1. Determine the value(s) of $$ x $$ in the equation below: $$ \frac{8}{x}=\frac{x}{2} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{8}{x} = \frac{x}{2}
$$

**Step 1: Eliminate the Fractions**

To eliminate the denominators, multiply both sides of the equation by $$ 2x $$ (assuming $$ x \neq 0 $$):

$$
2x \cdot \frac{8}{x} = 2x \cdot \frac{x}{2}
$$

**Step 2: Simplify Both Sides**

$$
2x \cdot \frac{8}{x} = 16 \quad \text{(since $$ x $$ cancels out)}
$$
$$
2x \cdot \frac{x}{2} = x^2 \quad \text{(since 2 and 2 cancel out)}
$$

Now the equation becomes:

$$
16 = x^2
$$

**Step 3: Solve for $$ x $$**

To find $$ x $$, take the square root of both sides:

$$
x = \sqrt{16} \quad \text{or} \quad x = -\sqrt{16}
$$
$$
x = 4 \quad \text{or} \quad x = -4
$$

**Step 4: Verify the Solutions**

It's good practice to substitute the solutions back into the original equation to ensure they are valid.

1. **For $$ x = 4 $$:**

$$
\frac{8}{4} = \frac{4}{2} \Rightarrow 2 = 2 \quad \text{(True)}
$$

2. **For $$ x = -4 $$:**

$$
\frac{8}{-4} = \frac{-4}{2} \Rightarrow -2 = -2 \quad \text{(True)}
$$

**Conclusion**

The solutions to the equation $$ \frac{8}{x} = \frac{x}{2} $$ are:

$$
x = 4 \quad \text{and} \quad x = -4
$$
The solutions are $$ x = 4 $$ and $$ x = -4 $$.

Determine the value(s) of in the equation below: