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Date: _____________
1.3.5 Writing, Solving, and Graphing Inequalities (p.164-193)
Algebra 1
Essential Question: How is solving inequalities different from solving equations?
Exercise 1: Consider the inequality 𝑥 2 + 4𝑥 ≥ 5.
a) Find 2 positive and 2 negative values to assign to x that makes this inequality a true statement.
b) Should your four values also be solutions to the inequality 𝑥(𝑥 + 4) ≥ 5? Explain why or why not.
c) Should your four values also be solutions to the inequality 4𝑥 + 𝑥 2 ≥ 5? Explain why or why not.
d) Should your four values also be solutions to the inequality 4𝑥 + 𝑥 2 − 6 ≥ −1? Explain why or why not.
e) Should your four values also be solutions to the inequality 12𝑥 + 3𝑥 2 ≥ 15? Explain why or why not.
Example 1:
What is the solution set to the inequality 5𝑞 + 10 > 20? Express the solution set in set notation, and
graphically on the number line.
Exercise 2: Find the solution set to each inequality. Express the solution in set notation and graphically
on the number line.
𝑚
3
a) 𝑥 + 4 ≤ 7
b)
d) 6(𝑥 − 5) ≥ 30
e) 4(𝑥 − 3) > 2(𝑥 − 2)
+8≠9
c) 8𝑦 + 4 < 7𝑦 − 2
Exercise 3: Recall the discussion on all the strange ideas for what could be done to both sides of an
equation. Let’s explore some of the same issues here but with inequalities. Recall, in this lesson we have
established that adding (or subtracting) and multiplying through by positive quantities does not change
the solution set of an inequality.
a) Squaring: Do 𝐵 ≤ 6 and 𝐵2 ≤ 36 have the same solution set? If not, give an example of a number
that is in one solution set but not the other.
b) Multiplying through by a negative number: Do 5 − 𝐶 > 2 and −5 + 𝐶 > −2 have the same solution
set? If not, give an example of a number that is in one solution set but not the other.
c) Ignoring exponents: Do 𝑦 2 < 52 and 𝑦 < 5 have the same solution set?
Example 2: Jojo was asked to solve 6𝑥 + 12 < 3𝑥 + 6 for x. She answered as follows:
6𝑥 + 12 < 3𝑥 + 6
6(𝑥 + 2) < 3(𝑥 + 2)
6<3
Apply the distributive property
Multiply both sides by
1
𝑥+2
(which is the same as dividing by x + 2)
a) Since the final line is a false statement, she deduced that there is no solution to this inequality
(that the solution set is empty). What is the solution set to 6𝑥 + 12 < 3𝑥 + 6 ?
b) Explain why Jojo came to an erroneous conclusion.
Example 3: During the last exercise, we saw that when both sides were multiplied by -1 the solution set
of the inequality changed. Since we can’t multiply both sides by a -1, but we can add the same number to
both sides of an inequality without changing the solution set (i.e. If 𝐴 > 𝐵, then 𝐴 + 𝑐 > 𝐵 + 𝑐 for any real
number c), can you figure out how to use this property in a way that is helpful?
Solve −𝑞 ≥ −7 for 𝑞.
Exercise 4: Find the solution set to each inequality. Express the solution in set notation and graphically
on the number line.
a. −2𝑓 < −16
𝑥
1
b. − 12 ≤ 4
c. 6 − 𝑎 ≥ 15
d. −3(2𝑥 + 4) > 0
Exercise 5: Use the properties of inequality to show that each of the following are true for any real
numbers p and q.
a. If 𝑝 ≥ 𝑞 then −𝑝 ≤ −𝑞
𝑝≥𝑞
b. If 𝑝 < 𝑞 then −5𝑝 > −5𝑞
𝑝<𝑞
𝑝−𝑞 ≥𝑞−𝑞
__________________________
𝑝−𝑞 ≥0
𝑝−𝑝−𝑞 ≥ 0−𝑝
__________________________
5𝑝 < 5𝑞
__________________________
5𝑝 − 5𝑝 − 5𝑞 < 5𝑞 − 5𝑝 − 5𝑞
__________________________
−5𝑞 < −5𝑝
−𝑞 ≥ −𝑝
c. Based on the results above, how might we expand the multiplication property of inequality?
If 𝐴 > 𝐵, then ____________________for any negative real number 𝑘.
Exercise 6: Solve −4 + 2𝑡 − 14 − 18𝑡 > −6 − 100𝑡, for 𝑡 in two different ways: 1. first without ever
1
multiplying through by a negative number, and then 2. by first multiplying through by − 2.
𝑥
1
Exercise 7: Solve − 4 + 8 < 2, for 𝑥 in two different ways: 1. first without ever multiplying through by a
negative number, and then 2. by first multiplying through by -4.
What moves do we know do not change the solution set of an inequality?
What moves did we see that did change the solution set?
Writing and Solving Inequalities:
8) The hard drive on your computer has a capacity of 120 gigabytes (GB). You have used 85 GB. You
want to save some home videos to your hard drive. What are the possible sizes of the home video
collection you can save?
9) A club has a goal to sell at least 25 plants for a fundraiser. Club members sell 8 plants on Wednesday
and 9 plants on Thursday. What are the possible numbers of plants the club can sell on Friday to meet
their goal?
10) A cyclist takes her bicycle on a chairlift to the top of a slope. The chairlift can safely carry 680 lbs.
and the bicycle weighs 32 lb. What are the possible additional weights the chairlift can safely carry?
11) Your goal is to take at least 10,000 steps per day. According to your pedometer, you have walked
5,274 steps. Write and solve an inequality to find the possible numbers of steps you can take to reach
your goal.
12) To avoid a service fee, your checking account balance must be at least $500 at the end of each
month. Your current balance is $536.45. You use your debit card to spend $125.19. What possible
amounts can you deposit into your account by the end of the month to avoid paying the service fee?