(3) $$ \frac{2}{x}=\frac{1}{x+2} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

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

**Step 1: Identify the Domain**

Before solving, determine the values of $$ x $$ that make the denominators zero:

1. $$ x \neq 0 $$ (since division by zero is undefined)
2. $$ x + 2 \neq 0 $$ $$\Rightarrow$$ $$ x \neq -2 $$

So, $$ x \neq 0 $$ and $$ x \neq -2 $$.

**Step 2: Cross-Multiply to Eliminate the Fractions**

$$
\frac{2}{x} = \frac{1}{x + 2} \implies 2(x + 2) = 1 \cdot x
$$

**Step 3: Expand and Simplify**

$$
2x + 4 = x
$$

$$
2x - x + 4 = 0
$$

$$
x + 4 = 0
$$

$$
x = -4
$$

**Step 4: Verify the Solution**

Ensure that $$ x = -4 $$ doesn't make any denominator zero:

- $$ x = -4 \neq 0 $$
- $$ x + 2 = -4 + 2 = -2 \neq 0 $$

Both conditions are satisfied.

**Step 5: Check the Original Equation**

$$
\frac{2}{-4} = -\frac{1}{2}
$$
$$
\frac{1}{-4 + 2} = \frac{1}{-2} = -\frac{1}{2}
$$

Both sides are equal, confirming that $$ x = -4 $$ is indeed the solution.

**Final Answer:**

$$
x = -4
$$
$$ x = -4 $$

(3) \( \frac{2}{x}=\frac{1}{x+2} \)