Solve |4x − 5| = 8
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −3/4, x = 13/4 |
Step-by-step solution
10 steps-
1 Given\left|{4 x - 5}\right| = 8
Solve for x.
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2 Split into two cases: the expression inside the bars is either 8 or -8
|u| is the distance of u from 0. A distance of 8 can be on either side of 0, so the inside is 8 or the opposite of 8, and each case is solved as its own equation.
4 x - 5 = 8\quad \text{or} \quad 4 x - 5 = -8 -
3 Case 14 x - 5 = 8
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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.
4 x = 13Move the constant terms to the right.
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5 Divide both sides by 4
Dividing both sides by 4 undoes the multiplication by 4, 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{13}{4} -
6 Case 24 x - 5 = -8
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7 Add 5 to both sides4 x = -3
Move the constant terms to the right.
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8 Divide both sides by 4x = \frac{-3}{4}
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9 Simplifyx = - \frac{3}{4}
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10 Check\begin{aligned}x = - \frac{3}{4}:\quad 8 = 8\quad\checkmark\\ x = \frac{13}{4}:\quad 8 = 8\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −3/4 | 8 | 8 | ✓ |
| 13/4 | 8 | 8 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.