Solve the following rational equations. Note that your solutions will be graded based on the correctness of your LCD (1 point), correctness of the steps prior to the final answer (1 point), correctness of the solutions/roots (1 point) and correctness of the checking part (1 point). (16 points)

1. x - 4         3
    ---------  =  -----
    3x + 2      7

### Problem: 

Solve the rational equation:

\[
\frac{x - 4}{3x + 2} = \frac{3}{7}
\]

### Step 1: Identify the LCD
Since the denominators are \(3x + 2\) and \(7\), the least common denominator (LCD) is \(7(3x + 2)\).

### Step 2: Eliminate the denominators
Multiply both sides of the equation by the LCD, \(7(3x + 2)\), to eliminate the fractions:

\[
7(3x + 2) \cdot \frac{x - 4}{3x + 2} = 7(3x + 2) \cdot \frac{3}{7}
\]

This simplifies to:

\[
7(x - 4) = 3(3x + 2)
\]

### Step 3: Simplify the equation
Now, distribute the constants on both sides:

\[
7x - 28 = 9x + 6
\]

### Step 4: Solve for \(x\)
Move all terms involving \(x\) to one side and constant terms to the other side:

\[
7x - 9x = 6 + 28
\]

Simplify:

\[
-2x = 34
\]

Now, solve for \(x\):

\[
x = \frac{34}{-2} = -17
\]

### Step 5: Check the solution
Substitute \(x = -17\) back into the original equation:

\[
\frac{-17 - 4}{3(-17) + 2} = \frac{-21}{-51 + 2} = \frac{-21}{-49} = \frac{3}{7}
\]

Both sides of the equation are equal, so the solution \(x = -17\) is correct.

### Final Answer:
The solution to the equation is \(x = -17\).

---

Do you want any further details or have any questions?

Here are 5 related questions:
1. How do you determine the least common denominator (LCD) in a rational equation?
2. Why do we multiply both sides of a rational equation by the LCD?
3. What happens if the denominator of a rational equation has no real solutions?
4. How do you check the solution of a rational equation?
5. What are extraneous solutions, and how do they affect rational equations?

**Tip**: Always verify your solution by substituting it back into the original equation to avoid extraneous solutions.

Solve the following rational equation. Note that your solutions will be graded based on the correctness of your LCD (1 point), correctness of the steps prior to the final answer (1 point), correctness of the solutions/roots (1 point) and correctness of the checking part (1 point). (16 points)

1. x - 4         3
    ---------  =  -----
    3x + 2      7

This page provides a step-by-step solution to the rational equation (x - 4) / (3x + 2) = 3 / 7. It explains how to find the least common denominator (LCD), eliminate the denominators, and solve for x. The solution includes checking the answer by substituting it back into the original equation. Suitable for students in Grades 9-11.

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