Bellwork
Solve each inequality. Write the solution
in interval notation.
1. 4x – 2(3x + 2) < 2x – 7
2. 4/3x – 2/9 > 2x + 5/6
3. 1/6(12x + ¾) < 1/6x + 2/3
Section 1.6/1.7
Compound Inequalities and Absolute Value
Equations and Inequalities
Compound Inequalities
• Two Inequalities connected by and or or
• And:
o Both statements must be true.
o When graphing, shade where the two
statements overlap
• Or:
o Either statement can be true.
o When graphing, shade anything that is
graphed
Compound Inequalities: And
Both must be true!
5. -2 < 3x - 8 < 10
+8
+ 8 +8
6. -9 < 2x + 3 < 10
-3
-3 -3
6<3x<18
3 3 3
-12<2x<7
2 2 2
2<x<6
-6<x<7/2
[2, 6]
(-6, 7/2)
Compound Inequalities: Or
Only one must be true
7. 2x + 3 < 5 or 4x - 7 > 9 8. 3x - 2 > 10 or
+2 +2
-3 -3
+ 7 +7
2X<2
X<1
or
4x > 16
or
x>4
(-∞, 1) U (4, ∞)
3X>12
x>4
or
-4x > 16
x < -4
x <-4
x < -4
(-∞, -4] U [4, ∞)
Solve each Compound Inequality
9. 13 > 3 – 4x > 3
10. 4x – 3 > -8 or 3 – 2x > 7
What is absolute Value?
Absolute Value
The distance a number is from zero on a
number line.
What value could make this statement true?
│x│ = 7
7 or – 7
│x + 1│ = 5
4 or -6
You will always answer two problems for
absolute value problems.

To solve: Isolate the │n│= c

Write two problems n = c and n = -c

Solve each problem

If it is an inequality, you must flip the
symbol of the second problem!

Always check your solution! They may not
work!

If the symbol is a:
= use or
< or < use and
> or > use or
Rewrite the absolute value problem as two
separate equations or inequalities.
1. │3x + 1│= 4
2. │x – 3 │= 7
3. │2 - 4x│< 10
4. │10x – 3 │> 4
Solve each absolute value
equation.
1. │2x – 5│ = 9
2. │6x - 3 │ = 15
2X – 5 = 9 or 2x – 5 = -9
6x-3 = 15 or 6x-3 = -15
2X = 14
2x = -4
X=7
x = -2
6x = 18
6x = -12
x=3
x = -2
Solve each absolute value
equation.
3. 7│x + 3│ = 42
│x + 3│ = 6
X + 3 = 6 or x + 3 = -6
X=3
x = -9
4. │ ½x + 3 │ = -7
½ x+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!
Solve each absolute value
equation.
5. │2 - 4x│ = 10
2 – 4x = 10 or 2 – 4x= -10
-4x = 8
X = -2
-4x = -12
6. -3│x – 5│+ 7 = -8
-3│x – 5│ = -15
│x – 5│ = 5
x=3
X – 5 = 5 or x – 5 = -5
X = 10
x=0
Solve and graph each absolute value
inequality, put answers in interval
notation.
7. 6│2x + 3│ < 24
8. │1 – 3x│ > 5
│2x + 3│ < 4
1 – 3x > 5 or 1 – 3x <
2X+3 < 4 and 2x+3 > -4
-5
2X < 1
2x > -7
-3x > 4
-3x < -6
X < ½ and x > -7/2
X < -4/3 or x > 2
(-7/2, ½)
(-∞, -4/3] U [2, ∞)
Solve and graph each absolute value
inequality, put answers in interval
notation.
9. │x – 8 │- 2 > 1
│x – 8 │ > 3
X – 8 > 3 or x – 8 < -3
X > 11 or
x<5
(-∞, 5] U [11, ∞)
10. -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!
Special Cases
• ││= -#
• ││< -#
• ││> -#
Word Problems
• │variable – ideal │{=,<, >} change
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.
• │variable – ideal │{=,<, >} change
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.
• │variable – ideal │{=,<, >} change
Jim keeps seedlings that are
within ½ inch of 4 inches. Write
an absolute value sentence that
describes the seedlings he will
discard.
• │variable – ideal │{=,<, >} change
Exit Pass
1. Solve: 4 + l 2x + 3l = 11
2. Solve and write answer in interval
notation: l ½x + 3 l < 12
3. Solve l 5x + 12 l > -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.
Closure
Murphy is joining a gym. He gets a 5 digit membership
number. When he turns his membership card upside
down, the original number is 7920 more than the new
number. What is his membership number. (The number
uses the digits 0 - 9 and no number is repeated.)