Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 2) = 1/3

Verified step by step guidance

1. Rewrite the equation to eliminate the fractions by finding the least common denominator (LCD). The LCD for the denominators \(x\), \(x + 2\), and \(3\) is \(3x(x + 2)\). Multiply through by the LCD to clear the fractions.
2. After multiplying through by \(3x(x + 2)\), simplify each term. The first term becomes \(3(x + 2)\), the second term becomes \(3x\), and the right-hand side becomes \(x(x + 2)\). This results in the equation \(3(x + 2) + 3x = x(x + 2)\).
3. Expand all terms in the equation. Distribute \(3\) in \(3(x + 2)\) to get \(3x + 6\), and distribute \(x\) in \(x(x + 2)\) to get \(x^2 + 2x\). The equation now becomes \(3x + 6 + 3x = x^2 + 2x\).
4. Combine like terms on the left-hand side. \(3x + 3x\) simplifies to \(6x\), so the equation becomes \(6x + 6 = x^2 + 2x\). Rearrange the equation to set it equal to zero: \(0 = x^2 + 2x - 6x - 6\), which simplifies to \(x^2 - 4x - 6 = 0\).
5. Solve the quadratic equation \(x^2 - 4x - 6 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = -6\). Substitute these values into the formula and simplify to find the solutions for \(x\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions is crucial for solving equations involving them. In this case, the equation contains rational expressions that need to be combined or simplified to find the value of x.

Finding a Common Denominator

To solve equations involving multiple fractions, it is often necessary to find a common denominator. This allows you to combine the fractions into a single equation, making it easier to isolate the variable. In the given equation, the common denominator will help eliminate the fractions and simplify the solving process.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving fractions. It involves multiplying the numerator of one fraction by the denominator of the other fraction, allowing for the elimination of the fractions. This method is particularly useful in this equation to simplify and solve for x efficiently.