Question

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

Accepted Answer

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. 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 $$3(x + 2) + 3x = x(x + 2). $$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 $$3x + 6 + 3x = x^2 + 2x. $$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 $$x^2 - 4x - 6 = 0. $$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 $$x.$$

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

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Key Concepts

Rational Expressions

Finding a Common Denominator

Cross-Multiplication