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

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.

  1. \(\color{blue}{\frac{10}{x+4}=\frac{15}{4x+4}}\)
  2. \(\color{blue}{\frac{x+4}{x+1}=\frac{x-6}{x-1}}\)
  3. \(\color{blue}{\frac{2x}{x+3}=\frac{x-6}{x+4}}\)
  4. \(\color{blue}{\frac{1}{x+5}-1=\frac{1}{1+x}}\)
  5. \(\color{blue}{\frac{1}{5x^2}-\frac{1}{x}=\frac{2}{x}}\)
  6. \(\color{blue}{\frac{2x}{2x-2}-\frac{2}{x}=\frac{1}{x-1}}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{x=\frac{4}{5}}\)
  2. \(\color{blue}{x=-\frac{1}{4}}\)
  3. \(\color{blue}{x=-9}\) or \(\color{blue}{x=-2}\)
  4. \(\color{blue}{x=-3}\)
  5. \(\color{blue}{x=\frac{1}{15}}\)
  6. \(\color{blue}{x=2}\)

Related to "How to Solve Rational Equations? (+FREE Worksheet!)"

How To Fall In Love With Math And StudyingHow To Fall In Love With Math And Studying
How to Apply Trigonometry to General Triangles?How to Apply Trigonometry to General Triangles?
How to Determine Limits Using the Squeeze Theorem?How to Determine Limits Using the Squeeze Theorem?
How to Determine Limits Using Algebraic Manipulation?How to Determine Limits Using Algebraic Manipulation?
How to Estimate Limit Values from the Graph?How to Estimate Limit Values from the Graph?
Properties of LimitsProperties of Limits
How to Find the Expected Value of a Random Variable?How to Find the Expected Value of a Random Variable?
How to Define Limits Analytically Using Correct Notation?How to Define Limits Analytically Using Correct Notation?
How to Solve Multiplication Rule for Probabilities?How to Solve Multiplication Rule for Probabilities?
How to Solve Venn Diagrams and the Addition Rule?How to Solve Venn Diagrams and the Addition Rule?

What people say about "How to Solve Rational Equations? (+FREE Worksheet!)"?

No one replied yet.

Leave a Reply