Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 3/(x + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

Accepted Answer

Identify the denominators in the equation: $$x + 2$$, $$x - 2$$, and $$(x + 2)(x - 2). To $$find the restrictions on the variable, set each denominator equal to zero and solve for $$x. $$This gives the values that make the denominators undefined. Set $$x + 2 = 0$$ which implies $$x = -2$$, and set $$x - 2 = 0$$ which implies $$x = 2. $$These values are restrictions and cannot be solutions because they make the denominators zero. Rewrite the original equation: $$\frac{3}{x + 2} + \frac{2}{x - 2} = \frac{8}{(x + 2)(x - 2)}. To $$solve, multiply every term by the least common denominator (LCD), which is $$(x + 2)(x - 2)$$, to eliminate the denominators. After multiplying through by the LCD, simplify each term: $$3(x - 2) + 2(x + 2) = 8. $$This step removes the fractions and results in a polynomial equation. Expand and combine like terms on the left side, then solve the resulting linear equation for $$x. $$Remember to check that your solutions do not include the restricted values $$x = -2 or$$ $$x = 2.$$

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주요 개념

Restrictions on the Variable

Solving Rational Equations

Factoring and Simplifying Expressions