Question

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

Answer 1

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

Explanation:

Recall the definition of the absolute value of a real number:
If `N >= 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: `5x+5 >= 0`,
which is equivalent to
`5x >= -5` or
`x >= -1`
In this case `|5x+5|=5x+5` and our equation looks like
`5x+5-7=18` or
`5x = 20` or
`x = 4` (which satisfies the initial condition of `x >= -1` and, therefore, is the real solution.
Check: `|5*5+5|-7=25-7=18` OK

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