# How do you solve abs(6x+7)=5x+2?

Apr 1, 2015

$x = - \frac{9}{11}$

The absolute value operation does:

1 - If the inner value of absolute value is non-negative, it returns the value.

2- If the inner value of absolute value is negative, it returns $\left(- 1\right)$ times of the value.

Our given equality is:

$\left\mid 6 x + 7 \right\mid = 5 x + 2$

• If $6 x + 7 \ge 0$ then absolute value opeariton will return $6 x + 7$

$6 x + 7 = 5 x + 2$
$x = - 5$

There is an important trick when it comes to absolute value operation. Always test the result with the if condition.

$6 \cdot \left(- 5\right) + 7 \ge 0$ fails. So $- 5$ is not in the solution set.

• If $6 x + 7 < 0$ then absolute value opeariton will return $- \left(6 x + 7\right)$

$- 6 x - 7 = 5 x + 2$
$- 9 = 11 x$
$- \frac{9}{11} = x$

Again, lets test the if condition.

$6 \cdot \left(- \frac{9}{11}\right) + 7 < 0$ success!

So $x = - \frac{9}{11}$