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,

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.

*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
.