Question

Solve the absolute value equation |12−3x|=|9+4x|
.(1 point)
Responses

no solution
no solution

x=37
and x=3
x equals Start Fraction 3 over 7 End Fraction and x equals 3

x=4
x equals 4

x=−21
and x=37

Answer 1

To solve the absolute value equation \( |12 - 3x| = |9 + 4x| \), we need to consider the cases for the expressions inside the absolute values.

### Case 1: \( 12 - 3x = 9 + 4x \)
1. Solve for \( x \):
\[
12 - 3x = 9 + 4x
\]
\[
12 - 9 = 4x + 3x
\]
\[
3 = 7x
\]
\[
x = \frac{3}{7}
\]

### Case 2: \( 12 - 3x = -(9 + 4x) \)
1. Solve for \( x \):
\[
12 - 3x = -9 - 4x
\]
\[
12 + 9 = -4x + 3x
\]
\[
21 = -x
\]
\[
x = -21
\]

### Case 3: \( -(12 - 3x) = 9 + 4x \)
1. Solve for \( x \):
\[
-12 + 3x = 9 + 4x
\]
\[
3x - 4x = 9 + 12
\]
\[
-x = 21
\]
\[
x = -21
\] (already found this in Case 2)

### Case 4: \( -(12 - 3x) = -(9 + 4x) \)
1. Solve for \( x \):
\[
-12 + 3x = -9 - 4x
\]
\[
-12 + 9 = -4x - 3x
\]
\[
-3 = -7x
\]
\[
x = \frac{3}{7}
\] (already found this in Case 1)

### Summary of Solutions
The solutions we have found are:
- \( x = \frac{3}{7} \)
- \( x = -21 \)

The complete solution set is:
- \( x = \frac{3}{7} \) and \( x = -21 \)

Thus, the answer is:
**x equals Start Fraction 3 over 7 End Fraction and x equals −21.**