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/(2x - 2) + 1/2 = 2/(x - 1)

Accepted Answer

Identify the denominators in the equation: $$2x - 2$$, $$2$$, and $$x - 1. $$Determine the restrictions on the variable by setting each denominator equal to zero and solving for $$x$$: $$2x - 2 = 0$$ gives $$x = 1$$, and $$x - 1 = 0$$ gives $$x = 1. $$Therefore, $$x = 1 is a $$restriction because it makes the denominators undefined. Rewrite the equation: $$\frac{3}{2x - 2} + \frac{1}{2} = \frac{2}{x - 1}. $$Notice that $$2x - 2$$ can be factored as $$2(x - 1)$$, so rewrite the first term as $$\frac{3}{2(x - 1)}. $$The equation becomes $$\frac{3}{2(x - 1)} + \frac{1}{2} = \frac{2}{x - 1}. $$Eliminate the fractions by multiplying through by the least common denominator (LCD), which is $$2(x - 1). $$Multiply each term by $$2(x - 1)$$: $$2(x - 1) \cdot \frac{3}{2(x - 1)} + 2(x - 1) \cdot \frac{1}{2} = 2(x - 1) \cdot \frac{2}{x - 1}. $$Simplify each term: The first term simplifies to $$3$$, the second term simplifies to $$(x - 1)$$, and the third term simplifies to $$4. $$The resulting equation is $$3 + (x - 1) = 4. $$Solve this linear equation for $$x$$, keeping in mind the restriction $$x 
eq 1.$$

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/(2x - 2) + 1/2 = 2/(x - 1)

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Key Concepts

Rational Equations

Restrictions on Variables

Solving for Variables