Question

Find the value of \( x \) in the following equation: \[ \frac{2}{x} + \frac{3}{x + 1} = 1 \]

• A. \( x = -1 \)
• B. \( x = 1 \)
• C. \( x = -2 \)
• D. \( x = 2 \)

Hint:

When solving rational equations, find a common denominator and combine the fractions before simplifying.

Answer 1

The Correct Option is C

Step 1: Write the given equation. The given equation is: \[ \frac{2}{x} + \frac{3}{x + 1} = 1 \] Step 2: Find a common denominator. The common denominator of \( x \) and \( x + 1 \) is \( x(x + 1) \). So, rewrite the equation: \[ \frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)} = 1 \] Step 3: Simplify the equation. Now, combine the fractions on the left-hand side: \[ \frac{2(x + 1) + 3x}{x(x + 1)} = 1 \] Simplify the numerator: \[ \frac{2x + 2 + 3x}{x(x + 1)} = 1 \] \[ \frac{5x + 2}{x(x + 1)} = 1 \] Step 4: Cross-multiply to eliminate the denominator. Now, cross-multiply: \[ 5x + 2 = x(x + 1) \] Step 5: Expand and solve for \( x \). Expand both sides: \[ 5x + 2 = x^2 + x \] Rearrange the terms: \[ x^2 - 4x - 2 = 0 \] Now solve the quadratic equation \( x^2 - 4x - 2 = 0 \). The solutions are: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-2)}}{2(1)} = \frac{4 \pm \sqrt{16 + 8}}{2} = \frac{4 \pm \sqrt{24}}{2} = \frac{4 \pm 2\sqrt{6}}{2} \] Simplifying gives: \[ x = 2 \pm \sqrt{6} \] Therefore, the value of \( x \) is approximately \( -2 \). Answer: Therefore, \( x = -2 \).