Solve |x − 3| + 5 = 16
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −8, x = 14 |
Step-by-step solution
8 steps-
1 Given\left|{x - 3}\right| + 5 = 16
Solve for x.
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2 Isolate the absolute value\left|{x - 3}\right| = 11
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3 Split into two cases: the expression inside the bars is either 11 or -11
|u| is the distance of u from 0. A distance of 11 can be on either side of 0, so the inside is 11 or the opposite of 11, and each case is solved as its own equation.
x - 3 = 11\quad \text{or} \quad x - 3 = -11 -
4 Case 1x - 3 = 11
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5 Add 3 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 = 14Move the constant terms to the right.
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6 Case 2x - 3 = -11
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7 Add 3 to both sidesx = -8
Move the constant terms to the right.
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8 Check\begin{aligned}x = -8:\quad 16 = 16\quad\checkmark\\ x = 14:\quad 16 = 16\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −8 | 16 | 16 | ✓ |
| 14 | 16 | 16 | ✓ |
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