Give step-by-step solution with explanation and final answer:
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Question

VIPSolutions · Accepted Answer

## Problem Statement

Solve step-by-step:  
$$
\frac{3x - 5}{2} + \frac{x + 7}{3} = 4
$$

## Step 1: Find a Common Denominator

The denominators are 2 and 3; the least common denominator (LCD) is 6. Multiply both sides by 6 to clear fractions:

$$
6 \left( \frac{3x - 5}{2} + \frac{x + 7}{3} \right) = 6 \cdot 4
$$

## Step 2: Distribute

Multiply out each term:

$$
6 \cdot \frac{3x - 5}{2} = 3 \cdot (3x - 5) = 9x - 15
$$
$$
6 \cdot \frac{x + 7}{3} = 2 \cdot (x + 7) = 2x + 14
$$
So the equation becomes:

$$
9x - 15 + 2x + 14 = 24
$$

## Step 3: Combine Like Terms

Combine $9x + 2x = 11x$ and $-15 + 14 = -1$:

$$
11x - 1 = 24
$$

## Step 4: Isolate $x$

Add 1 to both sides:

$$
11x - 1 + 1 = 24 + 1 \implies 11x = 25
$$

## Step 5: Solve for $x$

Divide by 11:

$$
x = \frac{25}{11}
$$

## Final Answer

$$
x = \frac{25}{11}
$$

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