Solve the rational equation. $$ \frac{1}{x-2}-\frac{2}{x+3}=0 $$

Question

UpStudy AI · Accepted Answer

To solve the rational equation:

$$
\frac{1}{x - 2} - \frac{2}{x + 3} = 0
$$

**Step 1: Identify the Domain**

First, determine the values of $$ x $$ that make the denominators zero, as these values are excluded from the domain.

- $$ x - 2 \neq 0 $$ ⟹ $$ x \neq 2 $$
- $$ x + 3 \neq 0 $$ ⟹ $$ x \neq -3 $$

So, $$ x $$ must satisfy $$ x \neq 2 $$ and $$ x \neq -3 $$.

**Step 2: Solve the Equation**

Set the equation equal to zero and solve for $$ x $$:

$$
\frac{1}{x - 2} = \frac{2}{x + 3}
$$

Cross-multiply to eliminate the denominators:

$$
1 \cdot (x + 3) = 2 \cdot (x - 2)
$$

Simplify both sides:

$$
x + 3 = 2x - 4
$$

Subtract $$ x $$ from both sides:

$$
3 = x - 4
$$

Add $$ 4 $$ to both sides:

$$
x = 7
$$

**Step 3: Verify the Solution**

Check if $$ x = 7 $$ is within the domain:

- $$ 7 \neq 2 $$ and $$ 7 \neq -3 $$

Since $$ x = 7 $$ does not violate the domain restrictions, it is a valid solution.

**Final Answer:**

$$\boxed{7}$$
$$ x = 7 $$

Solve the rational equation. \( \frac{1}{x-2}-\frac{2}{x+3}=0 \)