Rational Equations

Rational equations are equations containing rational expressions (or expressions with fractions that contain real numbers and/or variables). Some examples of rational equations:

• ${\frac {1}{8}}x+2={\frac {1}{4}}x$
• ${\frac {1}{x+1}}={\frac {2}{3}}$
• ${\frac {y^{2}}{3y+7}}={\frac {y}{6}}$

Steps to solving rational equations:

1. Note any value of the variable that would make any denominator zero.
2. Find the least common denominator of all denominators in the equation.
3. Clear the fractions by multiplying both sides of the equation by the LCD.
4. Solve the resulting equation.
5. Check: If any values found in step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation.

Example problem: $1-{\frac {1}{x}}={\frac {2}{x^{2}}}$

1. If x = 0, the denominator of ${\frac {1}{x}}$ and ${\frac {2}{x^{2}}}$ will be 0.
2. The least common denominator of all terms in the equation is $x^{2}$.
3. Multiplying each side of the equation $1-{\frac {1}{x}}={\frac {2}{x^{2}}}$ with $x^{2}$ gives us $x^{2}-x=2$
4. $x^{2}-x=2\;\;\;\to \;\;\;x^{2}-x-2=0\;\;\;\to \;\;\;(x-2)(x+1)=0\;\;\;\to \;\;\;x=-1,x=2$
5. None of these solutions were noted in step 1, so we can check our two solutions:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = -1} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 - \frac{1}{x} = \frac{2}{x^2} \;\;\;\to\;\;\; 1 - \frac{1}{-1} = \frac{2}{(-1)^2} \;\;\;\to\;\;\; 1 - (-1) = 2 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 2} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \;\;\, 1 - \frac{1}{x} = \frac{2}{x^2} \;\;\;\to\;\;\; 1 - \frac{1}{2} = \frac{2}{2^2} \;\;\;\to\;\;\; \frac{1}{2} = \frac{2}{4} }

Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = -1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 2} are solutions to our original rational equations.

Resources

• Solve Rational Equations, OpenStax
• Solving Rational Equations (Example), The Organic Chemistry Tutor
• Solving Rational Equations with Different Denominators (Example), The Organic Chemistry Tutor