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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