1. 
Solve and put the solution in interval notation:
4x – 2(3x + 2) < 2x – 7
2. 
Solve and put the solution in interval notation:
1
3
1
2
(12 x + ) < x +
6
4
6
3
3. 
Solve and put the solution in interval notation:
3x + 2 < 17 or ½ x – 2 > -4
4. 
Solve and put the solution in interval notation:
-10 < ¼ x – 1< 5
Algebra II
1
R.3
Absolute Value
Equations & Inequalities
Algebra II
The absolute value of a real number a,
denoted by |a|, is the distance between a and
0 on the number line.
| –4| = 4
Symbol for
absolute
value
|5| = 5
Distance of 4
–5 –4–3 –2 –1
0
Distance of 5
1
2
3
4
5
The distance a number is from zero.
Algebra II
3
│x│ = 7
7 or – 7
│x + 1│ = 5
4 or -6
Algebra II
4
You will always answer two problems for
absolute value problems.
¡ To
solve: Isolate the abs. value à│X│= c
** ”X” is whatever is in the bars **
¡ Write
two problems X = c and X = -c
¡ Solve
each problem
If it is an inequality, you must flip the symbol of the
second problem!
¡ Always
Algebra II
check your solution! They may not work!
5
If the symbol is:
= use or
< or < use and
> or > use or
Algebra II
6
3x – 1 = 4 or 3x – 1 = -4
½ x + 3 = 7 or ½ x + 3 = -7
Algebra II
2 – 4x < 10 and 2 – 4x > -10
10x – 3 > 4 or 10x – 3 < –4
7
Algebra II
8
1. │2x – 5│ = 9
2x – 5 = 9 or 2x – 5 = -9
2x = 14
x=7
Algebra II
2x = -4
2. 7│x + 3│ = 42
│x + 3│ = 6
x + 3 = 6 or x + 3 = -6
x=3
x = -9
x = -2
9
3. -3│x – 5│+ 7 = -8
-3│x – 5│ = -15
│x – 5│ = 5
x – 5 = 5 or x – 5 = -5
x = 10
Algebra II
x=0
4. | ½x + 3 | = -7
½x+3 = -7 or ½x+3 = 7
½x = -10
½x = 4
x = -20
x=8
neither one of these
solutions make the
statement true, so there
is no solution!
10
6.
6│2x + 3│ < 24
│2x + 3│ < 4
2x+3 < 4 and 2x+3 > -4
2x < 1
2x > -7
7. │x – 8 │- 2 ≥ 1
│x – 8 │ ≥ 3
x – 8 ≥ 3 or x – 8 ≤ -3
x ≥ 11
or
x≤5
x < ½ and x > -7/2
(-7/2, ½)
Algebra II
(-∞, 5] u [11, ∞)
11
8. -2│7 – 3x│- 6 > 2
-2│7 – 3x│ > 8
│7 – 3x│< -4
7–3x < -4 and 7–3x > 4
-3x < -11
-3x > -3
x > 11/3 and x < 1
no solution
Algebra II
12
¡  │anything│=
-#
absolute value can never be negative
¡  │anything│<
no solution
-#
absolute value cannot be less than a negative number
¡  │anything│>
no solution
-#
absolute value is always greater than a negative number
all real numbers
Algebra II
13
| variable – ideal | {=,<, >} change
¡ choose
a variable
¡ identify
¡ decide
the relationship à =, <, >
¡ identify
Algebra II
the “ideal” case
the acceptable change
14
1. A cereal manufacturer has a tolerance of .75
ounces for a box of cereal that is supposed to weigh
20 ounces. Write and absolute value inequality that
describes the acceptable boxes.
X is less than or equal to 20.75 and x is greater than or equal to 19.25
Algebra II
15
2. Ben's golf score is within 5 strokes of his average
of 75. Write an absolute value problem to represent
the maximum and minimum he could have scored.
Maximum: 80
Minimum: 70
Algebra II
16
3. Jim keeps seedlings that are within ½ inch of 4
inches. Write an absolute value sentence that
describes the seedlings he will discard.
X is greater than 4.5 or x is less than 3.5
Algebra II
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4. A type of mobile phone produced by NOKIA must be
less than 8 ounces in weight with a tolerance of 0.3
ounces. The mobiles that are not within the tolerated
weight must be recycled. Write and absolute value
statement describing the cell phones that would be
recycled.
X is greater than 8.3 or x is les than 7.7
Algebra II
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5. Girls selected for a dance team must be
165 cm in height with a tolerance of 5cm.
Write an absolute value statement that can be
used to assess which girls would be accepted
for the dance team.
X is less than or equal to 170 and x is greater than or equal to 160
Algebra II
19
6. The radius of the gears produced at a factory
must be 6 inches in length with a tolerance of 0.1
inches. The gears with radius beyond the
tolerated lengths will be thrown away. Write an
absolute value statement that represents the
longest and shortest acceptable gear.
Longest :6.1
Shortest: 5.9
Algebra II
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1. Solve: 4 + l 2x + 3l = 11
2. Solve and write answer in interval notation:
| ½x + 3 | < 12
3. Solve | 5x + 12 | > -8
4. A manufacturer is making doors for doll houses.
They will discard doors if they are not within 1/8 inch
of 5 inches. Write an absolute value inequality that
represents the doors they will discard.
Algebra II
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