We can solve *inequalities* using inverse operations in the same way we solve
*equations* using inverse operations with one exception: we have to pay attention to
the rules governing multiplication and division by a negative number and reciprocals, and
flip the inequality sign when appropriate.

Again, follow these steps to reverse the order of operations acting on the variable:

- Reverse addition and subtraction (by subtracting and adding)
outside parentheses.
- Reverse multiplication and division (by dividing and multiplying)
outside parentheses.
**When multiplying or dividing by a negative
number, flip the inequality sign**. It does not matter if the number
*being divided* is positive or negative.
- Remove (outermost) parentheses, and reverse the operations in order
according to these three steps.

The answer should be an inequality; for example,

*x* < 5.

To solve an inequality with a "≠" sign, change the "≠" sign
into an "=" sign, and solve the equation. Then, change the "="
sign in the answer to a "≠" sign. This works because determining
the values of *x* for which two expressions are *not equal* is
the same as determining the values for which they are *equal* and
excluding them from the replacement set.

*Example 1*: 5*x* - 8 < 12

5*x* - 8 + 8 < 12 + 8

5*x* < 20

<

*x* < 4

*Example 2*: 4 - 2*x*≤2*x* - 4

4 - 2*x* + 2*x*≤2*x* - 4 + 2*x*

4≤4*x* - 4

4 + 4≤4*x* - 4 + 4

8≤4*x*

≤4*x*4

2≤*x*

*x*≥2

*Example 3*: ≥ - 6

×5≥ -6×5

The number being divided is negative, but the number we are dividing by is
positive, so the sign does not flip.

*x* - 2≥ - 30
*x* - 2 + 2≥ - 30 + 2
*x*≥ - 28
*Example 4*: -3(*x* + 2) > 9

<

*x* + 2 < - 3

*x* + 2 - 2 < - 3 - 2

*x* < - 5

*Example 5*: +2≠ - 5

Solve +2 = - 5

+2 - = -5 -

2 = - 5

2 + 5 = - 5 + 5

7 =

7×6 = ×6

42 = *x*

*x* = 42

Thus, the solution to our original inequality is *x*≠42.