# Question #c041e

Aug 24, 2017

$x \in \left\{\emptyset\right\}$

#### Explanation:

Right from the start, the fact that you're dealing with the absolute value of an expression tells you that $x$ must be $\ge 0$.

That is the case because regardless of the sign of the expression inside the absolute value signs, the expression on the right side of the equation must be greater than or equal to $0$.

Since you have

$| 5 x + 8 | = x$

you can say that $x \ge 0$ because $| 5 x + 8 |$ must return a value that is $\ge 0$.

So, you know that you have two possible scenarios to look at

• $5 x + 8 \ge 0 \implies | 5 x + 8 | = 5 x + 8$

In this case, you have

$5 x + 8 = x$

$4 x = - 8 \implies x = \frac{- 8}{4} = - 2$

• $5 x + 8 < 0 \implies | 5 x + 8 | = - \left(5 x + 8\right)$

In this case, you have

$- \left(5 x + 8\right) = x$

$- 5 x - 8 = x$

$- 6 x = 8 \implies x = \frac{8}{- 6} = - \frac{4}{3}$

However, you already know that you need

$x \ge 0$

so you can say that $x = - 2$ or $x = - \frac{4}{3}$ will not be valid solutions to the original equation.

This means that the original equation has no solution when working with real numbers, or $x \in \left\{\emptyset\right\}$.

Aug 25, 2017

Simpler and Quicker version of the same thing.

#### Explanation:

$| 5 x + 8 | = x$
The solutions, if any exist, are found by solving the equations
$5 x + 8 = x$ or $5 x + 8 = - x$.
These are simple linear equations.
The first yields
$8 = - 4 x$
$- 2 = x$
This cannot be a solution since x must be positive.

The second equation yields
$5 x + 8 = - x$
$8 = - 6 x$
$x = - \frac{4}{3}$
This cannot be a solution since x must be positive.

The solution set is {}.