To solve the equation [tex]\(2x + 6 = |x|\)[/tex], we need to look at the scenarios separately, depending on the value of [tex]\(x\)[/tex], because the absolute value function splits into two cases.
### Case 1: [tex]\(x \geq 0\)[/tex]
In this case, [tex]\(|x| = x\)[/tex], so the equation becomes:
[tex]\[ 2x + 6 = x \][/tex]
Let's solve this:
[tex]\[ 2x - x + 6 = 0 \][/tex]
[tex]\[ x + 6 = 0 \][/tex]
[tex]\[ x = -6 \][/tex]
However, we assumed [tex]\(x \geq 0\)[/tex]. Since [tex]\(-6\)[/tex] does not satisfy this condition, there is no valid solution in this case.
### Case 2: [tex]\(x < 0\)[/tex]
In this case, [tex]\(|x| = -x\)[/tex], so the equation becomes:
[tex]\[ 2x + 6 = -x \][/tex]
Let's solve this:
[tex]\[ 2x + x + 6 = 0 \][/tex]
[tex]\[ 3x + 6 = 0 \][/tex]
[tex]\[ 3x = -6 \][/tex]
[tex]\[ x = -2 \][/tex]
We need to check if [tex]\(-2\)[/tex] satisfies the condition of [tex]\(x < 0\)[/tex]. Since [tex]\(-2 < 0\)[/tex], it is a valid solution.
### Summary
By examining both cases, the only solution that satisfies the equation is:
[tex]\[ x = -2 \][/tex]
Thus, the solution to the equation [tex]\(2x + 6 = |x|\)[/tex] is:
[tex]\[
\boxed{-2}
\][/tex]
