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ABSOLUTE VALUE EQUALITIES and INEQUALITIES Candace Moraczewski and Greg Fisher © April, 2004 x 3 An absolute value equation is an equation that contains a variable inside the absolute value sign. This absolute value equation represents the numbers on the number line whose distance from 0 is equal to 3. Two numbers satisfy this equation. Both 3 and -3 are 3 units from 0. Look at the number line and notice the distance from 0 of -3 and 3. 3 units 3 units -3 0 3 The absolute value of a number is its distance from 0 on a number line. -5 -3 0 -5 5 because -5 is 5 units from 0 -3 3 because -3 is 3 units from 0 Absolute Value Equalities Solve | x | = 7 x = 7 or x=-7 {-7, 7} Solve | x +2| = 7 x +2= 7 or x+2=-7 x=5 or x = -9 {5,-9} Solve 4|x – 3| + 2 = 10 4| x – 3 | = 8 |x–3|=2 x – 3 = 2 or x-3 = -2 x = 5 or x= 1 {1,5} Solve -2|2x + 1|-3 = 9 -2| 2x + 1| = 12 | 2x + 1| = -6 0 NO SOLUTION Because Abs. value cannot be negative Pause! • Try 1-4 on Absolute Value Worksheet MEMORIZE THIS: • GreatOR • Or statement, two inequalities • Less THAND • Sandwich, one inequality two signs x -3 0 3 If a number x is between -3 and 3 then this translates to: Absolute value notation: x 3 because all of the numbers between -3 and 3 have a distance from 0 less than 3 Inequality notation: -3 < x < 3 (a double inequality) because -3 is to the left of x and x is to the left of 3 x -3 0 3 If a number x is between -3 and 3, including the -3 and 3, then this translates to: Absolute value notation: Inequality notation: -3 x 3 x 3 (a double inequality) x x -3 0 3 If a number x is to the left of -3 or to the right of 3 then this translates to: - Absolute value notation: x 3 because the numbers to the left of -3 have a distance from 0 greater than 3 and the numbers to the right of 3 have a distance from 0 greater than 3 Inequality notation: x < -3 or x > 3 (a compound “or” inequality) because x is to the left of -3 or x is to the right of 3 x - x -3 0 3 If a number x is to the left of -3 or to the right of 3, including the -3 and 3, then this translates to: Inequality notation: x -3 or x 3 (a compound “or” inequality) Absolute value notation: x 3 x 2 This absolute value inequality represents all of the numbers on a number line whose distance from 0 is less than 2. See the red shaded line below. x -2 Inequality notation: 0 2 -2 < x < 2 x x 2 -2 0 2 This absolute value inequality represents all of the numbers on the number line whose distance from 0 is less than or equal to 2. Notice that both -2 and 2 are included on this interval. Inequality notation: 2 x 2 x - x 2 -2 0 x 2 This absolute value inequality represents all of the numbers on the number line whose distance from 0 is more than 2. Notice that the intervals satisfying this inequality are going in opposite directions. Inequality notation: x < -2 or x > 2 x 2 x - x -2 0 2 This absolute value inequality represents all of the numbers on the number line whose distance from 0 is more than or equal to 2. Notice that the intervals satisfying this inequality are going in opposite directions and that 2 and -2 are included on the intervals. Inequality notation: x -2 or x 2 TRY THE FOLLOWING PROBLEMS, CHECK YOUR ANSWERS WITH A PARTNER Solve the following absolute value inequalities. Write answer using both inequality notation and interval notation. 1. 2. 3. 4. 5. 6. 2x - 3 5 x -3 4 - 2 - 3x 5 - 7 - 2x 1 2x 1 1 4 - 3x 2 ANSWERS: 1. -1 x 4 , [ -1, 4 ] Click here to return to the problem set ANSWERS: 2. x -1 or x 7 , ( - , -1] [7, ) Click here to return to the problem set ANSWERS: 7 7 3. x 1 , ( , 1 ) 3 3 Click here to return to the problem set ANSWERS: 4. x - 4 or x -3 , ( - , - 4] [ -3, ) Click here to return to the problem set ANSWERS: Click here to return to the problem set 5. -1 x 0 , ( -1, 0) ANSWERS: 2 2 6. x or x 2 , ( - , ] [ 2, ) 3 3 Click here to return to the problem set Pause! • Try 5-8 on Absolute Value Worksheet on your own Can the absolute value of something be less than zero? • NO! Absolute value is always positive. • Cases: 2 x 1 5 8 x 3 All real numbers. The absolute value will always be greater than zero. No solution. The absolute value will never be less than zero. Just like absolute value cannot be = to a negative number. Pause! • More practice is on the back Word Problems • Pretend that you are allowed to go within 9 of the speed limit of 65mph without getting a ticket. Write an absolute value inequality that models this situation. |x – 65| 9 Desired amount Acceptable Range Check Answer: x-65 9 AND x-65 -9 X74 AND x 56 56 x 74 Word Problems • If a bag of chips is within .4 oz of 6 oz then it is allowed to go on the market. Write an inequality that models this situation. |x – 6| .4 Desired amount Acceptable Range Check Answer: x – 6 .4 AND x – 6 -.4 x 6.4 AND x 5.6 5.6 x 6.4 • In a poll of 100 people, Misty’s approval rating as a dog is 78% with a 3% of error. ticket. Write an absolute value inequality that models this situation. |x – 78| 3 Desired amount Acceptable Range Check answer: x-78 3 AND x-78 -3 X 81 AND x 75 75 x 81 Pause! • Try word problems from overhead