Solve |x − 6| + 5 = 8
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = 3, x = 9 |
Step-by-step solution
8 steps-
1 Given\left|{x - 6}\right| + 5 = 8
Solve for x.
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2 Isolate the absolute value\left|{x - 6}\right| = 3
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3 Split into two cases: the expression inside the bars is either 3 or -3
|u| is the distance of u from 0. A distance of 3 can be on either side of 0, so the inside is 3 or the opposite of 3, and each case is solved as its own equation.
x - 6 = 3\quad \text{or} \quad x - 6 = -3 -
4 Case 1x - 6 = 3
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5 Add 6 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 = 9Move the constant terms to the right.
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6 Case 2x - 6 = -3
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7 Add 6 to both sidesx = 3
Move the constant terms to the right.
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8 Check\begin{aligned}x = 3:\quad 8 = 8\quad\checkmark\\ x = 9:\quad 8 = 8\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 3 | 8 | 8 | ✓ |
| 9 | 8 | 8 | ✓ |
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