Exercises 137–139 will help you prepare for the material covered in the next section. Solve: (x + 2)/(4x + 3) = 1/x

Verified step by step guidance

1. Start with the given equation: $\frac{x+2}{4x+3}=\frac{1}{x}$.
2. To eliminate the fractions, cross-multiply both sides: multiply the numerator of the left fraction by the denominator of the right fraction and set it equal to the numerator of the right fraction times the denominator of the left fraction. This gives: $(x\; +\; 2)\; imes\; x\; =\; 1\; imes\; (4x\; +\; 3)$.
3. Expand both sides: on the left, distribute $x$ over $(x\; +\; 2)$ to get $x^2\; +\; 2x$; on the right, you have $4x\; +\; 3$.
4. Set up the equation: $x^2\; +\; 2x\; =\; 4x\; +\; 3$. Then, bring all terms to one side to set the equation equal to zero: $x^2\; +\; 2x\; -\; 4x\; -\; 3\; =\; 0$, which simplifies to $x^2\; -\; 2x\; -\; 3\; =\; 0$.
5. Solve the quadratic equation $x^2\; -\; 2x\; -\; 3\; =\; 0$ by factoring, completing the square, or using the quadratic formula to find the values of $x$.

Key Concepts

Solving Rational Equations

A rational equation involves fractions with polynomials in the numerator and denominator. To solve, find a common denominator or cross-multiply to eliminate fractions, then solve the resulting polynomial equation. Always check for values that make denominators zero, as these are excluded from the solution.

Cross-Multiplication

Cross-multiplication is a method used to solve equations where two fractions are set equal. Multiply the numerator of one fraction by the denominator of the other and set the products equal. This transforms the equation into a simpler polynomial form, making it easier to solve.

Domain Restrictions in Rational Expressions

The domain of a rational expression excludes values that make any denominator zero, as division by zero is undefined. When solving rational equations, identify and exclude these values from the solution set to ensure valid answers.