Let's solve the equation [tex]\(|6x + 18| = 12x\)[/tex] and determine which solutions are valid and which are extraneous. Here's a step-by-step breakdown:
1. Break Down the Absolute Value:
The equation [tex]\(|6x + 18| = 12x\)[/tex] can be split into two cases because the absolute value of [tex]\(6x + 18\)[/tex] depends on whether [tex]\(6x + 18\)[/tex] is non-negative or negative.
- Case 1: [tex]\(6x + 18 \geq 0\)[/tex]
In this case, [tex]\(|6x + 18| = 6x + 18\)[/tex].
Therefore, the equation becomes:
[tex]\[
6x + 18 = 12x
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
18 = 6x
\][/tex]
[tex]\[
x = 3
\][/tex]
- Case 2: [tex]\(6x + 18 < 0\)[/tex]
In this case, [tex]\(|6x + 18| = -(6x + 18) = -6x - 18\)[/tex].
Therefore, the equation becomes:
[tex]\[
-6x - 18 = 12x
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
-18 = 18x
\][/tex]
[tex]\[
x = -1
\][/tex]
2. Check for Validity:
Now, we need to check whether these solutions are valid by substituting them back into the original absolute value equation.
- For [tex]\(x = 3\)[/tex]:
[tex]\[
|6(3) + 18| = 12(3)
\][/tex]
[tex]\[
|18 + 18| = 36
\][/tex]
[tex]\[
|36| = 36 \quad \text{(True)}
\][/tex]
- For [tex]\(x = -1\)[/tex]:
[tex]\[
|6(-1) + 18| = 12(-1)
\][/tex]
[tex]\[
|-6 + 18| = -12
\][/tex]
[tex]\[
|12| = -12 \quad \text{(False, because absolute values cannot be negative)}
\][/tex]
Therefore, the correct solution to the equation is [tex]\(x = 3\)[/tex] and the extraneous solution is [tex]\(x = -1\)[/tex].
Thus, the answer is:
B. Solution: [tex]\(x=3\)[/tex]; extraneous solution: [tex]\(x=-1\)[/tex]
