Practice Questions
Solve each rational equation
\(\textbf{1)}\) \( \displaystyle \frac{1}{x+2}+ \frac{x+1}{5}= 1\)
\( \text{The answer is } x=-1 \text{ or } x=3\)
\(\displaystyle \frac{1}{x+2}+ \frac{x+1}{5}= 1\)
\(\displaystyle \frac{(5)(x+2)(1)}{x+2}+ \frac{(5)(x+2)(x+1)}{5}= (5)(x+2)(1)\,\,\, \left(\text{Multiply by LCD}\right)\)
\(\displaystyle (5)(1)+ (x+2)(x+1)= (5)(x+2)(1) \,\,\,\left(\text{Cancel any common factors}\right)\)
\( 5+ x^2+3x+2= 5x+10\,\,\, \left(\text{Distribute}\right)\)
\(\displaystyle x^2+3x+7= 5x+10\,\,\, \left(\text{Combine like terms}\right)\)
\(\displaystyle x^2-2x-3=0\,\,\, \left(\text{Set equal to zero}\right)\)
\((x-3)(x+1)=0\,\,\, \left(\text{Factor}\right)\)
\(x-3=0 \text{ or }x+1=0\)
\( \text{The answer is } x=-1 \text{ or } x=3\)
\(\textbf{2)}\) \( \displaystyle \frac{x-1}{6}- \frac{5}{x+2}= \frac{x-6}{3}\)
\( \text{The answer is } x=1 \text{ or } x=8\)
\(\textbf{3)}\) \( \displaystyle \frac{4}{x+1}+ \frac{2}{x+2}= 5\)
\( \text{The answer is } x=-\frac{9}{5} \text{ or } x=0\)
\(\textbf{4)}\) \( \displaystyle \frac{6}{x+1}+ \frac{4}{x-3}= 3\)
\( \text{The answer is } x=\frac{1}{3} \text{ or } x=5\)
\(\textbf{5)}\) \( \displaystyle \frac{4}{x+1}- \frac{8}{x+3}= 0\)
\( \text{The answer is } x=1\)
\(\textbf{6)}\) \( \displaystyle \frac{x}{x+1}- \frac{x-1}{3}= \frac{1}{x+1}\)
\( \text{The answer is } x=1 \text{ or } x=2\)
\(\textbf{7)}\) \( \displaystyle \frac{x}{x-3}- \frac{x+1}{12}= 2\)
\( \text{The answer is } x=-15 \text{ or } x=5\)
\(\textbf{8)}\) \( \displaystyle \frac{2}{x+2}- \frac{3}{x-1}= 4\)
\( \text{The answer is } x=-\frac{5}{4} \text{ or } x=0\)
Challenge Problems
\(\textbf{9)}\) Solve for x. \(\,\, \displaystyle 4= \frac{x^2-9}{3-x} \)
The answer is \(x=-7 \)
\(\textbf{10)}\) \( \displaystyle \frac{5}{x-4}+ \frac{1}{x+3}= 2\)
\( \text{The answers are } x=\displaystyle\frac{4+\sqrt{86}}{2} \text{ or } x=\displaystyle\frac{4-\sqrt{86}}{2} \)
See Related Pages\(\)
In Summary
A rational equation is an equation that involves one or more rational expressions. A rational expression is an expression that involves the division of one polynomial by another. When solving a rational equation, we need to follow a set of rules and procedures to isolate the rational expression and solve for the unknown variable. Rational equations are typically taught in an algebra class. In this class, students learn how to manipulate algebraic expressions and equations, and how to solve equations involving rational expressions is an important part of this topic, and it is often used to solve real-world problems involving ratios and proportions.
