Question

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

Answer 1

To solve the equation \( |12 - 3x| = |9 + 4x| \), we need to consider the different cases that arise from the properties of absolute value.

### Step 1: Identify Cases
We can break it down into different cases based on the expressions inside the absolute values.

**Case 1:** \( 12 - 3x \geq 0 \) and \( 9 + 4x \geq 0 \)
In this case, both absolute values can be removed:
\[
12 - 3x = 9 + 4x
\]
Solving for \( x \):
\[
12 - 9 = 4x + 3x
\]
\[
3 = 7x
\]
\[
x = \frac{3}{7}
\]

Next, we need to check if this solution satisfies the conditions of the case:
- For \( 12 - 3x \geq 0 \): \( 12 - 3 \cdot \frac{3}{7} = 12 - \frac{9}{7} = \frac{75}{7} \geq 0 \) (True)
- For \( 9 + 4x \geq 0 \): \( 9 + 4 \cdot \frac{3}{7} = 9 + \frac{12}{7} = \frac{75}{7} \geq 0 \) (True)

This case is valid.

---

**Case 2:** \( 12 - 3x \geq 0 \) and \( 9 + 4x < 0 \)
Here we have:
\[
12 - 3x = -(9 + 4x)
\]
Solving:
\[
12 - 3x = -9 - 4x
\]
\[
12 + 9 = -4x + 3x
\]
\[
21 = -x
\]
\[
x = -21
\]

Checking conditions:
- For \( 12 - 3(-21) = 12 + 63 = 75 \geq 0 \) (True)
- For \( 9 + 4(-21) = 9 - 84 = -75 < 0 \) (True)

This case is valid.

---

**Case 3:** \( 12 - 3x < 0 \) and \( 9 + 4x \geq 0 \)
Here:
\[
-(12 - 3x) = 9 + 4x
\]
\[
-12 + 3x = 9 + 4x
\]
\[
-12 - 9 = 4x - 3x
\]
\[
-21 = x
\]

Check conditions:
- For \( 12 - 3(-21) = 75 \geq 0 \) (False)
- This case is not valid.

---

**Case 4:** \( 12 - 3x < 0 \) and \( 9 + 4x < 0 \)
Here:
\[
-(12 - 3x) = -(9 + 4x)
\]
\[
-12 + 3x = -9 - 4x
\]
\[
-12 + 9 = -4x - 3x
\]
\[
-3 = -7x
\]
\[
x = \frac{3}{7}
\]

Check conditions:
- For \( 12 - 3(\frac{3}{7}) = 75 \geq 0 \) (False)
- This case is not valid.

---

### Step 2: Collect All Valid Solutions
The valid solutions are:
- From Case 1: \( x = \frac{3}{7} \)
- From Case 2: \( x = -21 \)

### Conclusion
The solutions to the equation \( |12 - 3x| = |9 + 4x| \) are:
\[
x = \frac{3}{7} \quad \text{and} \quad x = -21
\]