How to Solve Rational Equations? (+FREE Worksheet!)
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 solving 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.
- \(\color{blue}{\frac{10}{x+4}=\frac{15}{4x+4}}\)
- \(\color{blue}{\frac{x+4}{x+1}=\frac{x-6}{x-1}}\)
- \(\color{blue}{\frac{2x}{x+3}=\frac{x-6}{x+4}}\)
- \(\color{blue}{\frac{1}{x+5}-1=\frac{1}{1+x}}\)
- \(\color{blue}{\frac{1}{5x^2}-\frac{1}{x}=\frac{2}{x}}\)
- \(\color{blue}{\frac{2x}{2x-2}-\frac{2}{x}=\frac{1}{x-1}}\)
- \(\color{blue}{x=\frac{4}{5}}\)
- \(\color{blue}{x=-\frac{1}{4}}\)
- \(\color{blue}{x=-9}\) 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
- Matching a Model with a Ratio
- 5th Grade Common Core Math Worksheets: FREE & Printable
- Full-Length 6th Grade PSSA Math Practice Test
- Other Topics Puzzle – Challenge 97
- Top 10 4th Grade SBAC Math Practice Questions
- 10 Most Common 8th Grade Common Core Math Questions
- The Ultimate Common Core Algebra 1 Course (+FREE Worksheets)
- What is the Best Laptop for College Students?
- How to Find Factors of Numbers?
- 6th Grade New York State Assessments Math Worksheets: FREE & Printable
What people say about "How to Solve Rational Equations? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.