7.4 Solve Rational Equations

Learning objectives

Learning objectives

• Solve rational equations

• Use rational functions

• Solve a rational equation for a specific variable

After defining the terms ‘expression’ and ‘equation’ earlier, we have used them throughout this course. We have simplified many kinds of expressions and solved many kinds of equations. We have simplified many rational expressions so far in this chapter. Now we will solve a rational equation.

You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.

Solve Rational Equations

We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions.

We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then, we will have an equation that does not contain rational expressions and thus is much easier for us to solve. But because the original equation may have a variable in a denominator, we must be careful that we don’t end up with a solution that would make a denominator equal to zero.

So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.

An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution to a rational equation.

We note any possible extraneous solutions, c, by writing $x\ne{}c$ next to the equation.

Example 7.33

The steps of this method are shown.

We always start by noting the values that would cause any denominators to be zero.

Example 7.34

Solution

In the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.

Example 7.35

Solution

In the next example, the first denominator is a trinomial. Remember to factor it first to find the LCD.

Example 7.36

Solution

The equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.

Example 7.37

Solution

In some cases, all the algebraic solutions are extraneous.

Example 7.38

Solution

Example 7.39

Solution

Use Rational Functions

Working with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.

Example 7.40

Solution

ⓐ The domain of a rational function is all real numbers except those that make the rational expression undefined. So to find them, we will set the denominator equal to zero and solve.

${x}^{2}-8x+15=0$

Factor the trinomial. · $(x-3)(x-5)=0$ Use the Zero Product Property. · $x-3=0\,x-5=0$ Solve. · $x=3\,x=5$

The domain is all real numbers except $x\ne{}3,x\ne{}5.$

ⓑ

However, $x=3$ is outside the domain of this function, so we discard that root as extraneous.

ⓒ The value of the function is 1 when $x=7.$ So the points on the graph of this function when $f(x)=1$ is $(7,1))$

Solve a Rational Equation for a Specific Variable

When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.

When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.

$\begin{array}{cccrll} & & & m & = & \frac{y-{y}_{1}}{x-{x}_{1}} \\ \mathrm{Multiply both sides of the equation by}\,x-{x}_{1}. & & & m(x-{x}_{1}) & = & (\frac{y-{y}_{1}}{x-{x}_{1}})(x-{x}_{1}) \\ \mathrm{Simplify.} & & & m(x-{x}_{1}) & = & y-{y}_{1} \\ \mathrm{Rewrite the equation with the}\,y\,\mathrm{terms on the left.} & & & y-{y}_{1} & = & m(x-{x}_{1}) \\ \end{array}$

In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through a point in Chapter 3. We will add one more step to solve for y.

Example 7.41

Solution

Remember to multiply both sides by the LCD in the next example.

Example 7.42

Solution

Section 7.4 Exercises

Section 7.4 Exercises

Solve Rational Equations

In the following exercises, solve each rational equation.

Solve Rational Equations that Involve Functions

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of $1-10,$ how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

1. Solve: $\frac{1}{y}+\frac{2}{3}=\frac{1}{5}.$

2. Solve: $\frac{2}{3}+\frac{1}{5}=\frac{1}{x}.$

3. Solve: $1-\frac{2}{x}=\frac{15}{{x}^{2}}.$

1. Solve: $1-\frac{4}{y}=\frac{12}{{y}^{2}}.$

2. Solve: $\frac{2}{x+1}+\frac{1}{x-1}=\frac{1}{{x}^{2}-1}.$

3. Solve: $\frac{5}{y+3}+\frac{2}{y-3}=\frac{5}{{y}^{2}-9}.$

1. Solve: $\frac{x+13}{{x}^{2}-7x+10}=\frac{6}{x-5}-\frac{4}{x-2}.$

2. Solve: $\frac{y-6}{{y}^{2}+3y-4}=\frac{2}{y+4}+\frac{7}{y-1}.$

3. Solve: $\frac{x}{x+4}=\frac{32}{{x}^{2}-16}+5.$

1. Solve: $\frac{y}{y+8}=\frac{128}{{y}^{2}-64}+9.$

2. Solve: $\frac{y}{5y-10}-\frac{5}{3y+6}=\frac{2{y}^{2}-19y+54}{15{y}^{2}-60}.$

3. Solve: $\frac{z}{2z+8}-\frac{3}{4z-8}=\frac{3{z}^{2}-16z-16}{8{z}^{2}+16z-64}.$

1. Solve: $\frac{15}{{x}^{2}+x-6}-\frac{3}{x-2}=\frac{2}{x+3}.$

2. Solve: $\frac{5}{{x}^{2}+2x-3}-\frac{3}{{x}^{2}+x-2}=\frac{1}{{x}^{2}+5x+6}.$

3. For rational function, $f(x)=\frac{8-x}{{x}^{2}-7x+12},$ ⓐ find the domain of the function ⓑ solve $f(x)=3$ ⓒ find the points on the graph at this function value.

1. For rational function, $f(x)=\frac{x-1}{{x}^{2}-6x+5},$ ⓐ find the domain of the function ⓑ solve $f(x)=4$ ⓒ find the points on the graph at this function value.

2. Solve: $m=\frac{y-5}{x-4}$for $y.$

3. Solve: $m=\frac{y-1}{x+5}$ for $y.$

1. Solve: $\frac{1}{a}+\frac{1}{b}=c$ for a.

2. Solve: $\frac{2}{x}+\frac{1}{3}=\frac{1}{y}$ for y.

3. $\frac{1}{a}+\frac{2}{5}=\frac{1}{2}$

1. $\frac{4}{5}+\frac{1}{4}=\frac{2}{v}$

2. $1-\frac{2}{m}=\frac{8}{{m}^{2}}$

3. $1+\frac{9}{p}=\frac{-20}{{p}^{2}}$

1. $\frac{5}{3v-2}=\frac{7}{4v}$

2. $\frac{3}{x+4}+\frac{7}{x-4}=\frac{8}{{x}^{2}-16}$

3. $\frac{8}{z-10}-\frac{7}{z+10}=\frac{5}{{z}^{2}-100}$

1. $\frac{-10}{q-2}-\frac{7}{q+4}=1$

2. $\frac{v-10}{{v}^{2}-5v+4}=\frac{3}{v-1}-\frac{6}{v-4}$

3. $\frac{x-10}{{x}^{2}+8x+12}=\frac{3}{x+2}+\frac{4}{x+6}$

1. $\frac{b+3}{3b}+\frac{b}{24}=\frac{1}{b}$

2. $\frac{d}{d+3}=\frac{18}{{d}^{2}-9}+4$

3. $\frac{n}{n+2}-3=\frac{8}{{n}^{2}-4}$

1. $\frac{q}{3q-9}-\frac{3}{4q+12}=\frac{7{q}^{2}+6q+63}{24{q}^{2}-216}$

2. $\frac{s}{2s+6}-\frac{2}{5s+5}=\frac{5{s}^{2}-3s-7}{10{s}^{2}+40s+30}$

3. $\frac{2}{{x}^{2}+2x-8}-\frac{1}{{x}^{2}+9x+20}=\frac{4}{{x}^{2}+3x-10}$

1. $\frac{3}{{x}^{2}-5x-6}+\frac{3}{{x}^{2}-7x+6}=\frac{6}{{x}^{2}-1}$

2. For rational function, $f(x)=\frac{x-2}{{x}^{2}+6x+8},$ ⓐ find the domain of the function ⓑ solve $f(x)=5$ ⓒ find the points on the graph at this function value.

3. For rational function, $f(x)=\frac{2-x}{{x}^{2}-7x+10},$ ⓐ find the domain of the function ⓑ solve $f(x)=2$ ⓒ find the points on the graph at this function value.

1. $\frac{C}{r}=2\pi$ for $r.$

2. $\frac{v+3}{w-1}=\frac{1}{2}$ for $w.$

3. $a=\frac{b+3}{c-2}$ for $c.$

1. $\frac{1}{p}+\frac{2}{q}=4$ for $p.$

2. $\frac{2}{v}+\frac{1}{5}=\frac{3}{w}$ for $w.$

3. $\frac{m+3}{n-2}=\frac{4}{5}$ for $n.$

1. $\frac{E}{c}={m}^{2}$ for $c.$

2. $\frac{3}{x}-\frac{5}{y}=\frac{1}{4}$ for $y.$

3. Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.