1.4 Solving Absolute Value Equations

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1 Mrs. Townsend Algebra II Unit 1 Equations and Inequalities Name: Period: 1.4 Solving Absolute Value Equations Absolute Value: 6 14 x Evaluate Expressions with Absolute Value Note: When evaluating expressions, the absolute value bars act as a grouping symbol, so you always do what is inside of them first 1. Evaluate 3 6 2x if x 4 3. Evaluate y if y 2 2. Evaluate 1.4 5y 7 if y 3 4. Evaluate 4x if x 2. 8 Solve an Absolute Value Equation When solving equations with absolute value, you must remember that for a b or a b a b, either. Therefore, there will always be two possible solutions to absolute value equations. These solutions together are called the solution set. 1. x x x y 5

2 It is possible that an absolute value equation has no solution set. This happens when the absolute value of a number equals a number. Since this is not possible, the solution set is. 1. 5x a 2 6 One Solution It is important to ALWAYS check your answers when solving absolute value equations. Even if you did the problem correctly, the answers may not be true solutions to the original equation. Solve x 6 3x 2 Solve 2 x 1 x 3x Solving Inequalities Recall: The solution set to an inequality can be graphed on a number line. The point is either an or circle. The graph is completed by an arrow to either the left or right. Graph x Graph x For any two real numbers, a and b, exactly one of the following statements are true: *Note: These properties are also true for,, and 2

3 Solving Multi-Step Inequalities Solving multi-step inequalities is similar to solving multi-step equations k x 4 2. x q 12 6 k Solving Compound & Absolute Value Inequalities A compound inequality consists of two inequalities joined together by the words or. ~Inequalities with the word and are called. ~Compound inequalities with the word or are called. To solve a compound inequality, you must solve each part of the inequality and then graph both solutions. 3

4 Conjunctions ( and compound inequalities) A compound inequality with and is true if and only if inequalities are true. Example 1: Solve x 1 and x 2. Graph the solution set below. x 1 x 2 x 1 and 2 x The graph of an inequality with the word and is the of the two solutions. Example 2: Solve x. Graph the solution set on a number line. 1. Solve 8 3x Graph the solution set on a number line. 2. Solve 10 3y Graph the solution set on a number line. 4

5 Disjunctions ( or compound inequalities) A compound inequality containing or is true if of the inequalities are true. The graph of an inequality with the word or is the of the two solutions. 1. Solve x 1 or 4 x and graph the solution set on the number line. 2. Solve 2 3 y or y 4 3. Graph the solution set. 3. Solve y 5 7 or y 6 2 and graph the solution set. 4. Solve x 3 2 or x 4. Graph the solution set. 5

6 Absolute Value Inequalities Recall: The absolute value of a number is: Absolute value inequality using < Solve a 4 and graph the solution set on a number line. a 4 means that the between a and 0 on the number line is less than 4 units. To make a 4 true, substitute numbers for a that are than 4 units from 0. All of the numbers between and are less than 4 units from 0. Therefore, the solution set is. Solve x 3 and graph the solution set on a number line. Solution Set = Absolute value inequality using > Solve a 4 and graph the solution set. a 4 means the distance between a and o on a number line is greater than 4 units. Solution Set = Solve x 3 and graph the solution set on a number line. Solution Set = 6

7 An absolute value inequality can also be solved by rewriting it as a compound inequality. x then 5 2x 1 5 If If x then 2x 1 5 or 2x 1 5 Multi-Step Absolute Value Inequalities Solve 3x 12 6 and graph the solution set. 1. Solve 3x 4 10 and graph the solution set. 2x 2 2. Solve 4 3 and graph the solution set. 7

8 Write an Inequality 1. Sophia s goal is to score at least 200 points this basketball season. If she has already scored 122 points, how many points does Sophie have to score on average for the last 6 games to reach her goal? 2. Admission to the state fair is $12 per person. Bus parking costs $20. Write an inequality to determine how many people can go to the fair if the group has $600 and uses only one bus. 3. To prepare for a job interview, Megan researches the position s requirements and pay. She discovers that the average starting salary for the position is $38,500, but her actual starting salary could differ from the average by as much as $2450. Write and solve an absolute value inequality to find the range of Megan s starting salary. 4. Don is building a fence around a rectangular plot and wants the perimeter to be between 17 and 20 yards. The width of the plot is 5 yards. Write and solve a compound inequality to describe the range of possible measures for the length of the fence. 8