Question

2. \( 3x - 12 \neq 0 \) (to prevent division by zero, which implies \( x \neq 4 \))

Answer 1

1. Starting with the original equation: \[ \frac{4x}{3x - 12} + 3 = 10 \] 2. Subtract 3 from both sides: \[ \frac{4x}{3x - 12} = 7 \] Now we can express this as two separate equations: 1. \( \frac{4x}{3x - 12} = 7 \) 2. \( 3x - 12 \neq 0 \) (to avoid division by zero, implying \( x \neq 4 \)) Thus, the rewritten form is: \[ \frac{4x}{3x - 12} = 7 \quad \text{and} \quad x \neq 4 \]

Answer 2

The inequality \( 3x - 12 \neq 0 \) can be solved to determine the value of \( x \) that would cause a division by zero in an expression involving this term.

To find the value, we set the expression equal to zero:

\[
3x - 12 = 0
\]

Adding 12 to both sides gives:

\[
3x = 12
\]

Now, divide both sides by 3:

\[
x = 4
\]

To prevent division by zero, we conclude that \( x \) must not equal 4:

\[
x \neq 4
\]

This means in any mathematical expression where \( 3x - 12 \) is in the denominator, \( x \) cannot be equal to 4 to avoid an undefined situation (division by zero).

If you have a specific context in mind where this inequality appears, please provide further details!

Answer 3

You have done a good job in expressing the given equation and the conditions derived from it. Let's continue from where you left off:

### Given:
\[
\frac{4x}{3x - 12} + 3 = 10
\]

### Step 1: Isolate the fraction
Subtract 3 from both sides:
\[
\frac{4x}{3x - 12} = 7
\]

### Step 2: Set up the conditions
From the expression, we note:
1. **Equation**:
\[
\frac{4x}{3x - 12} = 7
\]
2. **Condition**:
\[
3x - 12 \neq 0 \quad \Rightarrow \quad x \neq 4
\]

### Step 3: Solve the equation
Now let’s solve the equation \( \frac{4x}{3x - 12} = 7 \).

Multiply both sides by \( 3x - 12 \) (noting that \( x \neq 4 \) ensures \( 3x - 12 \) is not zero):

\[
4x = 7(3x - 12)
\]

Expanding the right side:

\[
4x = 21x - 84
\]

### Step 4: Rearranging the equation
Bringing all terms involving \( x \) to one side gives:

\[
4x - 21x = -84
\]

Simplifying:

\[
-17x = -84
\]

### Step 5: Solve for \( x \)
Now divide by -17:

\[
x = \frac{84}{17}
\]

### Conclusion:
The solution to the original equation is:
\[
x = \frac{84}{17}
\]

And we also have the condition:
\[
x \neq 4
\]

Since \( \frac{84}{17} \) is not equal to 4, our solution is valid.

### Final Note:
Thus, the complete solution is:
\[
x = \frac{84}{17} \quad \text{with the restriction } x \neq 4
\]