctions  Compound inequalities with or are called disjunctions

1.6 – Solving Compound and Absolute Value Inequalities A compound inequality containing the word and is true if and only if both inequalities are true  Example  x ≥ -1  x < 2  x ≥ -1 and x < 2

1.6 – Solving Compound and Absolute Value Inequalities Example 1  Solve 10 ≤ 3y – 2 < 19. Graph the solution set on a number line

1.6 – Solving Compound and Absolute Value Inequalities The graph of a compound inequality containing or is the union of the solution sets of the two inequalities A compound inequality containing the word or is true if one or more of the inequalities is true  Example  x ≤ 1  x > 4  x ≤ 1 or x > 4

1.6 – Solving Compound and Absolute Value Inequalities Example 2  Solve x + 3 < 2 or –x ≤ -4. Graph the solution set on a number line.

1.6 – Solving Compound and Absolute Value Inequalities HOMEWORK Page 45 #12 – 15, 22 – 25, 32,

1.6 – Solving Compound and Absolute Value Inequalities Absolute Value Inequalities Example 1  Solve 3 > |d|. Graph the solution set on a number line

1.6 – Solving Compound and Absolute Value Inequalities Example 2  Solve 3 < |d|. Graph the solution set on a number line.

1.6 – Solving Compound and Absolute Value Inequalities An absolute value inequality can be solved by rewriting it as a compound inequality. For all real numbers a and b, b > 0, the following statements are true:  If |a| < b, then –b < a < b  If |2x + 1| < 5, then -5 < 2x + 1 < 5  If |a| > b, then a > b or a < -b  If |2x + 1| > 5, then 2x + 1 > 5 or 2x + 1 < -5 These statements are also true for ≤ and ≥

1.6 – Solving Compound and Absolute Value Inequalities Example 3  Solve |2x – 2| ≥ 4. Graph the solution set on a number line.

1.5 – Solving Inequalities Example 4  According to a recent survey, the average monthly rent for a one-bedroom apartment in one city neighborhood is $750. However, the actual rent for any given one-bedroom apartment in the area may vary as much as $250 from the average.  Write an absolute value inequality to describe this situation.  Solve the inequality to find the range of monthly rent.

1.5 – Solving Inequalities HOMEWORK Page 45 #16 – 21, 26 – 31, 33 – 39