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

Answer 1

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 2

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 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: \[ x = \frac{84}{17} \quad \text{with the restriction } x \neq 4 \]