Learning Objectives
In this section you will:
- 6.3.1 – Solve a rational equation
- 6.3.2 – Solve radical equations
We have solved linear equations, exponential equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at rational equations and radical equations. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.
6.3.1 – Solving a Rational Equation
In this section, we look at rational equations that, after some manipulation, result in a linear or quadratic equation. If an equation contains at least one rational expression, it is a considered a rational equation.
Recall that a rational number is the ratio of two numbers, such as [latex]\,\frac{2}{3}\,[/latex] or [latex]\,\frac{7}{2}.\,[/latex] A rational expression is the ratio, or quotient, of two polynomials. Here are three examples.
[latex]$$\frac{x+1}{{x}^{2}-4},\,\frac{1}{x-3},\,\text{or}\,\frac{4}{{x}^{2}+x-2}$$[/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.
Example 1 – Solving a Rational Equation
Solve the rational equation: [latex]\,\frac{7}{2x}-\frac{5}{3x}=\frac{22}{3}.[/latex]
We have three denominators; [latex]\,2x,3x,[/latex] and 3. The LCD must contain [latex]\,2x,3x,[/latex] and 3. An LCD of [latex]\,6x\,[/latex] contains all three denominators. In other words, each denominator can be divided evenly into the LCD. Next, multiply both sides of the equation by the LCD [latex]\,6x.[/latex]
[latex]$$\begin{array}{cccc}\hfill \left(6x\right)\left[\frac{7}{2x}-\frac{5}{3x}\right]& =& \left[\frac{22}{3}\right]\left(6x\right)\hfill & \\ \hfill \left(6x\right)\left(\frac{7}{2x}\right)-\left(6x\right)\left(\frac{5}{3x}\right)& =& \left(\frac{22}{3}\right)\left(6x\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Use the distributive property}.\hfill \\ \hfill 3\left(7\right)-2\left(5\right)& =& 22\left(2x\right)\hfill& \phantom{\rule{2em}{0ex}}\text{Cancel out the common factors}.\hfill \\\hfill 21-10& =& 44x\hfill & \\ \hfill 11& =& 44x\hfill & \\ \hfill \frac{11}{44}& =& x\hfill & \\ \hfill \frac{1}{4}& =& x\hfill & \end{array}$$[/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]\,\left(x+1\right).\,[/lat