How to Solve Rational Equations? (+FREE Worksheet!)
![How to Solve Rational Equations? (+FREE Worksheet!)](https://www.effortlessmath.com/wp-content/uploads/2021/03/Rational-Equations-512x240.png)
An equation that consists of at least one Rational expression is a Rational equation, and in this article, we will teach you how to solve this type of equation using two methods.
Related Topics
- How to Add and Subtract Rational Expressions
- How to Multiply Rational Expressions
- How to Divide Rational Expressions
- How to Simplify Complex Fractions
- How to Graph Rational Expressions
A step-by-step guide to solve Rational Equations
For solving rational equations, we can use following methods:
- Converting to a common denominator: In this method, you need to get a common denominator for both sides of the equation. Then, make numerators equal and solve for the variable.
- Cross-multiplying: This method is useful when there is only one fraction on each side of the equation. Simply multiply the first numerator by the second denominator and make the result equal to the product of the second numerator and the first denominator.
Examples
Rational Equations – Example 1:
Solve. \(\frac{x – 2}{x + 1 }=\frac{x + 4}{x – 2}\)
Solution:
Use cross multiply method: if \(\frac{a}{b}=\frac{c}{d}\), then: \(a×d=b×c \)
\(\frac{x – 2}{x + 1 }=\frac{x + 4}{x – 2}→(x-2)(x-2)=(x+4)(x+1)\)
Expand: \((x-2)^2=x^2-4x+4\) and \((x+4)(x+1)=x^2+5x+4\), Then:
\( x^2-4x+4=x^2+5x+4\), Now, simplify: \(x^2-4x=x^2+5x\), subtract both sides \((x^2+5x)\), Then: \(x^2-4x-(x^2+5x)=x^2+5x-(x^2+5x)→ -9x=0→x=0\)
Rational Equations – Example 2:
Solve. \(\frac{x – 3}{x + 1 }=\frac{x + 5}{x – 2}\)
Solution:
Use cross multiply method: if \(\frac{a}{b}=\frac{c}{d}\), then: \(a×d=b×c\)
Then: \((x-3)(x-2)=(x+5)(x+1)\)
Expand: \((x – 3)(x-2)=x^2-5x+6\)
Expand: \((x+5)(x+1)=x^2+6x+5\), Then: \(x^2-5x+6=x^2+6x+5\), Simplify: \(x^2-5x=x^2+6x-1\)
Subtract both sides \(x^2+6x ,Then: -11x=-1→x=\frac{1}{11}\)
Rational Equations – Example 3:
Solve. \(\frac{x +3}{x + 6 }=\frac{x + 2}{x – 4}\)
Solution:
Use cross multiply method: if \(\frac{a}{b}=\frac{c}{d}\), then: \(a×d=b×c \)
\(\frac{x+3}{x +6 }=\frac{x + 2}{x – 4}→(x+3)(x-4)=(x+2)(x+6)\)
Expand: \((x + 3)(x-4)=x^2-x-12\)
Expand: \((x+2)(x+6)=x^2+8x+12\), Then: \(x^2-x-12=x^2+8x+12\), Simplify: \(x^2-x=x^2+8x+24\)
Subtract both sides \(x^2+8x ,Then: -9x=24→x=-\frac{24}{9}=-\frac{8}{3}\)
Rational Equations – Example 4:
Solve. \(\frac{x +5}{x + 2 }=\frac{x -5}{x +3}\)
Solution:
Use cross multiply method: if \(\frac{a}{b}=\frac{c}{d}\), then: \(a×d=b×c \)
\(\frac{x+5}{x +2 }=\frac{x -5}{x+3}→(x+5)(x+3)=(x-5)(x+2)\)
Expand: \((x + 5)(x+3)=x^2+8x+15\)
Expand: \((x-5)(x+2)=x^2-3x-10\), Then: \(x^2+8x+15=x^2-3x-10\), Simplify: \(x^2+8x=x^2-3x-25\)
Subtract both sides \(x^2-3x ,Then: 11x=-25→x=-\frac{25}{11}\)
Exercises for Rational Equations
Solve Rational Equations.
- \(\color{blue}{\frac{10}{x+4}=\frac{15}{4x+4}}\)
- \(\color{blue}{\frac{x+4}{x+1}=\frac{x-6}{x-1}}\)
- \(\color{blue}{\frac{2x}{x+3}=\frac{x-6}{x+4}}\)
- \(\color{blue}{\frac{1}{x+5}-1=\frac{1}{1+x}}\)
- \(\color{blue}{\frac{1}{5x^2}-\frac{1}{x}=\frac{2}{x}}\)
- \(\color{blue}{\frac{2x}{2x-2}-\frac{2}{x}=\frac{1}{x-1}}\)
![This image has an empty alt attribute; its file name is answers.png](https://www.effortlessmath.com/wp-content/uploads/2019/12/answers.png)
- \(\color{blue}{x=\frac{4}{5}}\)
- \(\color{blue}{x=-\frac{1}{4}}\)
- \(\color{blue}{x=-9}\) or \(\color{blue}{x=-2}\)
- \(\color{blue}{x=-3}\)
- \(\color{blue}{x=\frac{1}{15}}\)
- \(\color{blue}{x=2}\)
The Absolute Best Book for the Algebra Test
Related to This Article
More math articles
- How to Find Complex Roots of the Quadratic Equation?
- How to Score 800 on SAT Math?
- Billionaire Basics: How to Master Addition and Subtraction of Massive Whole Numbers
- Full-Length 6th Grade STAAR Math Practice Test
- Cracking the Case: How to Understand Word Problems of Interpreting a Graph
- What Kind of Math Is on the PERT Test?
- 10 Most Common ASTB Math Questions
- Algebra Puzzle – Challenge 53
- Top 10 Math Books for Grade 4: Empowering Young Minds to Discover Numbers
- Remainder and Factor Theorems
What people say about "How to Solve Rational Equations? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.