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Home : Math & Science : Math Study Guides : Algebra I : Inequalities : Solving Inequalities Using Inverse Operations
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:
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 ≤4x42≤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≥ - 30x - 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≠ - 5Solve +2 = - 5 +2 - = -5 - ![]() 2 = - 52 + 5 = - 5 + 57 = ![]() 7×6 = ×642 = x x = 42 Thus, the solution to our original inequality is x≠42. |
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