If `(7)/(12)x-(1)/(3)x=(1)/(2)+(3)/(8)`, what is the value of x?

Answer

To solve the equation \(\frac{7}{12}x - \frac{1}{3}x = \frac{1}{2} + \frac{3}{8}\), we will follow these steps:

### Step 1: Find a common denominator for the left side.
The denominators are 12 and 3. The least common multiple (LCM) of 12 and 3 is 12.

### Step 2: Rewrite \(\frac{1}{3}x\) with a denominator of 12.
\[ \frac{1}{3}x = \frac{4}{12}x \]

### Step 3: Substitute back into the equation.
Now, we can rewrite the left side of the equation:
\[ \frac{7}{12}x - \frac{4}{12}x = \frac{1}{2} + \frac{3}{8} \]

### Step 4: Combine the left side.
\[ \frac{7}{12}x - \frac{4}{12}x = \frac{3}{12}x \]
So, the equation now looks like:
\[ \frac{3}{12}x = \frac{1}{2} + \frac{3}{8} \]

### Step 5: Find a common denominator for the right side.
The denominators are 2 and 8. The LCM of 2 and 8 is 8.

### Step 6: Rewrite \(\frac{1}{2}\) with a denominator of 8.
\[ \frac{1}{2} = \frac{4}{8} \]

### Step 7: Substitute back into the equation.
Now, we can rewrite the right side:
\[ \frac{3}{12}x = \frac{4}{8} + \frac{3}{8} \]

### Step 8: Combine the right side.
\[ \frac{4}{8} + \frac{3}{8} = \frac{7}{8} \]
So, the equation now looks like:
\[ \frac{3}{12}x = \frac{7}{8} \]

### Step 9: Solve for \(x\).
To isolate \(x\), multiply both sides by the reciprocal of \(\frac{3}{12}\), which is \(\frac{12}{3}\):
\[ x = \frac{7}{8} \cdot \frac{12}{3} \]

### Step 10: Simplify the right side.
\[ x = \frac{7 \cdot 12}{8 \cdot 3} = \frac{84}{24} \]
Now simplify \(\frac{84}{24}\):
\[ x = \frac{7}{2} \]

### Final Answer:
The value of \(x\) is \(\frac{7}{2}\).

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