ABSOLUTE VALUE EQUALITIES and INEQUALITIES Candace Moraczewski and

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

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 0 3

The absolute value of a number is its distance from 0 on a number line. -5 -3 0 because -5 is 5 units from 0 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|2 x + 1|-3 = 9 -2| 2 x + 1| = 12 | 2 x + 1| = -6 0 NO SOLUTION Because Abs. value cannot be negative

Pause! • Try 1 -4 on Absolute Value Worksheet

MEMORIZE THIS: • Great. OR • 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: 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 (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: 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 Absolute value notation: 3 (a compound “or” inequality)

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 -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:

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 2. Notice that the intervals satisfying this inequality are going in opposite directions. Inequality notation: x < -2 or 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:

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

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: 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

Compound Inequalities • Contains 2 parts 1. Intersection: intersection is a compound inequality that contains AND. • The solution must be a solution of BOTH inequalities to be true in the compound inequality – Ex: Graph the solution set of x < 3 and x ≥ 2. 0 1 2 3 NOTATION: (old) 2 ≤ x < 3 (new) x ≥ 2 x<3

Compound Inequalities cont’d 2. Union: intersection is a compound inequality that contains OR. • The solution must be a solution of EITHER inequality to be true in the compound inequality • Ex: Graph the solution set of x ≤ -1 or x > 4. -2 -1 0 1 2 3 4 5 NOTATION: (old) x ≤ -1 or x > 4 (new) x ≤ -1 x>4

Recap • Intersection: AND, , overlap • Union: OR, , opposite directions “U” for Union • Always write answers small to big (left to right)

How to solve compound inequalities • Think of it as solving two different inequalities and then combine their solutions as an intersection. • Ex: -5 < x – 4 < 2 Add four to each “side” +4 +4 9 < x +4 < 6 Ex: -16 < 5 – 3 q < 11 -5 -5 -21 < -3 q < -3 7 > q > -2 -3 -5 6 -3 Rewrite…. **Remember flip the sign if you multiply or divide by a negative number! -2 < q < 7

Pause! • Answer 5 -8 on page 6 in workbook (section 1. 6)

TO SOLVE A MORE COMPLICATED ABSOLUTE VALUE INEQUALITY, FOLLOW THESE STEPS AS ILLUSTRATED IN THE FOLLOWING EXAMPLES • 1. Draw a number line and identify the interval(s) which satisfy the inequality • 2. Write the expression in the absolute value sign over the designated interval(s) • 3. Translate this to either a double inequality or two inequalities going in opposite directions connected with the word “or” • 4. Remember to include the endpoint if the inequality also has an equal to symbol

Solve 1. Draw a number line and identify the interval(s) which satisfy the inequality: 2 x - 1 -4 0 4 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the double inequality

+1 +1 +1 ________ Divide every position by 2

Solve 1. Draw a number line and identify the interval(s) which satisfy the inequality 3 x + 2 -8 0 8 2. . Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the double inequality

-2 -2 -2 ________ Divide every position by 3

Solve 1. Draw a number line and identify the interval(s) which satisfy the inequality x+2 -5 0 5 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the “or” compound inequality

-2 -2

Solve 1. Draw a number line and identify the interval(s) which satisfy the inequality 4 – 3 x -2 0 2 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the “or” compound inequality

-4 -4 Divide both inequalities by -3. Remember to change the sense of the inequality signs because of division by a negative.

Pause! • Answer 9 -16 in your workbook (pg 6)

Word Problems • Pretend that you are allowed to go within 9 of the speed limit of 65 mph 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 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 – 6 > -. 4 x < 6. 4 AND x > 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