If `7/9x -1/6 x =1/3 +1/2,` what is the value of x ?

Options

A. `15/14`
B. `15/11`
C. `3/2`
D. `25/14`

Answer

To solve the equation \( \frac{7}{9}x - \frac{1}{6}x = \frac{1}{3} + \frac{1}{2} \), we will follow these steps:

### Step 1: Write down the equation
We start with the equation:
\[ \frac{7}{9}x - \frac{1}{6}x = \frac{1}{3} + \frac{1}{2} \]

### Step 2: Find a common denominator
To combine the fractions on the left side, we need to find a common denominator for \( \frac{7}{9} \) and \( \frac{1}{6} \). The least common multiple (LCM) of 9 and 6 is 18.

### Step 3: Rewrite the fractions
Now, we rewrite each fraction with the common denominator of 18:
\[ \frac{7}{9}x = \frac{7 \times 2}{9 \times 2}x = \frac{14}{18}x \]
\[ \frac{1}{6}x = \frac{1 \times 3}{6 \times 3}x = \frac{3}{18}x \]

### Step 4: Substitute back into the equation
Substituting these back into the equation gives us:
\[ \frac{14}{18}x - \frac{3}{18}x = \frac{1}{3} + \frac{1}{2} \]

### Step 5: Combine the left side
Now we can combine the left side:
\[ \frac{14 - 3}{18}x = \frac{11}{18}x \]

### Step 6: Simplify the right side
Next, we need to simplify the right side \( \frac{1}{3} + \frac{1}{2} \). The LCM of 3 and 2 is 6:
\[ \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6} \]
So,
\[ \frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \]

### Step 7: Set the equation
Now we have:
\[ \frac{11}{18}x = \frac{5}{6} \]

### Step 8: Solve for x
To isolate \( x \), we can multiply both sides by the reciprocal of \( \frac{11}{18} \):
\[ x = \frac{5}{6} \times \frac{18}{11} \]

### Step 9: Simplify the right side
Calculating this gives:
\[ x = \frac{5 \times 18}{6 \times 11} = \frac{90}{66} = \frac{15}{11} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{15}{11}} \]