ABSOLUTE VALUE EQUALITIES and INEQUALITIES Candace Moraczewski and Greg Fisher © April, 2004

-33 This absolute value equation represents the numbers on the number line whose distance from 0 is equal to units 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. An absolute value equation is an equation that contains a variable inside the absolute value sign.

The absolute value of a number is its distance from 0 on a number line because -5 is 5 units from 0 -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 {5,-9} x=5 or x = -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 NO SOLUTION Because Abs. value cannot be negative 0

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MEMORIZE THIS: Great OR Or statement, two inequalities Less TH AND Sandwich, one inequality two signs

-33 0 x If a number x is between -3 and 3 then this translates to: Inequality notation: -3 < x < 3 (a double inequality) Absolute value notation: because -3 is to the left of x and x is to the left of 3 because all of the numbers between -3 and 3 have a distance from 0 less than 3

-33 0 x If a number x is between -3 and 3, including the -3 and 3, then this translates to: Inequality notation: -3 x 3 (a double inequality) Absolute value notation:

-33 0 x If a number x is to the left of -3 or to the right of 3 then this translates to: Inequality notation: x 3 (a compound “or” inequality) Absolute value notation: x 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 because x is to the left of -3 or x is to the right of 3

-33 0 x 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

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 Inequality notation: -2 < x < 2 x

0 -22 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: x

0 -22 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 x x

0 -22 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 x

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.

ANSWERS: Click here to return to the problem set

ANSWERS: Click here to return to the problem set

ANSWERS: Click here to return to the problem set

ANSWERS: Click here to return to the problem set

ANSWERS: Click here to return to the problem set

ANSWERS: Click here to return to the problem set

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Can the absolute value of something be less than zero? NO! Absolute value is always positive. Cases: 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.

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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 X 74 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 – x 6.4 AND x 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 X 81 AND x 75  75 x 81

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