Solve |2x + 2| ≥ 3
Absolute value inequality, worked out line by line the way a teacher would write it.
Answer
| Solution | x ≤ −5/2 or x ≥ 1/2 |
| Interval notation | (−∞, −5/2] ∪ [1/2, ∞) |
Step-by-step solution
10 steps-
1 Given\left|{2 x + 2}\right| \ge 3
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 3 away from 0 means u is below −3 or above 3, so the answer has two separate pieces.
2 x + 2 \le -3\quad\text{or}\quad 2 x + 2 \ge 3 -
3 Case 12 x + 2 \le -3
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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 -5Move the constant terms to the right.
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5 Divide both sides by 2x \le - \frac{5}{2}
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6 Case 22 x + 2 \ge 3
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7 Subtract 2 from both sides2 x \ge 1
Move the constant terms to the right.
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8 Divide both sides by 2x \ge \frac{1}{2}
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9 Either case works, so combine the two answersx \le - \frac{5}{2} \;\text{ or }\; x \ge \frac{1}{2}
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10 Solutionx \le - \frac{5}{2} \;\text{ or }\; x \ge \frac{1}{2}\qquad \left( -\infty, - \frac{5}{2} \right] \cup \left[ \frac{1}{2}, \infty \right)
Check with test values
| x | Left side | Right side | Holds? |
|---|---|---|---|
| −3 | 4 | 3 | ✓ true |
| 1 | 4 | 3 | ✓ true |
| −2 | 2 | 3 | ✗ false |
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