# How do you solve abs(4x+3)=2x-1?

Jul 5, 2015

Solve |4x + 3| = 2x - 1

#### Explanation:

Jul 5, 2015

Split into cases $4 x + 3 \ge 0$ and $4 x + 3 < 0$ then solve the resulting linear equations.

We find that the resulting solutions are both spurious, so $\left\mid 4 x + 3 \right\mid = 2 x - 1$ has no solutions.

#### Explanation:

Split into cases $4 x + 3 \ge 0$ and $4 x + 3 < 0$

Case 1 : 4x+3 >= 0

If $4 x + 3 \ge 0$ then $\left\mid 4 x + 3 \right\mid = 4 x + 3$

Solve $4 x + 3 = 2 x - 1$

Subtract $2 x$ from both sides to get:

$2 x + 3 = - 1$

Subtract #3 from both sides to get:

$2 x = - 4$

Divide both sides by $2$ to get $x = - 2$

Then $4 x + 3 = 4 \left(- 2\right) + 3 = - 8 + 3 = - 5 < 0$

So $x = - 2$ is a spurious solution, since it does not satisfy $4 x + 3 \ge 0$

Case 2 : 4x+3 < 0

If $4 x + 3 < 0$ then $\left\mid 4 x + 3 \right\mid = - \left(4 x + 3\right)$

Solve $- \left(4 x + 3\right) = 2 x - 1$

Add $4 x$ to both sides to get:

$- 3 = 6 x - 1$

Add $1$ to both sides to get:

$- 2 = 6 x$

Divide both sides by $6$ to get $x = - \frac{1}{3}$

Then $4 x + 3 = - \frac{4}{3} + 3 = - \frac{4}{3} + \frac{9}{3} = \frac{5}{3} > 0$

So $x = - \frac{1}{3}$ is a spurious solution, since it does not satisfy $4 x + 3 < 0$