Solve |2x − 9| = 7
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = 1, x = 8 |
Step-by-step solution
11 steps-
1 Given\left|{2 x - 9}\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.
2 x - 9 = 7\quad \text{or} \quad 2 x - 9 = -7 -
3 Case 12 x - 9 = 7
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4 Add 9 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.
2 x = 16Move the constant terms to the right.
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5 Divide both sides by 2
Dividing both sides by 2 undoes the multiplication by 2, 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{16}{2} -
6 Simplifyx = 8
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7 Case 22 x - 9 = -7
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8 Add 9 to both sides2 x = 2
Move the constant terms to the right.
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9 Divide both sides by 2x = \frac{2}{2}
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10 Simplifyx = 1
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11 Check\begin{aligned}x = 1:\quad 7 = 7\quad\checkmark\\ x = 8:\quad 7 = 7\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 1 | 7 | 7 | ✓ |
| 8 | 7 | 7 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.