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