Solving Inequalities

In Lessons 4-2, 4-3 and 4-4 we have learned about graphing inequalities, solving when addition and subtraction are involved, solving when multiplication is involved and when there is a mix of operations on one or both sides of the inequality sign.

We always want to get our final answer simplified into the form of:

x > 3

When we are solving for a variable we work backwards through the order of operations:

Subtraction or Addition first

Division or Multiplication second

Clear any addition or subtraction first.

X + 4 < 12

subtract 4 from both sides

X + 4 - 4 < 12 - 4


X < 8


Clear any multiplication or division.


3x > 21


Multiply by the reciprocal

1/3 • 3x > 21 • 1/3


x > 7


For more complex problems such as

2(x - 3) < 4(x - 5)

Use the distributive property to clear the parenthesis first.

2x - 6 < 4x - 20

Then combine like terms by moving the terms with the variable to one side of the inequality and the terms without the variable to the other.

2x - 2x - 6 < 4x - 2x - 20

-6 < 2x - 20



-6 + 20 < 2x - 20 + 20

14 < 2x

Then multiply by the reciprocal.

1/2 • 14 < 2x • 1/2

7 < x or x > 7


Remember to always simplify each side of the inequality before you begin combining like terms.