Solving rational equations worksheet step by step

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.

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. \(\color{blue}{\frac{2x}{2x-2}-\frac{2}{x}=\frac{1}{x-1}}\)

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. \(\color{blue}{x=2}\)

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Examples, Solutions, videos, worksheets, and activities to help Algebra students learn how to solve rational equations.

The following figure shows how to solve rational equations. Scroll down the page for more examples and solutions.

Solving Rational Equations
In a rational equation, there will be a variable in the denominator of a fraction. Use cross multiplying when solving rational equations.
Examples:
3/(2x + 1) = 5
2/x - 3/(x + 1) = 9

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Advanced Rational Equations - Algebra Help
Students learn that when solving advanced rational equations, the first step is to factor each of the denominators, if possible, then multiply both sides of the equation by the common denominator for all the fractions in order to get rid of the fractions.
Note that in this lesson, once the fractions have been removed from the equation, the result will be a polynomial equation, so remember the rules for solving polynomial equations: set the equation equal to zero, then factor.
Finally, check each solution to see if it makes a denominator in the original equation equal to zero. If so, then it cannot be a solution to the equation.
Example:
(x - 2)/4 + 1 = 12/x

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Solving Rational Equations
5/(3x - 4) = 2/(x + 1)
-x/(x - 2) + (3x - 1)/(x + 4) = 1/(x2 + 2x - 8)

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Two more examples of solving rational equations
x + 1/x = 2
2/(x2 + 4x + 3) = 2 + (x -2)/(x + 3)

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