7.6: Solve Rational Equations (2024)

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    Learning Objectives

    By the end of this section, you will be able to:

    • Solve rational equations
    • Solve a rational equation for a specific variable
    Note

    Before you get started, take this readiness quiz.

    If you miss a problem, go back to the section listed and review the material.

    1. Solve: \(\frac{1}{6}x+\frac{1}{2}=\frac{1}{3}\).
      If you missed this problem, review Exercise 2.5.1.
    2. Solve: \(n^2−5n−36=0\).
      If you missed this problem, review Exercise 7.6.13.
    3. Solve for y in terms of x: 5x+2y=10 for y.
      If you missed this problem, review Exercise 2.6.22.

    After defining the terms expression and equation early in Foundations, we have used them throughout this book. 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 rational equations.

    The definition of a rational equation is similar to the definition of equation we used in Foundations.

    Definition: RATIONAL EQUATION

    A rational equation is two rational expressions connected by an equal sign.

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

    \[\begin{array}{cc} {\textbf{Rational Expression}}&{\textbf{Rational Equation}}\\ {\frac{1}{8}x+\frac{1}{2}}&{\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}}\\ {\frac{y+6}{y^2−36}}&{\frac{y+6}{y^2−36}=y+1}\\ {\frac{1}{n−3}+\frac{1}{n+4}}&{\frac{1}{n−3}+\frac{1}{n+4}=\frac{15}{n^2+n−12}}\\ \nonumber \end{array}\]

    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.

    Here is an example we did when we worked with linear equations:

    7.6: Solve Rational Equations (2) 7.6: Solve Rational Equations (3)
    We multiplied both sides by the LCD. 7.6: Solve Rational Equations (4)
    Then we distributed. 7.6: Solve Rational Equations (5)
    We simplified—and then we had an equation with no fractions. 7.6: Solve Rational Equations (6)
    Finally, we solved that equation. 7.6: Solve Rational Equations (7)
    7.6: Solve Rational Equations (8)

    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.

    Definition: EXTRANEOUS Solution TO A RATIONAL EQUATION

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

    We note any possible extraneous solutions, c, by writing \(x \ne c\) next to the equation.

    How to Solve Equations with Rational Expressions

    Example \(\PageIndex{1}\)

    Solve: \(\frac{1}{x}+\frac{1}{3}=\frac{5}{6}\).

    Answer

    7.6: Solve Rational Equations (9)7.6: Solve Rational Equations (10)7.6: Solve Rational Equations (11)7.6: Solve Rational Equations (12)7.6: Solve Rational Equations (13)

    Example \(\PageIndex{2}\)

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

    Answer

    \(−\frac{15}{7}\)

    Example \(\PageIndex{3}\)

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

    Answer

    \(\frac{15}{13}\)

    The steps of this method are shown below.

    Definition: SOLVE EQUATIONS WITH RATIONAL EXPRESSIONS.
    1. Note any value of the variable that would make any denominator zero.
    2. Find the least common denominator of all denominators in the equation.
    3. Clear the fractions by multiplying both sides of the equation by the LCD.
    4. Solve the resulting equation.
    5. Check.
      • If any values found in Step 1 are algebraic solutions, discard them.
      • Check any remaining solutions in the original equation.

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

    Example \(\PageIndex{4}\)

    Solve: \(1−\frac{5}{y}=−\frac{6}{y^2}\).

    Answer
    7.6: Solve Rational Equations (14)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (15)
    Find the least common denominator of all denominators in the equation. The LCD is \(y^2\)
    Clear the fractions by multiplying both sides of the equation by the LCD. 7.6: Solve Rational Equations (16)
    Distribute. 7.6: Solve Rational Equations (17)
    Multiply. 7.6: Solve Rational Equations (18)
    Solve the resulting equation. First write the quadratic equation in standard form. 7.6: Solve Rational Equations (19)
    Factor. 7.6: Solve Rational Equations (20)
    Use the Zero Product Property. 7.6: Solve Rational Equations (21)
    Solve. 7.6: Solve Rational Equations (22)
    Check.
    We did not get 0 as an algebraic solution.
    7.6: Solve Rational Equations (23)
    Example \(\PageIndex{5}\)

    Solve: \(1−\frac{2}{a}=\frac{15}{a^2}\).

    Answer

    5, −3

    Example \(\PageIndex{6}\)

    Solve: \(1−\frac{4}{b}=\frac{12}{b^2}\).

    Answer

    6, −2

    Example \(\PageIndex{7}\)

    Solve: \(\frac{5}{3u−2}=\frac{3}{2u}\).

    Answer
    7.6: Solve Rational Equations (24)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (25)
    Find the least common denominator of all denominators in the equation. The LCD is 2u(3u−2).
    Clear the fractions by multiplying both sides of the equation by the LCD. 7.6: Solve Rational Equations (26)
    Remove common factors. 7.6: Solve Rational Equations (27)
    Simplify. 7.6: Solve Rational Equations (28)
    Multiply. 7.6: Solve Rational Equations (29)
    Solve the resulting equation. 7.6: Solve Rational Equations (30)
    We did not get 0 or \(\frac{2}{3}\) as algebraic solutions.
    7.6: Solve Rational Equations (31)
    Example \(\PageIndex{8}\)

    Solve: \(\frac{1}{x−1}=\frac{2}{3x}\).

    Answer

    −2

    Example \(\PageIndex{9}\)

    Solve: \(\frac{3}{5n+1}=\frac{2}{3n}\).

    Answer

    −2

    When one of the denominators is a quadratic, remember to factor it first to find the LCD.

    Example \(\PageIndex{10}\)

    Solve: \(\frac{2}{p+2}+\frac{4}{p−2}=\frac{p−1}{p^2−4}\).

    Answer
    7.6: Solve Rational Equations (32)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (33)
    Find the least common denominator of all denominators in the equation. The LCD is (p+2)(p−2).
    Clear the fractions by multiplying both sides of the equation by the LCD. 7.6: Solve Rational Equations (34)
    Distribute. 7.6: Solve Rational Equations (35)
    Remove common factors. 7.6: Solve Rational Equations (36)
    Simplify. 7.6: Solve Rational Equations (37)
    Distribute. 7.6: Solve Rational Equations (38)
    Solve. 7.6: Solve Rational Equations (39)
    7.6: Solve Rational Equations (40)
    7.6: Solve Rational Equations (41)
    We did not get 2 or −2 as algebraic solutions.
    7.6: Solve Rational Equations (42)
    Example \(\PageIndex{11}\)

    Solve: \(\frac{2}{x+1}+\frac{1}{x−1}=\frac{1}{x^2−1}\).

    Answer

    \(\frac{2}{3}\)

    Example \(\PageIndex{12}\)

    Solve: \(\frac{5}{y+3}+\frac{2}{y−3}=\frac{5}{y^2−9}\)

    Answer

    2

    Example \(\PageIndex{13}\)

    Solve: \(\frac{4}{q−4}−\frac{3}{q−3}=1\).

    Answer
    7.6: Solve Rational Equations (43)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (44)
    Find the least common denominator of all denominators in the equation. The LCD is (q−4)(q−3).
    Clear the fractions by multiplying both sides of the equation by the LCD. 7.6: Solve Rational Equations (45)
    Distribute. 7.6: Solve Rational Equations (46)
    Remove common factors. 7.6: Solve Rational Equations (47)
    Simplify. 7.6: Solve Rational Equations (48)
    Simplify. 7.6: Solve Rational Equations (49)
    Combine like terms. 7.6: Solve Rational Equations (50)
    Solve. First write in standard form. 7.6: Solve Rational Equations (51)
    Factor. 7.6: Solve Rational Equations (52)
    Use the Zero Product Property. 7.6: Solve Rational Equations (53)
    We did not get 4 or 3 as algebraic solutions.
    7.6: Solve Rational Equations (54)
    Example \(\PageIndex{14}\)

    Solve: \(\frac{2}{x+5}−\frac{1}{x−1}=1\).

    Answer

    −1, −2

    Example \(\PageIndex{15}\)

    Solve: \(\frac{3}{x+8}−\frac{2}{x−2}=1\).

    Answer

    −2, −3

    Example \(\PageIndex{16}\)

    Solve: \(\frac{m+11}{m^2−5m+4}=\frac{5}{m−4}−\frac{3}{m−1}\).

    Answer
    7.6: Solve Rational Equations (55)
    Factor all the denominators, so we can note any value of the variable the would make any denominator zero. 7.6: Solve Rational Equations (56)
    Find the least common denominator of all denominators in the equation. The LCD is (m−4)(m−1)
    Clear the fractions. 7.6: Solve Rational Equations (57)
    Distribute. 7.6: Solve Rational Equations (58)
    Remove common factors. 7.6: Solve Rational Equations (59)
    Simplify. 7.6: Solve Rational Equations (60)
    Solve the resulting equation. 7.6: Solve Rational Equations (61)
    7.6: Solve Rational Equations (62)
    Check. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.
    Example \(\PageIndex{17}\)

    Solve: \(\frac{x+13}{x^2−7x+10}=\frac{6}{x−5}−\frac{4}{x−2}\).

    Answer

    no solution

    Example \(\PageIndex{18}\)

    Solve: \(\frac{y−14}{y^2+3y−4}=\frac{2}{y+4}+\frac{7}{y−1}\).

    Answer

    no solution

    The equation we solved in Example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution.

    Example \(\PageIndex{19}\)

    Solve: \(\frac{n}{12}+\frac{n+3}{3n}=\frac{1}{n}\).

    Answer
    7.6: Solve Rational Equations (63)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (64)
    Find the least common denominator of all denominators in the equation. The LCD is 12n.
    Clear the fractions by multiplying both sides of the equation by the LCD. 7.6: Solve Rational Equations (65)
    Distribute. 7.6: Solve Rational Equations (66)
    Remove common factors. 7.6: Solve Rational Equations (67)
    Simplify. 7.6: Solve Rational Equations (68)
    Solve the resulting equation. 7.6: Solve Rational Equations (69)
    7.6: Solve Rational Equations (70)
    7.6: Solve Rational Equations (71)
    7.6: Solve Rational Equations (72)
    Check.
    n=0 is an extraneous solution.
    7.6: Solve Rational Equations (73)
    Example \(\PageIndex{20}\)

    Solve: \(\frac{x}{18}+\frac{x+6}{9x}=\frac{2}{3x}\).

    Answer

    −2

    Example \(\PageIndex{21}\)

    Solve: \(\frac{y+5}{5y}+\frac{y}{15}=\frac{1}{y}\).

    Answer

    −3

    Example \(\PageIndex{22}\)

    Solve: \(\frac{y}{y+6}=\frac{72}{y^2−36}+4\).

    Answer
    7.6: Solve Rational Equations (74)
    Factor all the denominators, so we can note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (75)
    Find the least common denominator. The LCD is (y−6)(y+6).
    Clear the fractions. 7.6: Solve Rational Equations (76)
    Simplify. 7.6: Solve Rational Equations (77)
    Simplify. 7.6: Solve Rational Equations (78)
    Solve the resulting equation. 7.6: Solve Rational Equations (79)
    7.6: Solve Rational Equations (80)
    7.6: Solve Rational Equations (81)
    7.6: Solve Rational Equations (82)
    7.6: Solve Rational Equations (83)
    Check.
    y=−6 is an extraneous solution.
    7.6: Solve Rational Equations (84)
    Example \(\PageIndex{23}\)

    Solve: \(\frac{x}{x+4}=\frac{32}{x^2−16}+5\).

    Answer

    −4, 3

    Example \(\PageIndex{24}\)

    Solve: \(\frac{y}{y+8}=\frac{128}{y^2−64}+9\).

    Answer

    7

    Example \(\PageIndex{25}\)

    Solve: \(\frac{x}{2x−2}−\frac{2}{3x+3}=\frac{5x^2−2x+9}{12x^2−12}\).

    Answer
    7.6: Solve Rational Equations (85)
    We will start by factoring all denominators, to make it easier to identify extraneous solutions and the LCD. 7.6: Solve Rational Equations (86)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (87)
    Find the least common denominator.The LCD is 12(x−1)(x+1)
    Clear the fractions. 7.6: Solve Rational Equations (88)
    Simplify. 7.6: Solve Rational Equations (89)
    Simplify. 7.6: Solve Rational Equations (90)
    Solve the resulting equation. 7.6: Solve Rational Equations (91)
    7.6: Solve Rational Equations (92)
    7.6: Solve Rational Equations (93)
    7.6: Solve Rational Equations (94)
    Check.
    x=1 and x=−1 are extraneous solutions.
    The equation has no solution.
    Example \(\PageIndex{26}\)

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

    Answer

    no solution

    Example \(\PageIndex{27}\)

    Solve: \(\frac{z^2}{z+8}−\frac{3}{4z−8}=\frac{3z^2−16z−68}{z^2+8z−64}\).

    Answer

    no solution

    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.

    We’ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form.

    Example \(\PageIndex{28}\)

    Solve: \(\frac{D}{T}=R\) for T.

    Answer
    7.6: Solve Rational Equations (95)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (96)
    Clear the fractions by multiplying both sides of the equations by the LCD, T. 7.6: Solve Rational Equations (97)
    Simplify. 7.6: Solve Rational Equations (98)
    Divide both sides by R to isolate T. 7.6: Solve Rational Equations (99)
    Simplify. 7.6: Solve Rational Equations (100)
    Example \(\PageIndex{29}\)

    Solve: \(\frac{A}{L}=W\) for L.

    Answer

    \(L=\frac{A}{W}\)

    Example \(\PageIndex{30}\)

    Solve: \(\frac{F}{A}=M\) for A.

    Answer

    \(A=\frac{F}{M}\)

    Example uses the formula for slope that we used to get the point-slope form of an equation of a line.

    Example \(\PageIndex{31}\)

    Solve: \(m=\frac{x−2}{y−3}\) for y.

    Answer
    7.6: Solve Rational Equations (101)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (102)
    Clear the fractions by multiplying both sides of the equations by the LCD, y−3. 7.6: Solve Rational Equations (103)
    Simplify. 7.6: Solve Rational Equations (104)
    Isolate the term with y. 7.6: Solve Rational Equations (105)
    Divide both sides by m to isolate y. 7.6: Solve Rational Equations (106)
    Simplify. 7.6: Solve Rational Equations (107)
    Example \(\PageIndex{32}\)

    Solve: \(\frac{y−2}{x+1}=\frac{2}{3}\) for x.

    Answer

    \(x=\frac{3y−8}{2}\)

    Example \(\PageIndex{33}\)

    Solve: \(x=\frac{y}{1−y}\) for y.

    Answer

    \(y=\frac{x}{1+x}\)

    Be sure to follow all the steps in Example. It may look like a very simple formula, but we cannot solve it instantly for either denominator.

    Example \(\PageIndex{34}\)

    Solve \(\frac{1}{c}+\frac{1}{m}=1\) for c.

    Answer
    7.6: Solve Rational Equations (108)
    Note any value of the variable that would make any denominator zero. 7.6: Solve Rational Equations (109)
    Clear the fractions by multiplying both sides of the equations by the LCD, cm 7.6: Solve Rational Equations (110)
    Distribute. 7.6: Solve Rational Equations (111)
    Simplify. 7.6: Solve Rational Equations (112)
    Collect the terms with c to the right. 7.6: Solve Rational Equations (113)
    Factor the expression on the right. 7.6: Solve Rational Equations (114)
    To isolate c, divide both sides by m−1. 7.6: Solve Rational Equations (115)
    Simplify by removing common factors. 7.6: Solve Rational Equations (116)

    Notice that even though we excluded c=0 and m=0 from the original equation, we must also now state that \(m \ne 1\).

    Example \(\PageIndex{35}\)

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

    Answer

    \(a=\frac{b}{cb−1}\)

    Example \(\PageIndex{36}\)

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

    Answer

    \(y=\frac{3x}{6+x}\)

    Key Concepts

    • Strategy to Solve Equations with Rational Expressions
      1. Note any value of the variable that would make any denominator zero.
      2. Find the least common denominator of all denominators in the equation.
      3. Clear the fractions by multiplying both sides of the equation by the LCD.
      4. Solve the resulting equation.
      5. Check.
      • If any values found in Step 1 are algebraic solutions, discard them.
      • Check any remaining solutions in the original equation.

    Glossary

    rational equation
    A rational equation is two rational expressions connected by an equal sign.
    extraneous solution to a rational equation
    An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.
    7.6: Solve Rational Equations (2024)

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