Question

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

x=−21
𝑥
=
−
21
and x=37
𝑥
=
3
7
x equals negative 21 and x equals Start Fraction 3 over 7 End Fraction

no solution
no solution

x=4
𝑥
=
4
x equals 4

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

Answer 1

To solve the absolute value equation \( |12 - 3x| = |9 + 4x| \), we need to consider the different cases based on the definitions of absolute values.

### Case 1: \( 12 - 3x = 9 + 4x \)

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

### Case 2: \( 12 - 3x = -(9 + 4x) \)

1. Solve for \( x \):
\[
12 - 3x = -9 - 4x
\]
Combine like terms:
\[
12 + 9 = -4x + 3x
\]
\[
21 = -x
\]
\[
x = -21
\]

### Case 3: \( -(12 - 3x) = 9 + 4x \)

1. Solve for \( x \):
\[
-12 + 3x = 9 + 4x
\]
Combine like terms:
\[
-12 - 9 = 4x - 3x
\]
\[
-21 = x
\]

### Case 4: \( -(12 - 3x) = -(9 + 4x) \)

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

### Final Solutions:
The unique solutions obtained are \( x = \frac{3}{7} \) and \( x = -21 \).

Thus, the answer is:
**x = -21 and x = \(\frac{3}{7}\)**.