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Elementary Algebra

Katherine Yoshiwara

Section 8.5 Chapter 8 Summary and Review

Subsection Lesson 8.1 Algebraic Fractions

  • An algebraic fraction (or rational expression, as they are sometimes called) is a fraction in which both numerator and denominator are polynomials.
  • An algebraic fraction is undefined at any values of the variable that make the denominator equal to zero.
  • We use the fundamental principle of fractions to reduce or build fractions.

  • Fundamental Principle of Fractions.

    We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.
    \begin{equation*} \blert{\dfrac{a \cdot c}{b \cdot c} = \dfrac{a}{b}~~~\text{if}~~~b,~c \not= 0} \end{equation*}

  • To Reduce an Algebraic Fraction.

    1. Factor numerator and denominator completely.
    2. Divide numerator and denominator by any common factors.
  • We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).
  • A fraction is a negative number if either its numerator or its denominator is negative, but not both.

  • Negative of a Binomial.

    The opposite of \(a-b\) is
    \begin{equation*} \blert{-(a-b) = -a+b = b-a} \end{equation*}

Subsection Lesson 8.2 Operations on Algebraic Fractions

  • Product of Fractions.

    If \(~b \not= 0~\) and \(~d \not= 0~\text{,}\) then
    \begin{equation*} \blert{\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}} \end{equation*}

  • To Multiply Algebraic Fractions.

    1. Factor each numerator and denominator completely.
    2. If any factor appears in both a numerator and a denominator, divide out that factor.
    3. Multiply the remaining factors of the numerator and the remaining factors of the denominator.
    4. Reduce the product if necessary.

  • Quotient of Fractions.

    If \(b,~c,~,d \not= 0\text{,}\) then
    \begin{equation*} \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} \end{equation*}

  • To Divide One Fraction by Another.

    1. Take the reciprocal of the second fraction and change the division to multiplication.
    2. Follow the rules for multiplication of fractions.
  • Fractions with the same denominator are called like fractions.

  • Sum or Difference of Like Fractions.

    If \(c \not= 0\text{,}\) then
    \begin{equation*} \blert{\dfrac{a}{c}+\dfrac{b}{c}= \dfrac{a+b}{c}}~~~~~~\text{and}~~~~~~ \blert{\dfrac{a}{c}-\dfrac{b}{c}= \dfrac{a-b}{c}} \end{equation*}

  • To Add or Subtract Like Fractions.

    1. Add or subtract the numerators.
    2. Keep the same denominator.
    3. Reduce the sum or difference if necessary.

Subsection Lesson 8.3 Lowest Common Denominator

  • To add or subtract fractions with unlike denominators, we must first convert the fractions to equivalent forms with the same denominator.
  • The lowest common denominator for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator.

  • To Find the LCD.

    1. Factor each denominator completely, and arrange the factors in order.
    2. For each factor,
      1. Which denominator has the most copies of that factor? Circle them. (If there is a tie, either denominator will do.)
      2. Include all the circled factors in the LCD.
    3. Multiply together the factors of the LCD.

  • To Add or Subtract Algebraic Fractions.

    1. Find the lowest common denominator (LCD) for the fractions.
    2. Build each fraction to an equivalent one with the LCD as denominator.
    3. Add or subtract the resulting like fractions: Add or subtract their numerators, and keep the same denominator.
    4. Reduce the sum or difference if necessary.

Subsection Lesson 8.4 Equations with Fractions

  • To solve an equation that contains algebraic fractions, we first clear the denominators by multiplying both sides of the equation by the LCD of the fractions.
  • When clearing fractions from am equation, we must be sure to multiply each term of the equation by the LCD.
  • Whenever we multiply an equation by an expression containing the variable, we should check for extraneous solutions.

  • Work Formula.

    \begin{align*} \blert{\text{work rate} \times \text{time}} \amp \blert{= \text{work completed}}\\ \blert{rt} \amp \blert{= w} \end{align*}

Subsection Review Questions

Use complete sentences to answer the questions.
  1. When reducing an algebraic fraction, we should always before we .
  2. The fundamental principle of fractions says that we can cancel , but not .
  3. Delbert says that to multiply two algebraic fractions, we just multiply the numerators together and multiply the denominators together. Comment on Delbert’s method.
  4. To divide by a fraction is the same as to by its.
  5. Describe how to add or subtract unlike fractions in three steps.
  6. Why do we need to find an LCD when adding unlike fractions?
  7. Francine says that to solve an equation containing algebraic fractions, we build each fraction to an equivalent one with the LCD. Comment on Francine’s method.
  8. When might you expect to encounter extraneous solutions?
  9. Which of the operations listed below use an LCD?
    1. add fractions
    2. multiply fractions
    3. solve an equation with fractions
    4. subtract fractions
    5. divide fractions
  10. Which of the operations listed above use building factors?

Subsection Review Problems

Exercises Exercises

1.
  1. Evaluate the fraction \(~\dfrac{s^2-s}{s^2+3s-10}~~\)for \(~s=-2\)
  2. For what value(s) of \(s\) is the fraction undefined?
Answer.
  1. \(\displaystyle \dfrac{-1}{2} \)
  2. \(-5\text{,}\) \(~2\)
2.
Ed’s Diner uses a package of coffee filters every \(x+5\) days.
  1. What fraction of a package does Ed use every day?
  2. What fraction of a package does Ed use in one week?
Exercise Group.
For Problems 3–10, reduce the fraction if possible.
3.
\(\dfrac{a+3}{b+3}\)
Answer.
Cannot be reduced
4.
\(\dfrac{5x+7}{5x}\)
5.
\(\dfrac{10+2y}{2y}\)
Answer.
\(\dfrac{5+y}{y} \)
6.
\(\dfrac{3x^2-1}{1-3x^2}\)
7.
\(\dfrac{v-2}{v^2-4}\)
Answer.
\(\dfrac{1}{v+2} \)
8.
\(\dfrac{q^5-q^4}{q^4}\)
9.
\(\dfrac{-3x}{6x^2+9x}\)
Answer.
\(\dfrac{-1}{2x+3} \)
10.
\(\dfrac{x^2+5x+6}{x^2-4}\)
Exercise Group.
For Problems 11–16,
  1. add the fractions,
  2. multiply the fractions.
11.
\(\dfrac{3}{8},~\dfrac{5}{12}\)
Answer.
  1. \(\displaystyle \dfrac{19}{24} \)
  2. \(\displaystyle \dfrac{5}{32} \)
12.
\(\dfrac{2}{x},~\dfrac{1}{x+2}\)
13.
\(\dfrac{3x}{2x+2},~\dfrac{x+1}{6x}\)
Answer.
  1. \(\displaystyle \dfrac{10x^2+2x+1}{6x(x+1)} \)
  2. \(\displaystyle \dfrac{1}{4} \)
14.
\(\dfrac{x+1}{x-1},~\dfrac{1}{x^2-1}\)
15.
\(2,~\dfrac{1}{x}\)
Answer.
  1. \(\displaystyle \dfrac{2x+1}{x} \)
  2. \(\displaystyle \dfrac{2}{x} \)
16.
\(x,~\dfrac{1}{x+2}\)
Exercise Group.
For Problems 17–22, write the expression as a single fraction in lowest terms.
17.
\(\dfrac{4c^2d}{3}\div(6cd^2)\)
Answer.
\(\dfrac{2c}{9d} \)
18.
\(\dfrac{u^2-2uv}{uv} \div \dfrac{3u-6v}{2uv}\)
19.
\(\dfrac{2m^2-m-1}{m+1}-\dfrac{m^2-m}{m+1}\)
Answer.
\(m-1 \)
20.
\(\dfrac{3}{2p}+\dfrac{7}{6p^2}\)
21.
\(\dfrac{5q}{q-3}-\dfrac{7}{q}+3\)
Answer.
\(\dfrac{8q^2-16q+21}{q(q-3)} \)
22.
\(\dfrac{2w}{w^2-4}+\dfrac{4}{w^2+4w+4}\)
23.
On Saturday mornings, Olive takes her motorboat 5 miles upstream to the general store for supplies and then returns home. The current in the river is 2 miles per hour. Let \(x\) represent Olive’s speed in still water, and write algebraic fractions to answer each question.
  1. How long does it take Olive to get to the store?
  2. How long does the return trip take?
  3. How long does the round trip take?
Answer.
  1. \(\displaystyle \dfrac{5}{x-2} \)
  2. \(\displaystyle \dfrac{5}{x+2} \)
  3. \(\displaystyle \dfrac{10x}{x^2-4} \)
24.
On spring break, Johann and Sebastian both walk from the university to the next town. Johann leaves at noon and Sebastian leaves one hour later, but Sebastian walks 1 mile per hour faster. Let \(r\) stand for Johann’s walking speed. Write polynomials to answer the following questions.
  1. What is Sebastian’s walking speed?
  2. How far has Sebastian walked at 3 pm?
  3. How far has Johann walked at 3 pm?
  4. How far apart are Johann and Sebastian at 3 pm?
Exercise Group.
For Problems 25–28, solve the equation.
25.
\(q-\dfrac{16}{q}=6\)
Answer.
\(-2, ~8\)
26.
\(\dfrac{2-x}{5x} =\dfrac{4}{15x}-\dfrac{1}{6}\)
27.
\(\dfrac{9}{m+2}+\dfrac{2}{m}=2\)
Answer.
\(\dfrac{-1}{2}, ~4 \)
28.
\(\dfrac{15}{x^2-3x}+\dfrac{4}{x}=\dfrac{5}{x-3}\)
Exercise Group.
For Problems 29–30, solve solve for the indicated variable.
29.
\(\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{z}~~~~\)for \(~z\)
Answer.
\(\dfrac{yz}{y-z} \)
30.
\(y=\dfrac{2x+3}{1-x}~~~~\)for \(~x\)
Exercise Group.
For Problems 31–32,
  1. Write an equation to model the problem.
  2. Solve your equation and answer the question posed in the problem.
31.
On a walking tour, Nora walks uphill 5 miles to an inn where she has lunch. After lunch, she increases her speed by 2 miles per hour and walks for 8 more miles. If she walked for 1 hour longer before lunch than after lunch, what was her speed before lunch?
Answer.
  1. \(\displaystyle \dfrac{5}{r}=\dfrac{8}{r+2}+1 \)
  2. 1.53 mph
32.
Brenda can fill her pool in 30 hours using the normal intake pipe. She can instead fill the pool in 45 hours using the garden hose. How long will it take to fill the empty pool if both the pipe and garden hose are running?
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