To solve the absolute value equation [tex]\( |x + 6| = 4 \)[/tex], we can break it down into two separate equations because the absolute value expression will be equal to 4 in two scenarios: when the expression inside the absolute value is 4 or when it is -4.
Here are the two cases we need to consider:
1. [tex]\( x + 6 = 4 \)[/tex]
2. [tex]\( x + 6 = -4 \)[/tex]
Let's solve each case separately:
### Case 1: [tex]\( x + 6 = 4 \)[/tex]
1. Subtract 6 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x + 6 - 6 = 4 - 6
\][/tex]
2. Simplify the equation:
[tex]\[
x = -2
\][/tex]
### Case 2: [tex]\( x + 6 = -4 \)[/tex]
1. Subtract 6 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x + 6 - 6 = -4 - 6
\][/tex]
2. Simplify the equation:
[tex]\[
x = -10
\][/tex]
So, the solutions to the equation [tex]\( |x+6| = 4 \)[/tex] are:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = -10 \][/tex]
Among these solutions, the most negative one is:
[tex]\[ x = -10 \][/tex]
Therefore, the most negative solution for [tex]\( x \)[/tex] is:
[tex]\[ x = -10 \][/tex]
