Solving Open Sentences Involving Absolute Value 3 2

Solving Open Sentences Involving Absolute Value | | | – 3 – 2 – 1 | | | | | 0 1 2 3 4 5 6 | | – 5 – 4 – 3 – 2 – 1 | | | 0 1 2 3 4

Section 6 -5 SOLVING OPEN SENTENCES INVOLVING ABSOLUTE VALE

Solving Open Sentences Involving Absolute Value There are three types of open sentences that can involve absolute value. Consider the case | x | = n. | x | = 5 means the distance between 0 and x is 5 units If | x | = 5, then x = – 5 or x = 5. The solution set is {– 5, 5}.

Solving Open Sentences Involving Absolute Value When solving equations that involve absolute value, there are two cases to consider: Case 1 The value inside the absolute value symbols is positive. Case 2 The value inside the absolute value symbols is negative. Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it.

Solve an Absolute Value Equation Method 1 Graphing means that the distance between b and – 6 is 5 units. To find b on the number line, start at – 6 and move 5 units in either direction. The distance from – 6 to – 11 is 5 units. The distance from – 6 to – 1 is 5 units. Answer: The solution set is

Solve an Absolute Value Equation Method 2 Compound Sentence Write as or Case 1 Case 2 Original inequality Subtract 6 from each side. Simplify. Answer: The solution set is

Solve an Absolute Value Equation Answer: {12, – 2}

Write an Absolute Value Equation Write an equation involving the absolute value for the graph. Find the point that is the same distance from – 4 as the distance from 6. The midpoint between – 4 and 6 is 1. The distance from 1 to – 4 is 5 units. The distance from 1 to 6 is 5 units. So, an equation is.

Write an Absolute Value Equation Answer: Check Substitute – 4 and 6 into

Write an Absolute Value Equation Write an equation involving the absolute value for the graph. Answer:

Solving Open Sentences Involving Absolute Value Consider the case | x | < n. | x | < 5 means the distance between 0 and x is LESS than 5 units If | x | < 5, then x > – 5 and x < 5. The solution set is {x| – 5 < x < 5}.

Solving Open Sentences Involving Absolute Value When solving equations of the form | x | < n, find the intersection of these two cases. Case 1 The value inside the absolute value symbols is less than the positive value of n. Case 2 The value inside the absolute value symbols is greater than negative value of n.

Solve an Absolute Value Inequality (<) Then graph the solution set. Write as and Case 2 Case 1 Original inequality Add 3 to each side. Simplify. Answer: The solution set is

Solve an Absolute Value Inequality (<) Then graph the solution set. Answer:

Solving Open Sentences Involving Absolute Value Consider the case | x | > n. | x | > 5 means the distance between 0 and x is GREATER than 5 units If | x | > 5, then x < – 5 or x > 5. The solution set is {x| x < – 5 or x > 5}.

Solving Open Sentences Involving Absolute Value When solving equations of the form | x | > n, find the union of these two cases. Case 1 The value inside the absolute value symbols is greater than the positive value of n. Case 2 The value inside the absolute value symbols is less than negative value of n.

Solve an Absolute Value Inequality (>) Then graph the solution set. Write as or Case 2 Case 1 Original inequality Add 3 to each side. Simplify. Divide each side by 3. Simplify.

Solve an Absolute Value Inequality (>) Answer: The solution set is

Solve an Absolute Value Inequality (>) Then graph the solution set. Answer:

Solving Open Sentences Involving Absolute Value In general, there are three rules to remember when solving equations and inequalities involving absolute value: 1. If then or (solution set of two numbers) 2. If then and (intersection of inequalities) 3. If then or (union of inequalities)

Assignment • Study Guide 6 -5 (In-Class) • Pages 349 -350 #’s 14 -19, 24 -35, 40, 41. (Homework)