Solving Inequalities Using Inverse Operations

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:

1. Reverse addition and subtraction (by subtracting and adding) outside parentheses.
2. 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.
3. 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: 5x - 8 < 12

5x - 8 + 8 < 12 + 8
5x < 20
<
x < 4

Example 2: 4 - 2x≤2x - 4

4 - 2x + 2x≤2x - 4 + 2x
4≤4x - 4
4 + 4≤4x - 4 + 4
8≤4x
≤4x4
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 .