| Solving Inequalities in One Variable |
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SOLVING INEQUALITIES in one variable is much like solving a linear equation. You isolate the variable by using the "transformations" of addition, subtraction, multiplication and division to affect both sides of the inequality.
| The one important
difference is that when you multiply or divide
both sides of an inequality by a negative number,
the
inequality symbol reverses (flips) direction!
|
THINK ABOUT IT!
| IF... -x > 3 |  |
| THEN... x < -3 |
Test a few examples like the one above on a number line. You can then save this concept as a trusted tool!
Example 1 (No Reversal)
3x + 7 < -14 << subtract 7 from both sides >>
3x < -21 << divide both sides by 3 >>
x < -7
Check your solution by plugging in a qualifying "x" value such as x = -10 into the original inequality:
3(-10) + 7 < -14 ??
-30 + 7 < -14 ??
-23 < -14 TRUE!!
Example 2 (Reversal!)
-3x + 7 < -14 << subtract 7 from both sides >>
-3x < -21 << divide both sides by -3 >>
x > 7
Check your solution by plugging in a qualifying " x" value such as x = 10 into the original inequality:
-3(10) + 7 < -14 ??
-30 + 7 < -14 ??
-23 < -14 TRUE!!
Let's solve the following inequality AND express the answer using set-builder notation:
30 - x < 68
- 30 - 30 << subtract 30 from both sides >>
- x < 38 << multiply both sides by -1 >>
x > - 38
Do Now
1) x - 7 > 40
2) 3b + 4 < 31
3) -7m - 3 < 32
4) - 1/4 x + 7 > 12