Certainly! Let's find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = -5|x+1| + 3 \)[/tex] equals [tex]\(-12\)[/tex].
Starting with the given equation:
[tex]\[ f(x) = -5|x+1| + 3 \][/tex]
We need to solve for [tex]\( x \)[/tex] such that:
[tex]\[ -5|x+1| + 3 = -12 \][/tex]
First, isolate the absolute value term:
[tex]\[ -5|x+1| + 3 = -12 \][/tex]
[tex]\[ -5|x+1| = -12 - 3 \][/tex]
[tex]\[ -5|x+1| = -15 \][/tex]
Divide both sides by [tex]\(-5\)[/tex]:
[tex]\[ |x+1| = 3 \][/tex]
The absolute value equation [tex]\( |x+1| = 3 \)[/tex] has two possible solutions:
[tex]\[ x+1 = 3 \][/tex]
and
[tex]\[ x+1 = -3 \][/tex]
Solving each equation individually:
1. For [tex]\( x + 1 = 3 \)[/tex]:
[tex]\[ x = 3 - 1 \][/tex]
[tex]\[ x = 2 \][/tex]
2. For [tex]\( x + 1 = -3 \)[/tex]:
[tex]\[ x = -3 - 1 \][/tex]
[tex]\[ x = -4 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = -12 \)[/tex] are:
[tex]\[ x = 2 \][/tex]
[tex]\[ x = -4 \][/tex]
Therefore, the correct answer is:
[tex]\[ x = 2, x = -4 \][/tex]
So, the values of [tex]\( x \)[/tex] are:
[tex]\[ \boxed{x = 2, x = -4} \][/tex]
