How do you solve -10| h + 5| - 3= - 83?

Oct 16, 2016

$h \in \left\{- 13 , 3\right\}$

Explanation:

Your first goal here is to isolate the modulus on one side of the equation. To do that, add $3$ to both sides and divide both sides by $- 10$

$- 10 \cdot | h + 5 | - \textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} = - 83 + 3$

$- 10 \cdot | h + 5 | = - 80$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 10}}} \cdot | h + 5 |}{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 10}}}} = \frac{- 80}{- 10}$

$| h + 5 | = 8$

At this point, you have two possible cases to look at

• $h + 5 \ge 0 \implies | h + 5 | = h + 5$

This gets you

$h + 5 = 8 \implies h = 3$

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

This time, you have

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

$- h - 5 = 8$

$h = - 13$

Therefore, the original equation has two possible solutions

$h \in \left\{- 13 , 3\right\}$

Do a quick double-check to make sure that the calculations are correct

$- 10 \cdot | 3 + 5 | - 3 = - 83$

$- 10 \cdot 8 - 3 = - 83 \text{ } \textcolor{g r e e n}{\sqrt{}}$

and

$- 10 \cdot | - 13 + 5 | - 3 = - 83$

$- 10 \cdot | - 8 | - 3 = - 83$

$- 10 \cdot 8 - 3 = - 83 \text{ } \textcolor{g r e e n}{\sqrt{}}$