Solve |2x − 3| = 5
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −1, x = 4 |
Step-by-step solution
11 steps-
1 Given\left|{2 x - 3}\right| = 5
Solve for x.
-
2 Split into two cases: the expression inside the bars is either 5 or -5
|u| is the distance of u from 0. A distance of 5 can be on either side of 0, so the inside is 5 or the opposite of 5, and each case is solved as its own equation.
2 x - 3 = 5\quad \text{or} \quad 2 x - 3 = -5 -
3 Case 12 x - 3 = 5
-
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.
2 x = 8Move the constant terms to the right.
-
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{8}{2} -
6 Simplifyx = 4
-
7 Case 22 x - 3 = -5
-
8 Add 3 to both sides2 x = -2
Move the constant terms to the right.
-
9 Divide both sides by 2x = \frac{-2}{2}
-
10 Simplifyx = -1
-
11 Check\begin{aligned}x = -1:\quad 5 = 5\quad\checkmark\\ x = 4:\quad 5 = 5\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −1 | 5 | 5 | ✓ |
| 4 | 5 | 5 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.