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B-05 Compound Inequalities: And Warm-Up · PDF file B-05 Compound Inequalities: “And” Warm-Up: Solve each of the following inequalities. Graph your solutions on a number line

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  • Name _______________________________________ Date ____________________ Hour ________

    B-05 Compound Inequalities: “And”

    Warm-Up: Solve each of the following inequalities. Graph your solutions on a number line. Write the

    interval notation for each solution set. Some are basic inequalities, and some are compound “or”

    inequalities. There might be special solutions in this warm-up.

    1. 2(𝑥 − 4) ≤ −12 2. 3𝑥 + 7 ≥ 2(11 − 𝑥)

    3. 8 ≤ 4(𝑥 − 9) or −2(𝑥 + 3) > −6 4. 𝑥−7

    3 > −3 or 2(4𝑥 − 5) < 46

  • Last class we talked about compound “or” inequalities; today we will focus on “and” inequalities. The

    following is an example of a completely solved “and” inequality.

    −2 < 𝑥 < 4

    Remember that compound inequalities are made up of two inequalities. What two separate

    inequalities make up the “and” compound above?

    Graph the above separate inequalities on the same number line:

    Solutions have to work for both parts of the inequality, so they will lay wherever ______________________

    _________________________. You need to shade this portion of the graph to indicate where the solutions

    are. Go back and shade the portion of the graph above where the solutions for both inequalities are.

    Solving a compound “and” inequality is relatively easy. The goal is to isolate the variable

    _________________________________, using reverse PEMDAS. Any operation you do to the middle of the

    inequality, you must do to __________________________ of the inequality as well.

    You will need to provide interval notation for “and” compound inequalities. Since these inequalities

    typically have solutions that are restricted between two numbers, your interval notation will most likely

    be between two numbers. It will look like a coordinate point, but remember that interval notation is

    not a coordinate point!

    Let’s try solving a few “and” inequalities! For each example, you should solve each inequality and

    graph your solutions. We’re also going to pick one solution and one non-solution.

    Example 1) −3 < 𝑥 − 4 < −1 Solution:

    Non-solution:

    Example 2) −3 < −2𝑥 + 5 < 9 Solution:

    Non-solution:

  • Example 3) −3 ≤ 𝑥

    2 < 0 Solution:

    Non-solution:

    Example 4) 𝑥 + 8 ≥ 9 𝑎𝑛𝑑 𝑥

    7 ≤ 1 Solution:

    Non-solution:

    Example 5) −33 ≤ −7𝑥 − 12 < −26 Solution:

    Non-solution:

    Example 6) −4 ≤ − 3

    4 𝑥 + 11 > 14 Solution:

    Non-solution:

  • Special solutions can occur with compound “and” inequalities. You will find the special solution after

    graphing the inequality. Let’s try a few!

    Example 7) 19 ≤ 4 − 3𝑥 < 1

    Example 8) 5 ≤ 2𝑥 − 1 ≤ 3

    Notice that both of our special solutions are _______________________________. We will never get

    ________________________________ for a compound “and” inequality.

    Conclusions…

    When solving a single inequality, you have three options:

    1)

    2)

    3)

    When solving a compound “or” inequality, you have two options:

    1)

    2)

    When solving a compound “and” inequality, you have two options:

    1)

    2)

  • Name ___________________________________________ Date ___________________ Hour _______________

    B-05 Compound Inequalities: “And” Homework

    Directions:

    a) Solve each inequality.

    b) Graph your solutions.

    c) Provide interval notation for each solution set.

    If there is no solution, you have to write that as your final answer.

    1) 8 < 3𝑥 − 1 ≤ 11

    2) −9 < 𝑥 − 10 < −5

    3) −4 ≤ 3𝑛 + 5 < 13

  • 4) −47 > 1 − 8𝑥 > −63

    5) 1 ≤ 𝑥

    8 ≤ 0

    6) −16 ≤ 2𝑥 − 10 ≤ −22

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