8.8 Solving Multi-Step Equations
and Inequalities
Steps to solve multi-step linear equations
1) If there are ( ) distribute
2) Collect like terms on same side.
3) Move variables to same side of
equation.
4) Move the constant to the other side of
the equation
5) Solve for x using the
Ex :
3(n - 2) – 1 = 2 – 5(n + 5)
3n - 6 – 1 = 2 – 5(n + 5)
3n - 6 – 1 = 2 – 5n - 25
5n +
3n - 7
= -23 – 5n +5n
8n - 7 +7 = -23 +7
8n
1

8
= -16
1

8
n =-2
No Solution
• Also referred to as: “Null Set” or “Empty Set”
• Happens when the variable cancels out and
leaves you with a false equation
14 + 8w = 4(8+2w)
14  8w  4  8  4  2w
14  8w  32  8w
8w
Empty Set
or
Null set
8w
14  32

false
or
{}
Identity
• An equation that is true for every value of the
variable
• The variable cancels out and you are left with
a true equation
-2(3r+4) = -5r-8-r
2(3r )  2(4)  6r  8
6r  8  6r  8
6r
6r
all
numbers
work
8  8
identity
many
Solutions
Steps to solve multi-step linear inequality
1) If there are ( ) distribute
2) Collect like terms on same side.
3) Move variables to Left hand side
of inequality.
4) Move the constant to the right side of
the inequality
5) Solve for x using the (If you multiply or
divide by a negative number to both
sides of the inequality then flip the
inequality sign)
Example :
4
x
9
-6
-6x + 7 – x + 1 < 2x + 4
-7x + 8 < 2x + 4
-2x
-2x
-9x + 8 < 4
-8
-8
-9x
< -4
-9
-9
-4
-2
0
2
4
6
Example :
5(12-3t) ≥15(t+4)
60-15t ≥ 15t +60
15t 15t
60  30t  60
60
-6
-4
60
-30t ≥ 0
30 30
-2
0
2
t≤0
4
6
Homework
Page 377 (18-35) all