Question

Solve for $x$: \[ \frac{4}{x} - \frac{5}{2x + 3} = 3. \]

Answer 1

Step 1: Understand the equation:
We are given the equation:
\[ \frac{4}{x} - \frac{5}{2x + 3} = 3 \] We need to solve for \( x \).

Step 2: Find the least common denominator (LCD):
The denominators are \( x \) and \( 2x + 3 \). The least common denominator (LCD) is \( x(2x + 3) \).

Step 3: Multiply the entire equation by the LCD:
Multiply every term in the equation by \( x(2x + 3) \) to eliminate the fractions:
\[ x(2x + 3) \left( \frac{4}{x} \right) - x(2x + 3) \left( \frac{5}{2x + 3} \right) = x(2x + 3) \times 3 \] Simplifying each term:
\[ 4(2x + 3) - 5x = 3x(2x + 3) \]

Step 4: Simplify the equation:
Expand each term:
\[ 4(2x + 3) = 8x + 12 \] \[ 3x(2x + 3) = 6x^2 + 9x \] So, the equation becomes:
\[ 8x + 12 - 5x = 6x^2 + 9x \] Simplify the left side:
\[ 3x + 12 = 6x^2 + 9x \]

Step 5: Rearrange the equation:
Move all terms to one side of the equation:
\[ 0 = 6x^2 + 9x - 3x - 12 \] Simplify the equation:
\[ 0 = 6x^2 + 6x - 12 \] Divide through by 6 to simplify further:
\[ 0 = x^2 + x - 2 \]

Step 6: Factor the quadratic equation:
Factor \( x^2 + x - 2 \):
\[ x^2 + x - 2 = (x - 1)(x + 2) \] So, the equation becomes:
\[ (x - 1)(x + 2) = 0 \]

Step 7: Solve for \( x \):
Set each factor equal to zero:
\[ x - 1 = 0 \quad \text{or} \quad x + 2 = 0 \] Solving these gives:
\[ x = 1 \quad \text{or} \quad x = -2 \]

Step 8: Check for extraneous solutions:
Substitute \( x = 1 \) into the original equation:
\[ \frac{4}{1} - \frac{5}{2(1) + 3} = 3 \quad \Rightarrow \quad 4 - \frac{5}{5} = 3 \quad \Rightarrow \quad 4 - 1 = 3 \quad \Rightarrow \quad 3 = 3 \] This is true, so \( x = 1 \) is a valid solution.
Now, substitute \( x = -2 \) into the original equation:
\[ \frac{4}{-2} - \frac{5}{2(-2) + 3} = 3 \quad \Rightarrow \quad -2 - \frac{5}{-1} = 3 \quad \Rightarrow \quad -2 + 5 = 3 \quad \Rightarrow \quad 3 = 3 \] This is also true, so \( x = -2 \) is a valid solution.

Conclusion:
The solutions are \( x = 1 \) and \( x = -2 \).