Solve |2x + 2| ≥ 8
Absolute value inequality, worked out line by line the way a teacher would write it.
Answer
| Solution | x ≤ −5 or x ≥ 3 |
| Interval notation | (−∞, −5] ∪ [3, ∞) |
Step-by-step solution
10 steps-
1 Given\left|{2 x + 2}\right| \ge 8
Solve for x.
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2 |u| > d means u is below −d or above d
|u| is the distance from u to 0. Being more than 8 away from 0 means u is below −8 or above 8, so the answer has two separate pieces.
2 x + 2 \le -8\quad\text{or}\quad 2 x + 2 \ge 8 -
3 Case 12 x + 2 \le -8
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4 Subtract 2 from both sides
Like an equation, an inequality stays true when the same amount is added to or subtracted from both sides. Only multiplying or dividing by a negative number needs extra care: it reverses the sign.
2 x \le -10Move the constant terms to the right.
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5 Divide both sides by 2x \le -5
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6 Case 22 x + 2 \ge 8
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7 Subtract 2 from both sides2 x \ge 6
Move the constant terms to the right.
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8 Divide both sides by 2x \ge 3
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9 Either case works, so combine the two answersx \le -5 \;\text{ or }\; x \ge 3
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10 Solutionx \le -5 \;\text{ or }\; x \ge 3\qquad \left( -\infty, -5 \right] \cup \left[ 3, \infty \right)
Check with test values
| x | Left side | Right side | Holds? |
|---|---|---|---|
| −6 | 10 | 8 | ✓ true |
| 4 | 10 | 8 | ✓ true |
| −4 | 6 | 8 | ✗ false |
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