Solve:`2/3(x-5)-1/4(x-2)=9/2`

Options

A. `88/5`
B. `88/10`
C. `77/5`
D. none of these

Answer

To solve the equation \( \frac{2}{3}(x - 5) - \frac{1}{4}(x - 2) = \frac{9}{2} \), we will follow these steps:

### Step 1: Distribute the fractions
Distribute \( \frac{2}{3} \) and \( -\frac{1}{4} \) to the terms inside the parentheses.

\[ \frac{2}{3} \cdot x - \frac{2}{3} \cdot 5 - \frac{1}{4} \cdot x + \frac{1}{4} \cdot 2 = \frac{9}{2} \]

This simplifies to:

\[ \frac{2}{3}x - \frac{10}{3} - \frac{1}{4}x + \frac{1}{2} = \frac{9}{2} \]

### Step 2: Combine like terms
Next, we need to combine the \( x \) terms and the constant terms.

First, we will convert \( \frac{1}{2} \) to a fraction with a denominator of 3 and 4 for easy addition:

\[ \frac{1}{2} = \frac{3}{6} = \frac{2}{4} \]

Now, we have:

\[ \frac{2}{3}x - \frac{1}{4}x - \frac{10}{3} + \frac{2}{4} = \frac{9}{2} \]

### Step 3: Find a common denominator for \( x \) terms
The common denominator for \( \frac{2}{3} \) and \( -\frac{1}{4} \) is 12.

Convert each term:

\[ \frac{2}{3}x = \frac{8}{12}x \quad \text{and} \quad -\frac{1}{4}x = -\frac{3}{12}x \]

Now combine them:

\[ \frac{8}{12}x - \frac{3}{12}x = \frac{5}{12}x \]

### Step 4: Combine constant terms
Now, we need to combine the constant terms:

Convert \( -\frac{10}{3} \) to a fraction with a denominator of 12:

\[ -\frac{10}{3} = -\frac{40}{12} \]

And convert \( \frac{2}{4} \):

\[ \frac{2}{4} = \frac{6}{12} \]

Now combine:

\[ -\frac{40}{12} + \frac{6}{12} = -\frac{34}{12} \]

### Step 5: Set up the equation
Now we can rewrite the equation:

\[ \frac{5}{12}x - \frac{34}{12} = \frac{9}{2} \]

### Step 6: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 12:

\[ 5x - 34 = 54 \]

### Step 7: Solve for \( x \)
Now, add 34 to both sides:

\[ 5x = 54 + 34 \]
\[ 5x = 88 \]

Now divide by 5:

\[ x = \frac{88}{5} \]

### Final Answer
Thus, the solution to the equation is:

\[ x = 17.6 \quad \text{or} \quad x = \frac{88}{5} \]

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