1.6/1.7 Solving Linear
Inequalities
Inequality Symbols
    
Less
than
Less
than or
equal to
Greater
than
Not
equal to
Greater
than or
equal to
Linear Inequality
• Inequality with one variable to the first
power.
for example: 2x-3<8
• A solution is a value of the variable that
makes the inequality true.
x could equal -3, 0, 1, etc.
Transformations for Inequalities
• Add/subtract the same number on each side
of an inequality
• Multiply/divide by the same positive number
on each side of an inequality
• If you multiply or divide by a negative number,
you MUST flip the inequality sign!
Ex: Solve the inequality.
2x-3<8
+3 +3
2x<11
2 2
x< 11
 3x  7  13
 3x  6
x  2
2
Flip the sign after
dividing by the -3!
Graphing Linear Inequalities
• Remember:
< and > signs will have an open dot o
 and  signs will have a closed dot 
graph of
4
5
6
x
7
11
2
graph of x  2
-3
-2
-1
0
Example: Solve and graph the solution.
7 x  9  10 x 12
9  3x  12
21  3x
7 x
6
7
8
9
Compound Inequality
• An inequality joined by “and” or “or”.
Examples
“and”
“or”
3  x 1
-4 -3 -2 -1 0
1
x  2 or x  4
2
-3 -2 -1
0
1
2
3
4
5
think between
think oars on a boat
Example: Solve & graph.
-9 < t+4 < 10
-13 < t < 6
Think between!
-13
6
Last example! Solve & graph.
-6x+9 < 3 or -3x-8 > 13
-6x < -6
-3x > 21
x > 1 or
x < -7
Flip signs
Think oars
-7
1
Compound Inequality: Try these
• 2x>-10 and 9x<18
• 3x≥-12 and 8x≤16
• 4x<16 or 12x>144
Answers
• -5<x<2
• -4≤x≤2
• X<4 or x>12
Try these:
• 2-3z≥7(8-2z)+12
• 2/3(x-12)≤x+8
• 3[4x-(2x-7)]<2(3x-5)
Answers
• Z≥6
• X≥-48
• No solution
1.7 Solving Absolute Value
Equations & Inequalities
Absolute Value (of x)
• Symbol lxl
• The distance x is from 0 on the number
line.
• Always positive
• Ex: l-3l=3
-4
-3
-2
-1
0
1
2
Ex: x = 5
• What are the possible values of x?
x=5
or
x = -5
To solve an absolute value equation:
ax+b = c, where c>0
To solve, set up 2 new equations, then
solve each equation.
ax+b = c or ax+b = -c
** make sure the absolute value is by
itself before you split to solve.
Also, make sure it is not equal to a
negative number.
Ex: Solve 6x-3 = 15
6x-3 = 15 or
6x = 18 or
x = 3 or
6x-3 = -15
6x = -12
x = -2
* Plug in answers to check your solutions!
Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.
Solving Absolute Value Inequalities
1. ax+b < c, where c>0
Becomes an “and” problem
Changes to: –c<ax+b<c
2. ax+b > c, where c>0
Becomes an “or” problem
Changes to: ax+b>c or ax+b<-c
Ex: Solve & graph.
4 x  9  21
• Becomes an “and” problem
 21  4x  9  21
12  4x  30
15
3 x 
2
-3
7
8
Solve & graph.
3x  2  3  11
• Get absolute value by itself first.
3x  2  8
• Becomes an “or” problem
3x  2  8 or 3x  2  8
3x  10 or
3x  6
10
x
or x  2
3
-2
3
4
Assignment