Question

Saw the absolute value |12 - 3x| = 9 + 4x

Answer 1

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

### Case 1: \(12 - 3x \geq 0\)

In this case, we can remove the absolute value signs:
\[
12 - 3x = 9 + 4x
\]
Now, rearranging the equation:
1. Add \(3x\) to both sides:
\[
12 = 9 + 7x
\]
2. Subtract 9 from both sides:
\[
3 = 7x
\]
3. Divide by 7:
\[
x = \frac{3}{7}
\]

Next, we check if this solution satisfies the condition \(12 - 3x \geq 0\):
\[
12 - 3\left(\frac{3}{7}\right) = 12 - \frac{9}{7} = \frac{84 - 9}{7} = \frac{75}{7} \geq 0
\]
This condition is satisfied.

### Case 2: \(12 - 3x < 0\)

In this case, we need to negate the expression inside the absolute value:
\[
-(12 - 3x) = 9 + 4x
\]
This simplifies to:
\[
-12 + 3x = 9 + 4x
\]
Rearranging the equation:
1. Subtract \(3x\) from both sides:
\[
-12 = 9 + x
\]
2. Subtract 9 from both sides:
\[
-21 = x
\]

Now, we check if this solution satisfies the condition \(12 - 3x < 0\):
\[
12 - 3(-21) = 12 + 63 = 75 \geq 0
\]
Since this does not satisfy the condition \(12 - 3x < 0\), we discard this solution.

### Conclusion

The only valid solution to the equation \(|12 - 3x| = 9 + 4x\) is:
\[
\boxed{\frac{3}{7}}
\]