Question

How do you solve $|v + 8| - 5 = 2$?

Answer 1

Hint: In this question we have the term in the form of an absolute value therefore we will first simplify the equation and then solve for both the positive and the negative equivalents to get the final solutions and write them together in the form of a solution set.

Complete step-by-step answer:
We have the given expression as:
$ \Rightarrow |v + 8| - 5 = 2$
On transferring the term $2$ from the right-hand side of the equation to the left-hand side of the equation, we get:
$ \Rightarrow |v + 8| - 5 - 2 = 0$
On simplifying the expression, we get:
$ \Rightarrow |v + 8| - 7 = 0$
Now we know that abstract value function takes a term and transforms the term into the non-negative form therefore, we will solve the expression for both the positive and the negative equivalent.
Therefore, solution $1$ can be found out as:
$ \Rightarrow v + 8 = 7$
On transferring the term $8$ from the right-hand side to the left-hand side, we get:
$ \Rightarrow v = 7 - 8$
On simplifying, we get:
$ \Rightarrow v = - 1$, which is the first solution.
Now solution $2$ can be found out as:
$ \Rightarrow v + 8 = - 7$
On transferring the term $8$ from the right-hand side to the left-hand side, we get:
$ \Rightarrow v = - 7 - 8$
On simplifying, we get:
$ \Rightarrow v = - 15$, which is the second solution.

Therefore, the solution set is: $v = \{ - 15, - 1\} $.

Note:
The absolute value function is defined as \[f(x) = |x| = \left\{ {\begin{array}{*{20}{c}}
  {x,{\text{if }}x \geqslant 0} \\
  { - x,{\text{if }}x < 0}
\end{array}} \right\}\].
It is also written as $abs(x)$
The absolute value function only considers the positive value of a term if it is negative.
The real-life application of using the absolute value function is when there are quantities which cannot have negative value, for example length, time, resistance etc. these quantities cannot be negative otherwise it would result in a calculation fallacy, therefore absolute value function is used.