Solving Absolute Value Equations and Inequalities

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1 Solving Absolute Value Equations and Inequalities Say Thanks to the Authors Click (No sign in required)

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3 Chapter 1. Solving Absolute Value Equations and Inequalities CHAPTER 1 Solving Absolute Value Equations and Inequalities Objective To solve equations and inequalities that involved absolute value. Review Queue Solve the following equations and inequalities 1. 5x 14 = 1. 1 < x 5 < x (x + 7) 15 = 53 Solving Absolute Value Equations Objective To solve absolute value equations. Watch This Watch the first part of this video. MEDIA Click image to the left for use the URL below. URL: Khan Academy: Absolute Value Equations Guidance Absolute value is the distance a number is from zero. Because distance is always positive, the absolute value will always be positive. Absolute value is denoted with two vertical lines around a number, x. 5 = 5 9 = 9 0 = 0 1 = 1 1

4 When solving an absolute value equation, the value of x could be two different possibilities; whatever makes the absolute value positive OR whatever makes it negative. Therefore, there will always be TWO answers for an absolute value equation. If x = 1, then x can be 1 or -1 because 1 = 1 and 1 = 1. If x = 15, then x can be 15 or -15 because 15 = 15 and 15 = 15. From these statements we can conclude: Example A Determine if x = 1 is a solution to x 5 = 9. Solution: Plug in -1 for x to see if it works. ( 1) 5 = = 9 9 = 9-1 is a solution to this absolute value equation. Example B Solve x + 4 = 11. Solution: There are going to be two answers for this equation. x + 4 can equal 11 or -11. x + 4 = 11 x + 4 = 11 x + 4 = 11 or x = 7 x = 15 Test the solutions: = = = = 11 Example C Solve 3 x 5 = 17. Solution: Here, what is inside the absolute value can be equal to 17 or -17.

5 Chapter 1. Solving Absolute Value Equations and Inequalities 3 x 5 = 17 3 x 5 = 17 3 x 5 = 17 3 x = or 3 x = 1 x = 3 x = 1 3 x = 33 x = 18 Test the solutions: 3 = 17 3 ( 18) 5 = 17 5 = = = = 17 Guided Practice 1. Is x = 5 a solution to 3x + = 6? Solve the following absolute value equations.. 6x 11 + = x + 3 = 9 Answers 1. Plug in -5 for x to see if it works. 3( 5) + = = is not a solution because 7 = 7, not 6.. Find the two solutions. Because there is a being added to the left-side of the equation, we first need to subtract it from both sides so the absolute value is by itself. 3

6 6x 11 + = 41 6x 11 = 39 6x = 50 6x 11 = 39 6x 11 = 39 6x = 8 x = 50 or x = = 5 3 or 81 = or 4 3 Check both solutions. It is easier to check solutions when they are improper fractions. 3. What is inside the absolute value is equal to 9 or -9. Test solutions: Vocabulary ( ) ( = ) 11 3 = = 39 and 8 11 = = = 39 1 x + 3 = 9 1 x + 3 = 9 1 x + 3 = 9 1 x = 6 or 1 x = 1 x = 1 x = 4 1 (1) + 3 = 9 1 ( 4) + 3 = 9 Absolute Value The positive distance from zero a given number is. Problem Set = = 9 9 = 9 9 = 9 Determine if the following numbers are solutions to the given absolute value equations x 7 = 16;9

7 Chapter 1. Solving Absolute Value Equations and Inequalities. 1 4x + 1 = 4; x = 7; 1 Solve the following absolute value equations. 4. x + 3 = 8 5. x = 9 6. x + 15 = x 5 = 8. x = x 1 = x + = x = x + 0 = Solve 1x 18 = 0. What happens? 14. Challenge When would an absolute value equation have no solution? Give an example. Absolute Value Inequalities Objective To solve absolute value inequalities. Watch This MEDIA Click image to the left for use the URL below. URL: Khan Academy: Absolute Value Inequalities Guidance Like absolute value equations, absolute value inequalities also will have two answers. However, they will have a range of answers, just like compound inequalities. x > 1 This inequality will have two answers, when x is 1 and when x is 1. But, what about the inequality sign? The two possibilities would be: 5

8 Notice in the second inequality, we did not write x > 1. This is because what is inside the absolute value sign can be positive or negative. Therefore, if x is negative, then x > 1. It is a very important difference between the two inequalities. Therefore, for the first solution, we leave the inequality sign the same and for the second solution we need to change the sign of the answer AND flip the inequality sign. Example A Solve x Solution: There will be two solutions, one with the answer and sign unchanged and the other with the inequality sign flipped and the answer with the opposite sign. x + 10 x + 10 x + 10 x 8 x 1 Test a solution, x = 0 : When graphing this inequality, we have Notice that this particular absolute value inequality has a solution that is an and inequality because the solution is between two numbers. If ax + b < c where a > 0 and c > 0, then c < ax + b < c. If ax + b c where a > 0 and c > 0, then c ax + b c. If ax + b > c where a > 0 and c > 0, then ax + b < c or ax + b > c. If ax + b c where a > 0 and c > 0, then ax + b c or ax + b c. If a < 0, we will have to divide by a negative and have to flip the inequality sign. This would change the end result. If you are ever confused by the rules above, always test one or two solutions and graph it. Example B Solve and graph 4x 3 > 9. Solution: Break apart the absolute value inequality to find the two solutions. 4x 3 > 9 4x 3 > 9 4x 3 < 9 4x > 1 4x < 6 x > 3 x < 3 6

9 Chapter 1. Solving Absolute Value Equations and Inequalities Test a solution, x = 5 : 4(5) 3 > > 9 17 > 9 The graph is: Example C Solve x + 5 < 11. Solution: In this example, the rules above do not apply because a < 0. At first glance, this should become an and inequality. But, because we will have to divide by a negative number, a, the answer will be in the form of an or compound inequality. We can still solve it the same way we have solved the other examples. x + 5 < 11 x + 5 < 11 x + 5 > 11 x < 6 x > 16 x > 3 x < 8 The solution is less than -8 or greater than -3. The graph is: When a < 0 for an absolute value inequality, it switches the results of the rules listed above. Guided Practice 1. Is x = 4 a solution to 15 x > 9?. Solve and graph 3 x Answers 1. Plug in -4 for x to see if it works. 15 ( 4) > > 9 3 > 9 3 > 9 7

10 Yes, -4 works, so it is a solution to this absolute value inequality.. Split apart the inequality to find the two answers. Test a solution, x = 0 : Problem Set 3 x x x x 1 3 x x 1 3 x 3 x 18 x 33 3 (0) Determine if the following numbers are solutions to the given absolute value inequalities. 1. x 9 > 4;10. 1 x 5 1;8 3. 5x ; 8 Solve and graph the following absolute value inequalities. 4. x + 6 > x x x 5 < x + 1 > x x 8 > x x 46 Solve the following absolute value inequalities. a is greater than zero. 13. x a > a 14. x + a a 15. a x a 8