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

ABSOLUTE VALUE EQUALITIES and INEQUALITIES

Candace Moraczewski and Greg Fisher

© April, 2004

3 x

-3 3

This absolute value equation represents the numbers on the number line whose distance from 0 is equal to 3.

0

3 units 3 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 containsa variable inside the absolute value sign.

The absolute value of a number is its distance from 0 on a number line.

-5 0

5 5- because -5 is 5 units from 0

-3

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

{5,-9}

x=5 or x = -9

Solve 4|x – 3| + 2 = 104| 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

Pause!

• Try 1-4 on Absolute Value Worksheet

MEMORIZE THIS:

• GreatOR• Or statement, two inequalities

• Less THAND• Sandwich, one inequality two signs

-3 3

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: 3 x

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 adistance from 0 less than 3

-3 3

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)


-3 3

0

x

If a number x is to the left of -3 or to the right of 3 thenthis translates to:

Inequality notation: x < -3 or x > 3 (a compound “or” inequality)


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 adistance from 0 greater than 3

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

-

-3 3

0

x

If a number x is to the left of -3 or to the right of 3, includingthe -3 and 3, then this translates to:

Inequality notation: x -3 or x 3 (a compound “or” inequality)


x

-

2 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.

0 -2 2

Inequality notation: -2 < x < 2

x

2 x

0 -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: 2x2

x

2 x

0 -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.

Inequality notation: x < -2 or x > 2

x x

-

2 x

0 -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:

2or x -2x

x x

-

TRY THE FOLLOWING PROBLEMS, CHECK YOUR ANSWERS WITH A PARTNER

2 3x - 4 .6

1 1 2x .5

1 2x - 7- .4

5 3x - 2- .3

4 3 - x .2

5 3 - 2 .1

x

Solve the following absolute value inequalities. Write answer using both inequality notation and interval notation.

ANSWERS:

] 4 1,- [ , 4 x 1- .1

Click here to returnto the problem set

) [7, 1]- ,- ( , 7 or x 1- x 2.

ANSWERS:


) 1 , 37- ( , 1 x

37- 3.

ANSWERS: Click here to returnto the problem set

) 3,- [ 4]- ,- ( , 3- or x 4- x 4.

ANSWERS:


0) 1,- ( , 0 x 1- 5.

ANSWERS:Click here to returnto the problem set

) 2, [ ] 32 ,- ( , 2 or x

32 x 6.

ANSWERS:


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:

512 xAll real numbers. The

absolute value will always be greater than zero.

38 x 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.

NOTATION: (old) 2 ≤ x < 3 (new) x ≥ 2 x < 3 0 1 2 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 3 4 52NOTATION: (old) x ≤ -1 or x > 4 (new) x ≤ -1 x > 4

Recap

• 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”

Ex: -16 < 5 – 3q < 11

- 5 -5 -5

-21 < -3q < 6

**Remember flip the sign if you multiply or divide by a negative number!-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

Solve 4 1 -x 2

0-4 4

2x - 1

4 1-2x 4

Now solve the double inequality

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

4 1 -2x 4

Divide every position by 2

25 x

23

+1 +1 +1 ________________

5 2x 3

Solve 8 2 3x

0-8 8

3x + 2

8 2 3x 8

Now solve the double inequality


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)

8 2 3x 8

Divide every position by 3

2 x 310

-2 -2 -2 ________________

6 3x 10

Solve 5 2 x

0-5 5

x + 2

5 2 or x 5- 2 x

Now solve the “or” compound inequality

x + 2




-

5 2 or x 5- 2 x -2 -2 -2 -2

3 or x 7- x

Solve 2 3x - 4

0-2 2

4 – 3x

2 3x 4or 2- 3x - 4

Now solve the “or” compound inequality

4 – 3x




-

-4 -4 -4 -4

2- 3x -or 6- 3x -

2 3x 4or 2- 3x - 4

Divide both inequalities by -3. Remember to changethe sense of the inequality signs because of divisionby a negative.

32 or x 2 x

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 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> -9x<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 > -.4x < 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 RangeCheck answer: x-78 < 3 AND x-78>-3

x<81 AND x>75 75<x<81

Pause!

• Try word problems from overhead

Documents

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