Problem 136 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. Solve: \(\frac{x+2}{4 x+3}=\frac{1}{x}\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 3\) and \(x = -1\).

Step 1: Identifying the least common multiple (LCM)

First, identify the least common multiple of the denominators, which in this case is \(4x^2+3x\).

Step 2: Multiplying both sides by the LCM

Multiply both sides of the equation by the LCM to get \(x^2 + 2x = 4x + 3\).

Step 3: Simplify the Equation

Rearrange the equation by subtracting \(2x\) and \(3\) from both sides to get \(x^2 - 2x - 3 = 0\).

Step 4: Factoring the Equation

Factor this quadratic equation to get \((x - 3)(x + 1) = 0\).

Step 5: Solve for \(x\)

Set each factor equal to zero and solve for \(x\). This results in two solutions: \(x = 3\) and \(x = -1\).