Inequalities

Warm-UpSolve the following equations.

6y = 120

b + 15 = 24

c – 18 = 6

What is an inequality?

Definition: a mathematical statement that compares two expressions and includes an inequality symbol

5 > 2 x > 7y < 18 y ≤ 24

Inequalities can have multiple solutions (sometimes infinitely many) which makes them different from equations that only have one solution.

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What do the inequality symbols mean?

≥ means “is greater than or equal to”

< means “is less than”

> means “is greater than”

≤ means “is less than or equal to”

Key words for inequalities

> < ≥ ≤- greater

than- more than

- less than

- fewer than

- at least- no less

than

- at most- no more

than

Writing Inequalities #1Write an inequality to represent this phrase:

a number less than 5

What does your variable represent?

x < 5

The set of all numbers less than 5.

Writing Inequalities #2A pastry chef wants to spend no more than $100

on almonds. Almonds cost $4 per pound. Write an inequality to represent p, the number of pounds of almonds the chef could buy.

4p ≤ 100

Writing Inequalities #2 (continued)

Now use substitution to determine which of the following numbers are solutions to the inequality (it can be more than one!)

10.5, 25, 30

Substitute 10.5 for p

4(10.5) ≤ 100

42 ≤ 100

10.5 is a solution

Substitute 25 for p

4(25) ≤ 100

100 ≤ 100

25 is a solution

Substitute 30 for p

4(30) ≤ 100

120 ≤ 100

30 isn’t a

solution

Writing Inequalities #3You must be at least 16 years old to receive a

driver’s license. Write an inequality to represent this situation.

y ≥ 16

Warm-Up1. Which equation best represents the relationship

between x and y in the table?

a) y = x + 1 b) y = 5x c) y = 4x + 1 d) y = 5x + 1

x 0 1 2 3

y 1 6 11 16

2. Which ordered pair does NOT make the following equation true: y = x + 3?

a. (1, 4) b. (3, 9) c. (12, 15) d. (18, 21)

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Solving InequalitiesIt’s exactly like solving equations!!

While solving inequalities, just imagine that the inequality sign is an equal sign.

1.What is the variable?2.What is attached to the variable?

3.How is it attached? 4.What is the inverse operation?

5.WYDTOSDTTO says…

Addition Inequality

y + 5 ≥ 16−5 −5

y ≥ 11

What are the solutions?

Any number that is 11 or higher.

Subtraction Inequality

t − 12 < 5+ 12 +12

t < 17

What are the solutions?

Any number that is less than 17.

Multiplication Inequality

4x ≤ 1204 4

x ≤ 30

What are the solutions?

Any number that is 30 or less.

Division Inequality

q > 123

131

31

q > 36

What are the solutions?

Any number that is greater than 36.

Warm-UpWrite an equation for the following situations and

then solve.

Carrie makes $8.50 per hour for working at McDonald’s. She also got her Christmas bonus this month of $50. If she made $645 this month, how many hours did she work?

Daniel baked 50 cookies to give out to his students on the last day of school before Christmas break. If he needs to make 120 cookies, how many more does he have to bake?

Graphing Inequalities

Steps for graphing inequalities

1.Put an open or closed circle on the number in the inequality

2.Shade the number line to left for less than and right for greater than

Open or Closed Circle?Open Circle: Greater than > or Less than <

Closed Circle: Greater than or equal to ≥ or less than or equal to ≤

x > 10

x ≤ 2x ≥ 10

x < 2

10 11 1298

10 11 1298

2 3 410

2 3 410

Solving and Graphing #1Solve and graph the following inequality.

b + 13 > 17

b > 4

4 5 632

Solving and Graphing #2Solve and graph the following inequality.

≥ 12

x ≥ 24

24 25 262322

x2

Warm-Up1. Jamie has 6 apples and 7 oranges. What is the

ratio of apples to total pieces of fruit. (Write the ratio in all 3 forms: in words, with a colon, as a fraction)

2. You have to be at least 4 feet tall to ride the roller coaster. Write an inequality to represent how tall you must be to ride the roller coaster.