Practice Questions
Solve each rational equation
\(\textbf{1)}\) \( \displaystyle \frac{1}{x+2}+ \frac{x+1}{5}= 1\)\( \text{The answer is } x=-1 \text{ or } x=3\)
\(\displaystyle \frac{1}{x+2}+ \frac{x+1}{5}= 1\)
\(\displaystyle \frac{(5)(x+2)(1)}{x+2}+ \frac{(5)(x+2)(x+1)}{5}= (5)(x+2)(1)\,\,\, \left(\text{Multiply by LCD}\right)\)
\(\displaystyle (5)(1)+ (x+2)(x+1)= (5)(x+2)(1) \,\,\,\left(\text{Cancel any common factors}\right)\)
\( 5+ x^2+3x+2= 5x+10\,\,\, \left(\text{Distribute}\right)\)
\(\displaystyle x^2+3x+7= 5x+10\,\,\, \left(\text{Combine like terms}\right)\)
\(\displaystyle x^2-2x-3=0\,\,\, \left(\text{Set equal to zero}\right)\)
\((x-3)(x+1)=0\,\,\, \left(\text{Factor}\right)\)
\(x-3=0 \text{ or }x+1=0\)
\( \text{The answer is } x=-1 \text{ or } x=3\)
\(\displaystyle \frac{1}{x+2}+ \frac{x+1}{5}= 1\)
\(\displaystyle \frac{(5)(x+2)(1)}{x+2}+ \frac{(5)(x+2)(x+1)}{5}= (5)(x+2)(1)\,\,\, \left(\text{Multiply by LCD}\right)\)
\(\displaystyle (5)(1)+ (x+2)(x+1)= (5)(x+2)(1) \,\,\,\left(\text{Cancel any common factors}\right)\)
\( 5+ x^2+3x+2= 5x+10\,\,\, \left(\text{Distribute}\right)\)
\(\displaystyle x^2+3x+7= 5x+10\,\,\, \left(\text{Combine like terms}\right)\)
\(\displaystyle x^2-2x-3=0\,\,\, \left(\text{Set equal to zero}\right)\)
\((x-3)(x+1)=0\,\,\, \left(\text{Factor}\right)\)
\(x-3=0 \text{ or }x+1=0\)
\( \text{The answer is } x=-1 \text{ or } x=3\)
\(\textbf{2)}\) \( \displaystyle \frac{x-1}{6}- \frac{5}{x+2}= \frac{x-6}{3}\)\( \text{The answer is } x=1 \text{ or } x=8\)
\(\textbf{3)}\) \( \displaystyle \frac{4}{x+1}+ \frac{2}{x+2}= 5\)\( \text{The answer is } x=-\frac{9}{5} \text{ or } x=0\)
\(\textbf{4)}\) \( \displaystyle \frac{6}{x+1}+ \frac{4}{x-3}= 3\)\( \text{The answer is } x=\frac{1}{3} \text{ or } x=5\)
\(\textbf{5)}\) \( \displaystyle \frac{4}{x+1}- \frac{8}{x+3}= 0\)\( \text{The answer is } x=1\)
\(\textbf{6)}\) \( \displaystyle \frac{x}{x+1}- \frac{x-1}{3}= \frac{1}{x+1}\)\( \text{The answer is } x=1 \text{ or } x=2\)
\(\textbf{7)}\) \( \displaystyle \frac{x}{x-3}- \frac{x+1}{12}= 2\)\( \text{The answer is } x=-15 \text{ or } x=5\)
\(\textbf{8)}\) \( \displaystyle \frac{2}{x+2}- \frac{3}{x-1}= 4\)\( \text{The answer is } x=-\frac{5}{4} \text{ or } x=0\)
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\(\)
\(\bullet\text{ Rational Expressions- Multiplying and Dividing}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{x^2+3x-4}{(x+4)(x+5)}\cdot \displaystyle\frac{x+5}{x-1}…\)
\(\bullet\text{ Rational Expressions- Adding and Subtracting}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{x-5}{x+3}+\frac{x+2}{x^2+5x+6}…\)