Topic 13: Writing and Solving Inequalities
    for use after
    Moving Straight Ahead
    Investigation 3
    You just explored what it means for two quantities to be equal. When you
    write an equation you are comparing the value of two equal quantities.
    Sometimes you need to compare two quantities that are not equal.
    An
    is a mathematical sentence that compares the values of two
    expressions that are not equal. Instead of using an equal sign, you use an
    inequality symbol.
    Inequality Symbols
    ,
    less than
    .
    greater than
    #
    less than or equal to
    $
    greater than or equal to
    2
    not equal to
    Writing Inequalities
    A. For each situation, first define a variable. Then represent the situation
    with an inequality.
    1. The height of a child must be at least 48 inches to ride a
    roller coaster.
    2. The speed limit on the road is less than or equal to 45 miles
    per hour.
    3. A piece of luggage must be less than 60 pounds.
    4. You must be at least 13 years old to see a movie rated PG-13.
    Exercises
    For each situation, define a variable. Then write an inequality to model
    each situation.
    1. Ana has a car. She wants to limit herself to driving at most 500 miles
    per month.
    2. The Simon family’s car emits 0.75 pounds of CO
    2
    per mile. It emits
    2 pounds of CO
    2
    when it is started. The Simons want to limit their
    emissions to at most 100 pounds of CO
    2
    per use of the car.
    3. An online discount costs $50 per month. It decreases $2 for every
    person you recommend to sign-up. You want to keep the total cost
    below $35.
    inequality
    Topic 13: Writing and Solving Inequalities
    Teaching Guide
    Mathematical Goals
    • Write inequalities to represent situations
    • Solve inequalities in one variable
    Vocabulary
    •
    inequality
    At a Glance
    Students should have a clear understanding of equations before they start
    working with inequalities, as many of the same rules apply. While students
    should be familiar with the similarities between equations and inequalities,
    it is also crucial that they recognize the differences. Remind students
    throughout the lesson to use the correct inequality symbols since some
    students may use equal signs out of habit.
    Many students will also need to practice writing verbal statements as
    algebraic expressions. Sufficient time should be spent familiarizing students
    with the meanings of the inequality symbols. One possible classroom
    activity is to divide the students into pairs and instruct them to take turns
    writing situations for their partner to translate into algebraic expressions.
    When solving inequalities, many students struggle with solving by
    multiplication and division. You may choose to do several examples as a
    class to highlight the importance of reversing the inequality sign when
    multiplying or dividing by a negative number.
    After Problem 13.1, ask:
    •
    What are some common phrases that are used to describe “greater than
    or equal to”? To describe “less than or equal to”?
    •
    How would you rewrite an inequality with the variable on the opposite
    side of the inequality symbol?
    Explain to students what it means for a number to satisfy an inequality,
    or be a solution to the inequality. Have students find several different
    solutions to the same inequality. Point out to students the difference
    between the solutions for pairs of inequalities like
    x
    , 2 and
    x
    # 2. Then
    have students discuss what the solutions of the inequality mean in a real-
    world situation.
    Summarize the example before Problem 13.2 by asking:
    •
    What values could you use to check your solution to the inequality?
    During Problem 13.2, ask:
    •
    What information does the number 825 represent?
    •
    What information does the number 3.25 represent?
    •
    For which variable in the inequality should you substitute 2,200?
    Homework Check
    When reviewing Exercise 6, ask:
    •
    What was your first step in solving this inequality?
    •
    Could you still get the correct answer without doing this first?
    PACING
    1 day
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    Assignment Guide for Topic 13
    Core
    Problem 13.1 Exercises 1–3, Problem
    13.2–13.3 Exercises 1–6
    Advanced
    Problem 13.2–13.3 Exercise 7
    Answers to Topic 13
    Problem 13.1
    A. 1.
    Let
    h
    5 the child’s height in inches;
    h
    ? 48.
    2.
    Let
    s
    5 speed in miles per hour;
    s
    ? 45.
    3.
    Let
    w
    5 the weight of the piece of luggage
    in pounds;
    w
    ? 60.
    4.
    Let
    a
    5 your age in years;
    a
    ? 13.
    Exercises
    1.
    Let
    m
    5 the number of miles Ana drives;
    m
    ? 500
    2.
    Let
    m
    5 the number of miles the Simons
    drive; 2 1 0.75
    m
    ? 100
    3.
    Let
    p
    5 the number of people you
    recommend; 50 2 2
    p
    ? 35
    Problem 13.2
    A. 1.
    yes
    2.
    yes; no
    3.
    Inequalities remain true when you add or
    subtract each side by the same number, and
    when you multiply or divide by a positive
    number.
    B.
    Yes; Sally reversed the inequality symbol
    when she divided each side by a negative
    number.
    Problem 13.3
    A. 1.
    2,200 ? 825 1 3.25
    n
    2.
    n
    ? 423.08; the bakery must make fewer
    than 423 cakes this month.
    B.
    Check students’ work.
    Exercises
    1.
    x
    ? 10
    2.
    x
    ? 4
    3.
    t
    ? 6.5
    4.
    m
    ? –45
    5.
    c
    ?
    6.
    d
    ? 203.95
    7. a.
    1.25
    m
    ? 600
    b.
    m
    ? 480; Vince’s family must drive no more
    than 480 miles per month.
    c.
    Check students’ work; students should
    show that they checked at least one value
    that makes the inequality true, as well as
    one that makes the inequality false.
    22
    2
    3
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