To determine the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = -5|x+1| + 3 \)[/tex] equals [tex]\(-12\)[/tex], we'll follow a systematic approach:
1. Set the function equal to [tex]\(-12\)[/tex]:
[tex]\[ -5|x+1| + 3 = -12 \][/tex]
2. Isolate the absolute value term:
First, move the constant term 3 to the other side by subtracting 3 from both sides:
[tex]\[ -5|x+1| = -12 - 3 \][/tex]
[tex]\[ -5|x+1| = -15 \][/tex]
3. Divide both sides by -5 to solve for [tex]\( |x+1| \)[/tex]:
[tex]\[ |x+1| = \frac{-15}{-5} \][/tex]
[tex]\[ |x+1| = 3 \][/tex]
4. Solve the absolute value equation:
The absolute value equation [tex]\( |x+1| = 3 \)[/tex] implies two cases:
- Case 1: [tex]\( x+1 = 3 \)[/tex]
- Case 2: [tex]\( x+1 = -3 \)[/tex]
5. Solve each case separately:
- For [tex]\( x+1 = 3 \)[/tex]:
[tex]\[ x = 3 - 1 \][/tex]
[tex]\[ x = 2 \][/tex]
- For [tex]\( x+1 = -3 \)[/tex]:
[tex]\[ x = -3 - 1 \][/tex]
[tex]\[ x = -4 \][/tex]
Hence, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = -12 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = -4 \)[/tex].
Therefore, the correct answer is:
[tex]\[ x = 2, x = -4 \][/tex]
So, the correct option from the given choices is:
[tex]\[ x = 2, x = -4 \][/tex]
