# How do you solve abs(5x + 5) - 7 = 18?

Aug 2, 2015

There are two solutions:
$x = 4$ and $x = - 6$

#### Explanation:

Recall the definition of the absolute value of a real number:
If $N \ge 0$ then $| N | = N$
If $N < 0$ then $| N | = - N$.

Since absolute value of a number is evaluated differently, depending on this number's sign, let's consider two different cases:

Case A is when the absolute value is taken of the positive number or zero (and in this case absolute value of a positive number or zero is this number itself);

Case B is when the absolute value is taken of the negative number (and in this case absolute value of a negative number is its negation).

Here is the solution in each of these cases.

Case A: $5 x + 5 \ge 0$,
which is equivalent to
$5 x \ge - 5$ or
$x \ge - 1$
In this case $| 5 x + 5 | = 5 x + 5$ and our equation looks like
$5 x + 5 - 7 = 18$ or
$5 x = 20$ or
$x = 4$ (which satisfies the initial condition of $x \ge - 1$ and, therefore, is the real solution.
Check: $| 5 \cdot 5 + 5 | - 7 = 25 - 7 = 18$ OK

Case B: $5 x + 5 < 0$,
which is equivalent to
$5 x < - 5$ or
$x < - 1$
In this case $| 5 x + 5 | = - \left(5 x + 5\right)$ and our equation looks like
$- \left(5 x + 5\right) - 7 = 18$ or
$- 5 x = 30$ or
$x = - 6$ (which satisfies the initial condition of $x < - 1$ and, therefore, is the real solution.
Check: $| - 6 \cdot 5 + 5 | - 7 = | - 25 | - 7 = 25 - 7 = 18$ OK