Solve 3|x − 5| − 1 = 20
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −2, x = 12 |
Step-by-step solution
8 steps-
1 Given3 \left|{x - 5}\right| - 1 = 20
Solve for x.
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2 Isolate the absolute value\left|{x - 5}\right| = 7
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3 Split into two cases: the expression inside the bars is either 7 or -7
|u| is the distance of u from 0. A distance of 7 can be on either side of 0, so the inside is 7 or the opposite of 7, and each case is solved as its own equation.
x - 5 = 7\quad \text{or} \quad x - 5 = -7 -
4 Case 1x - 5 = 7
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5 Add 5 to both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
x = 12Move the constant terms to the right.
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6 Case 2x - 5 = -7
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7 Add 5 to both sidesx = -2
Move the constant terms to the right.
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8 Check\begin{aligned}x = -2:\quad 20 = 20\quad\checkmark\\ x = 12:\quad 20 = 20\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −2 | 20 | 20 | ✓ |
| 12 | 20 | 20 | ✓ |
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