The positive value of x in the equation $\frac{24}{18-x} - \frac{24}{18+x} = 1$ is:

Question

Accepted Answer

To solve the equation $\frac{24}{18-x} - \frac{24}{18+x} = 1$ and verify that the positive value of $x$ is indeed 6, we can follow these steps:

1. **Combine the Fractions:**
   The equation is $\frac{24}{18-x} - \frac{24}{18+x} = 1$. To combine the fractions, we need a common denominator, which is $(18-x)(18+x)$.

$$
   \frac{24(18+x) - 24(18-x)}{(18-x)(18+x)} = 1
   $$

2. **Simplify the Numerator:**
   Simplify the expression in the numerator:

$$
   24(18+x) - 24(18-x) = 24 \times 18 + 24x - 24 \times 18 + 24x = 48x
   $$

So, the equation becomes:

$$
   \frac{48x}{(18-x)(18+x)} = 1
   $$

3. **Cross-Multiply:**
   Cross-multiply to eliminate the fraction:

$$
   48x = (18-x)(18+x)
   $$

4. **Expand the Right Side:**
   Expand the right side using the difference of squares:

$$
   48x = 18^2 - x^2 = 324 - x^2
   $$

5. **Rearrange the Equation:**
   Rearrange the equation to form a quadratic equation:

$$
   x^2 + 48x - 324 = 0
   $$

6. **Solve the Quadratic Equation:**
   Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 48$, and $c = -324$:

$$
   x = \frac{-48 \pm \sqrt{48^2 - 4 \times 1 \times (-324)}}{2 \times 1}
   $$

$$
   x = \frac{-48 \pm \sqrt{2304 + 1296}}{2}
   $$

$$
   x = \frac{-48 \pm \sqrt{3600}}{2}
   $$

$$
   x = \frac{-48 \pm 60}{2}
   $$

This gives two solutions:

$$
   x = \frac{12}{2} = 6 \quad \text{and} \quad x = \frac{-108}{2} = -54
   $$

Since we are looking for the positive value of $x$, the solution is $x = 6$.

7. **Verification:**
   Substitute $x = 6$ back into the original equation to verify:

$$
   \frac{24}{18-6} - \frac{24}{18+6} = \frac{24}{12} - \frac{24}{24} = 2 - 1 = 1
   $$

The equation holds true, confirming that $x = 6$ is correct.

**Explanation for Other Options:**

- **Option 1 (5):** Substituting $x = 5$ into the equation does not satisfy the equation, as the left side does not equal 1.
- **Option 3 (9):** Substituting $x = 9$ results in a division by zero in the denominator, making it undefined.
- **Option 4 (11):** Substituting $x = 11$ does not satisfy the equation, as the left side does not equal 1.

Thus, the correct answer is indeed Option 2, $x = 6$.