Equations with Rational Expressions - Tutorial

This is a tutorial with detailed explanations on how to solve equations with rational expressions. The main idea is to multiply all terms of the equation by the lowest common denaminator.

Example 1: Solve the following equation.

1 - 1 / (x - 2) = 4

Solution to example 1

  • This equation has a rational expression. Multiply all the terms of the equation by the denominator in the rational expression

    (x - 2)(1 - 1 / (x - 2)) = (x - 2)4

  • Simplify.
    (x - 2) - 1 = 4 (x - 2)

  • Mutliply factors and group like terms.
    x - 3 = 4x - 8

  • Add -4x to both sides and simplify

    -3x - 3 = - 8

  • Add + 3 to both sides and simplify

    -3x = - 5

  • Solve for x

    x = 5 /3
conclusion:The solution to the above equation is x = 5 / 3.

Example 2: Find all real solutions to the equation.

1 - 1 / (x - 2) = -4 / (x 2 - 4)

Solution to example 2

  • This is an equation with two rational expressions. Multiply all the terms on the left side of the equation and all the terms in the right side of the equation by the lowest common denominator (x - 2)(x + 2).

    (x - 2)(x + 2)(1 - 1 / (x - 2)) = (x - 2)(x + 2)( -4 / (x2 - 4) )

  • Cancel common factors.
    (x - 2) (x + 2) - (x + 2) = - 4

  • multiply factors and group like terms.
    x 2 - x - 6 = -4

  • add 4 to both sides.
    x 2 - x - 2 = 0

  • factor left side of equation.
    (x + 1) (x - 2) = 0

  • 2 values make the LS zero.

    x1 = -1
    x2 = 2

  • check:
    x = -1
    Left Side of equation = 1 -1 / (-3)
    = 4/3

    Right Side of Equation = -4 / (1-4)

    = 4/3

    x = 2

    this value of x is not a solution because it makes each of the denominators in the given equation equal to zero.
conclusion:The solution to the above equation is x = -1.


More references and links to equations.

Equations With Rational Expressions - Problems

Tutorial on Equations of the Quadratic Form.

Solve Quadratic Equations Using the Discriminant.

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