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
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 – 3q < 11
-5
-5
-21 <
-3q <
-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
2x - 1  4
Solve
1. Draw a number line and identify the interval(s) which
satisfy the inequality:
2x - 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
 4  2x -1  4
Now solve the double inequality
 4  2x - 1  4
+1
+1 +1
________________
 3  2x  5
Divide every position by 2
3
5
 x 
2
2
Solve
3x  2  8
1. Draw a number line and identify the interval(s)
which satisfy the inequality
3x + 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
 8  3x  2  8
Now solve the double inequality
 8  3x  2  8
-2
-2 -2
________________
10  3x  6
Divide every position by 3
10
 x  2
3
x2  5
Solve
1. Draw a number line and identify the interval(s) which
satisfy the inequality
x+2
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
x  2  -5 or x  2  5
Now solve the “or” compound inequality
x  2  - 5 or x  2  5
-2
-2
-2
x  - 7 or x  3
-2
Solve
4 - 3x  2
1. Draw a number line and identify the interval(s) which
satisfy the inequality
4 – 3x
4 – 3x
-
-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
4 - 3x  - 2 or 4  3x  2
Now solve the “or” compound inequality
4 - 3x  - 2 or 4  3x  2
-4
-4
-4
-4
- 3x  - 6 or - 3x  - 2
Divide both inequalities by -3. Remember to change
the sense of the inequality signs because of division
by a negative.
2
x  2 or x 
3
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> -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
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