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