Question

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

Answer 1

Step 1: Equation Identification:
The given equation is: \[ \frac{4}{x} - \frac{5}{2x + 3} = 3 \] The objective is to determine the value(s) of \( x \).

Step 2: Least Common Denominator Determination:
The denominators are \( x \) and \( 2x + 3 \). The LCD is \( x(2x + 3) \).

Step 3: Equation Transformation by LCD Multiplication:
Multiply all terms by \( x(2x + 3) \) to clear the fractions: \[ x(2x + 3) \left( \frac{4}{x} \right) - x(2x + 3) \left( \frac{5}{2x + 3} \right) = x(2x + 3) \times 3 \] Simplified form: \[ 4(2x + 3) - 5x = 3x(2x + 3) \]

Step 4: Equation Simplification:
Expand the terms: \[ 4(2x + 3) = 8x + 12 \] \[ 3x(2x + 3) = 6x^2 + 9x \] The equation transforms to: \[ 8x + 12 - 5x = 6x^2 + 9x \] Simplify the left side: \[ 3x + 12 = 6x^2 + 9x \]

Step 5: Equation Rearrangement:
Consolidate all terms to one side: \[ 0 = 6x^2 + 9x - 3x - 12 \] Resulting equation: \[ 0 = 6x^2 + 6x - 12 \] Divide by 6 for further simplification: \[ 0 = x^2 + x - 2 \]

Step 6: Quadratic Factoring:
Factor the quadratic expression \( x^2 + x - 2 \): \[ x^2 + x - 2 = (x - 1)(x + 2) \] The factored equation is: \[ (x - 1)(x + 2) = 0 \]

Step 7: Solution Derivation:
Equate each factor to zero: \[ x - 1 = 0 \quad \text{or} \quad x + 2 = 0 \] The solutions are: \[ x = 1 \quad \text{or} \quad x = -2 \]

Step 8: Extraneous Solution Verification:
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 confirms \( x = 1 \) is a valid solution.
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 confirms \( x = -2 \) is also a valid solution.

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