Solve |3x − 5| = 7
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −2/3, x = 4 |
Step-by-step solution
11 steps-
1 Given\left|{3 x - 5}\right| = 7
Solve for x.
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2 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.
3 x - 5 = 7\quad \text{or} \quad 3 x - 5 = -7 -
3 Case 13 x - 5 = 7
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4 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.
3 x = 12Move 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{12}{3} -
6 Simplifyx = 4
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7 Case 23 x - 5 = -7
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8 Add 5 to both sides3 x = -2
Move the constant terms to the right.
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9 Divide both sides by 3x = \frac{-2}{3}
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10 Simplifyx = - \frac{2}{3}
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11 Check\begin{aligned}x = - \frac{2}{3}:\quad 7 = 7\quad\checkmark\\ x = 4:\quad 7 = 7\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −2/3 | 7 | 7 | ✓ |
| 4 | 7 | 7 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.