Chapter 4 Study Guide /Notes
4.1 Inequalities and Their Graphs pp 200-202
Terms
A solution of an inequality - any number that makes the inequality true.
○ use when your inequality has the symbol < or >. This means that the inequality cannot
equal the endpoint. x < 3
-1 0 1 2 3
use when your inequality has the symbol ≤ or ≥. This means that the inequality can equal
the endpoint. x ≤ 3

-1 0 1 2 3
If x is on the left the greater than or less than sign will point the way the line should go!
4.2 Solving Inequalities Using Addition and Subtraction pp 206-208
Terms
Equivalent inequalities – inequalities with the same solutions.
Addition Property of Inequalities - If a > b, then a + c > b + c. If 6 > 2, then 6 + 5 > 2 + 5 (11
> 7).
Subtraction Property of Inequalities - If a > b, then a - c > b - c. If 7 > 5, then 7 – 3 > 5 – 3
(4 > 2).
You can solve inequalities using the same addition and subtraction methods you used for
solving equations.
x–6< 8
+6 +6
x < 14
add 6 to both sides
y + 5 ≥ 12
-5 -5
y ≥ 7
subtract 5 from both sides
4.3 Solving Inequalities Using Multiplication and Division pp 212-215
Terms
Multiplication Property of Inequality (for positive numbers)
If a > b, then a•c >b•c. If 10 > 5, then 10 • 4 > 5 • 4 (40> 20)
Multiplication Property of Inequality (for negative numbers)
If a > b, then a•c <b•c. If 10 > 5, then 10 •(- 4) < 5 •(-4) (-40<- 20)
Division Property of Inequality (for positive numbers)
a
b
15
10
If a > b, then > . If 15 > 10, then
> (3 > 2)
c
c
5
5
Division Property of Inequality (for negative numbers)
a
b
15
10
If a > b, then < . If 15 > 10, then
<
(- 3 < - 2)
c
c
−5
−5
4.4 Solving Multi-Step Inequalities pp 219-221
Remembering the rules for solving inequalities are much like the rules for solving equations
with the exception of multiplying or dividing by a negative!
5x – 3(x -2) < 6x – 10
distribute
5x – 3x + 6 < 6x – 10
combine like terms on each side of the inequality sign
2x + 6 < 6x – 10
-6x
-6x
-4x + 6 < -10
-6
-6
-4 x < -16
-4
-4
x > 4
get the variables on one side
the constants on the other
divide by -4 --- don’t forget to flip the sign!!!
Check a number greater than 4 to see if you solved correctly.
5(8) – 3((8) -2) < 6(8) – 10
40 – 3 (6) < 48 – 10
40 – 18 < 38
22
< 38
Correct!
4.5 Compound Inequalities pp 227-229
Terms
Compound Inequality – inequalities than are joined by the words and or or.
x < 5 and x ≥ -2
-3 -2 -1 0 1 2 3 4
5
6 7
8
“and “ means both inequalities must be true, so the overlapping parts of the lines are the
solution.
-3 -2 -1 0 1 2 3 4
5
6 7
8
The inequality can be written -2 ≤ x < 5.
x < -2 or x ≥ 1
-3 -2 -1 0 1 2 3 4
5
6 7
8
“or “ means either inequalities can be true, so all the parts of both lines are the solution.
-3 -2 -1 0 1 2 3 4
5
6 7
8
4.6 Absolute Value Equations and Inequalities pp 235-237
Absolute Value Equations
To solve an absolute value equation, first get just the absolute value portion of the equation
on one side.
6 + |2x – 3| = 13 subtract 6
-6
-6
|2x – 3| = 7
2x – 3 could equal either 7 or – 7 so set up two equations from the original one.
2x – 3 = 7
2x – 3 =- 7
+3 +3 add 3 to both sides
+3
2x = 10
2 2
divide by 2
x=5
or
+3
2x = -4
2 2
x = -2
Check your work!
6 + |2(5) – 3| = 13
6 + |2(-2) – 3| = 13
6 + |10 - 3| = 13
6 + |(-4) – 3| = 13
6 + |7| = 13
6 + |-7| = 13
6 + 7 = 13
6 + 7 = 13
13 = 13
13 = 13
Absolute Value Inequalities
To solve an absolute value inequality, first get just the absolute value portion of the inequality
on one side.
3 + |5x – 10| < 13subtract 3
-3
-3
|5x – 10| < 10
5x -10 must be less than 10 and greater than -10. (Between -10 and 10)
5x – 10 < 10
5x – 10 > - 10
+10 +10
add 10 to both sides
+10 +10
5x < 20
5 5
divide by 5
x<4
and
5x > 0
5 5
x>0
Check your work! A number that is less than 4 and greater than 0 would be 1.
( 2 or 3 would also work.)
3 + |5(1) – 10| < 13
3 + |5 - 10| < 13
3 + |-5| < 13
-3 -2 -1 0 1 2 3 4
3 + 5 < 13
8 < 13 True!
5
6 7
8
Here is another:
2 + |3x + 15| > 8 subtract 2
-2
-2
|3x + 15| > 6
-8 -7 -6 -5 -4 -3 -2 -1
3x + 15 must be bigger than 6 or smaller than47-6. 8
3x + 15 > 6
-15 -15
3x > -9
3
3
x > -3
Check with 0!
2 + |3(0) + 15| > 8
2 + |0 + 15| > 8
2 + |15| > 8
2 + 15 > 8
17 > 8
subtract 15
divide by 3
or
True!
3x + 15 < - 6
-15 -15
3x < -21
3
3
x < -7
Check with -10!
2 + |3(-10) + 15| > 8
2 + | -30 + 15| > 8
2 + |-15|> 8
2 + 15 > 8
17 > 8
0
1
2
3