Solve:`1/4x+1/6x=1/2x+3/4.`

Options

A. `x=9`
B. `x=-9`
C. `x=-12`
D. `x=12`

Answer

To solve the equation \( \frac{1}{4}x + \frac{1}{6}x = \frac{1}{2}x + \frac{3}{4} \), we will follow these steps:

### Step 1: Combine like terms on the left side
We start with the equation:
\[ \frac{1}{4}x + \frac{1}{6}x = \frac{1}{2}x + \frac{3}{4} \]
To combine the \( x \) terms on the left side, we need a common denominator for \( \frac{1}{4} \) and \( \frac{1}{6} \). The least common multiple (LCM) of 4 and 6 is 12.

### Step 2: Rewrite each term with the common denominator
Convert each fraction:
\[ \frac{1}{4}x = \frac{3}{12}x, \quad \frac{1}{6}x = \frac{2}{12}x, \quad \frac{1}{2}x = \frac{6}{12}x \]
Now, substituting these values back into the equation gives:
\[ \frac{3}{12}x + \frac{2}{12}x = \frac{6}{12}x + \frac{3}{4} \]

### Step 3: Combine the left side
Now combine the \( x \) terms on the left:
\[ \frac{3}{12}x + \frac{2}{12}x = \frac{5}{12}x \]
So the equation now looks like:
\[ \frac{5}{12}x = \frac{6}{12}x + \frac{3}{4} \]

### Step 4: Move all \( x \) terms to one side
Subtract \( \frac{6}{12}x \) from both sides:
\[ \frac{5}{12}x - \frac{6}{12}x = \frac{3}{4} \]
This simplifies to:
\[ -\frac{1}{12}x = \frac{3}{4} \]

### Step 5: Solve for \( x \)
To isolate \( x \), multiply both sides by -12:
\[ x = -12 \times \frac{3}{4} \]
Calculating the right side:
\[ x = -9 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{-9} \]

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