Other Types of Equations: Learn It 1

  • Solve equations that include fractions with variables.
  • Solve equations with roots and fractional powers.
  • Use factoring to find solutions to polynomial equations.
  • Find solutions to equations that involve absolute values.
  • Solve absolute value inequalities.

Solving a Rational Equation

A rational equation is an equation that contains at least one rational expression—a fraction where the numerator, denominator, or both are polynomials. Although these equations may appear more complex at first glance, many can be manipulated to reveal a underlying linear structure.

rational equation

A rational equation contains at least one rational expression where the variable appears in at least one of the denominators.

Recall that a rational number is the ratio of two numbers, such as [latex]\dfrac{3}{4}[/latex] or [latex]\dfrac{7}{2}[/latex]. A rational expression is the ratio, or quotient, of two polynomials – [latex]\dfrac{x+1}{x^2-4}, \dfrac{1}{x-3}, or \dfrac{4}{x^2+x-2}[/latex].

Rational equations have a variable in the denominator in at least one of the terms. Our goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).

Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.

Solve the rational equation:

[latex]\dfrac{7}{2x} - \dfrac{5}{3x} = \dfrac{22}{3}[/latex]

A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial—two terms added or subtracted—such as [latex](x + 1)[/latex]. Always consider a binomial as an individual factor—the terms cannot be separated.

For example, suppose a problem has three terms and the denominators are [latex]x[/latex], [latex]1[/latex], and [latex]3x - 3[/latex].

First, factor all denominators. We then have [latex]x[/latex], [latex]1[/latex], and [latex]3(x - 1)[/latex] as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of [latex](x - 1)[/latex].

The [latex]x[/latex] in the first denominator is separate from the [latex]x[/latex] in the [latex](x - 1)[/latex] denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor.

The LCD in this instance is found by multiplying together the [latex]x[/latex], one factor of [latex](x - 1)[/latex], and the 3. Thus, the LCD is the following:

[latex]x(x - 1)3 = 3x(x - 1)[/latex]

So, both sides of the equation would be multiplied by [latex]3x(x - 1)[/latex]. Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.

Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.

[latex]\dfrac{a}{b} = \dfrac{c}{d}[/latex]

We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.

[latex]\text{If } \dfrac{a}{b} = \dfrac{c}{d} \text{, then } a \cdot d = b \cdot c[/latex]

Multiply [latex]a(d)[/latex] and [latex]b(c)[/latex], which results in [latex]ad = bc[/latex].

Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.
How to: Given a rational equation, solve it.

    1. Factor all denominators in the equation.
    2. Find and exclude values that set each denominator equal to zero.
    3. Find the LCD.
    4. Multiply the whole equation by the LCD. If the LCD is correct, there will be no denominators left.
    5. Solve the remaining equation.
    6. Make sure to check solutions back in the original equations to avoid a solution producing zero in a denominator.

    Solve the following rational equation:

    [latex]\dfrac{2}{x} - \dfrac{3}{2} = \dfrac{7}{2x}[/latex]

    Solve the following rational equation:

    [latex]\dfrac{1}{x} = \dfrac{1}{10} - \dfrac{3}{4x}[/latex]

    Solve the following rational equations and state the excluded values:

    1. [latex]\dfrac{3}{x-6} = \dfrac{5}{x}[/latex]
    2. [latex]\dfrac{x}{x-3} = \dfrac{5}{x-3} - \dfrac{1}{2}[/latex]
    3. [latex]\dfrac{x}{x-2} = \dfrac{5}{x-2} - \dfrac{1}{2}[/latex]