By definition, we have
[tex] |p+2| = \begin{cases} p+2 &\text{ if } p+2 \geq 0 \\-p-2 &\text{ if } p+2 < 0 \end{cases} [/tex]
So, we have to solve two different equations, depending of the possible range for the variable. We have to remember about these ranges when we decide to accept or discard the solutions:
Suppose that [tex] p+2\geq 0 \iff p \geq -2 [/tex]
In this case, the absolute value doesn't do anything: the equation is
[tex] p+2 = 10 \iff p = 10-2 = 8 [/tex]
We are supposing [tex] p \geq -2 [/tex], so we can accept this solution.
Now, suppose that [tex] p+2 < 0 \iff p < -2 [/tex]. Now the sign of the expression is flipped by the absolute value, and the equation becomes
[tex] -p-2 = 10 \iff -p = 12 \iff p = -12 [/tex]
Again, the solution is coherent with the assumption, so we can accept this value as well.
