A. 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). (20 points)

1. x   =   10
   -----       -----
  x + 2     15

Solution Answer: 

Checking:

Let's solve the rational equation step by step:

### Problem:
\[
\frac{x}{x + 2} = \frac{10}{15}
\]

### Step 1: Simplify the right side
The right side of the equation can be simplified:
\[
\frac{10}{15} = \frac{2}{3}
\]

Now, the equation becomes:
\[
\frac{x}{x + 2} = \frac{2}{3}
\]

### Step 2: Cross-multiply
To eliminate the fractions, we use cross-multiplication:
\[
x \cdot 3 = (x + 2) \cdot 2
\]

Expanding both sides:
\[
3x = 2(x + 2)
\]
\[
3x = 2x + 4
\]

### Step 3: Solve for \(x\)
Now, let's isolate \(x\) by subtracting \(2x\) from both sides:
\[
3x - 2x = 4
\]
\[
x = 4
\]

### Solution Answer:
The solution is \(x = 4\).

---

### Step 4: Checking the solution
We substitute \(x = 4\) back into the original equation to check if it holds:

The original equation is:
\[
\frac{x}{x + 2} = \frac{10}{15}
\]

Substitute \(x = 4\):
\[
\frac{4}{4 + 2} = \frac{10}{15}
\]
\[
\frac{4}{6} = \frac{10}{15}
\]

Simplify both sides:
\[
\frac{2}{3} = \frac{2}{3}
\]

Since both sides are equal, the solution \(x = 4\) is correct.

---

### Final Answer:
- **Solution:** \(x = 4\)
- **Checking:** The solution satisfies the original equation.

---

Would you like any further details or clarification? 

Here are 5 related questions to expand on this problem:
1. How do you find the LCD (Least Common Denominator) in rational equations with multiple terms?
2. Can cross-multiplication always be used to solve rational equations?
3. How can extraneous solutions arise when solving rational equations?
4. What happens if \(x + 2 = 0\) in the original equation?
5. How would you solve the equation if there were multiple rational expressions involved?

**Tip:** Always simplify fractions at the earliest step to avoid unnecessary complexity!

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). (20 points)

1. x / (x + 2) = 10 / 15

Learn how to solve the rational equation x / (x + 2) = 10 / 15 through cross-multiplication and simplification. This guide walks through the solution process, including checking the solution for correctness. Ideal for students in Grades 7-9 learning algebra and rational equations.

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