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 units -3 3

The absolute value of a number is its distance from 0 on a number line. -5 -3 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}

4| x – 3 | = 8 | x – 3 | = 2 x – 3 = 2 or x-3 = -2 x = 5 or x= 1 {1,5} 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}

Abs. value cannot be negative Solve -2|2x + 1|-3 = 9 -2| 2x + 1| = 12 | 2x + 1| = -6 NO SOLUTION Because Abs. value cannot be negative

Try 1-4 on Absolute Value Worksheet Pause! Try 1-4 on Absolute Value Worksheet

MEMORIZE THIS: GreatOR Less THAND Or statement, two inequalities Sandwich, one inequality two signs

x -3 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 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)

because x is to the left of -3 or x is to the right of 3 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 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:

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 2 -2 < x < 2 Inequality notation:

x -2 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 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. x < -2 or x > 2 Inequality notation:

x x -2 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

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

Try 5-8 on Absolute Value Worksheet on your own 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.

More practice is on the back 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. 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 1 2 3 4 5 NOTATION: (old) x ≤ -1 or x > 4 (new) x ≤ -1 x > 4

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

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 +4 +4 +4 9 < x < 6 Add four to each “side” **Remember flip the sign if you multiply or divide by a negative number! Ex: -16 < 5 – 3q < 11 - 5 -5 -5 -21 < -3q < 6 -3 -3 -3 7 > q > -2 Rewrite…. -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

Now solve the double inequality 1. Draw a number line and identify the interval(s) which satisfy the inequality: 2x - 1 -4 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

Divide every position by 2 +1 +1 +1 ________________ Divide every position by 2

Now solve the double inequality 1. Draw a number line and identify the interval(s) which satisfy the inequality 3x + 2 8 -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

Divide every position by 3 -2 -2 -2 ________________ Divide every position by 3

Now solve the “or” compound inequality 1. Draw a number line and identify the interval(s) which satisfy the inequality x + 2 x + 2 -5 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 -2 -2

Now solve the “or” compound inequality 1. Draw a number line and identify the interval(s) which satisfy the inequality 4 – 3x 4 – 3x -2 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 -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)

|x – 65| < 9 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 x<74 AND x >56  56<x<74

|x – 6| < .4 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