Solve |3x + 6| = 12
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −6, x = 2 |
Step-by-step solution
11 steps-
1 Given\left|{3 x + 6}\right| = 12
Solve for x.
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2 Split into two cases: the expression inside the bars is either 12 or -12
|u| is the distance of u from 0. A distance of 12 can be on either side of 0, so the inside is 12 or the opposite of 12, and each case is solved as its own equation.
3 x + 6 = 12\quad \text{or} \quad 3 x + 6 = -12 -
3 Case 13 x + 6 = 12
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4 Subtract 6 from 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.
3 x = 6Move the constant terms to the right.
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5 Divide both sides by 3
Dividing both sides by 3 undoes the multiplication by 3, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{6}{3} -
6 Simplifyx = 2
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7 Case 23 x + 6 = -12
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8 Subtract 6 from both sides3 x = -18
Move the constant terms to the right.
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9 Divide both sides by 3x = \frac{-18}{3}
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10 Simplifyx = -6
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11 Check\begin{aligned}x = -6:\quad 12 = 12\quad\checkmark\\ x = 2:\quad 12 = 12\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −6 | 12 | 12 | ✓ |
| 2 | 12 | 12 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.