Solving & Graphing Inequalities

Open Ended

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Multiple Choice

Solve:

3p + 2(p -2) = 21

Remember to do distributive property & combine any like-terms before doing your opposite operations

• p = $$\frac{17}{5}$$

• p = 11

• p = 5

• p = 21

SOLVING INEQUALITIES

We're going to watch a short video to help us understand the basics of inequalities...

​INTRODUCING INEQUALITIES

INEQUALITY SIGNS

An inequality is like an equation, but

instead of an equal sign (=) it has one of

these signs:

< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to

SOLVING INEQUALITIES!

•SOLVING INEQUALITIES IS THE SAME AS SOLVING EQUATIONS.

•THERE ARE ONLY 2 THINGS YOU NEED TO KNOW…

• 1.) IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER, YOU MUST SWITCH THE SIGN.

• 2.) YOU WILL GRAPH YOUR SOLUTIONS.

Multiple Choice

Think of the equation x - 15 = 73

What inverse operation would you use to solve this?

• Addition

• Subtraction

• Multiplication

• Division

We can use the same inverse

operation to solve this inequality

X-15<73

Multiple Choice

Think of the equation y + 29 = 25

What inverse operation would you use to solve this?

• Addition

• Subtraction

• Multiplication

• Division

We can use the same inverse

operation to solve this inequality

y+29<25

Solving One-Step

Inequalities

Let’s try some on our
own …… ready?

Draw

Solve: x + 6 ≤ 7

Solving One-Step

Inequalities #1

Did you get:

X ≤ 1

Draw

Solve: 3 ≤ x - 5

Solving One-Step

Inequalities #1

Did you get:

8 ≤ x

SPECIAL CASE 1: SWITCHING THE SIGNS

WHEN SOLVING INEQUALITIES, IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE YOU MUST SWITCH THE SIGNS.

•SWITCHING THE SIGNS:

• LESS THAN BECOMES GREATER THAN

< SWITCHES TO >

• GREATER THAN BECOMES LESS THAN

> SWITCHES TO <

• LESS THAN OR EQUAL TO BECOMES GREATER THAN OR EQUAL TO

≤ SWITCHES TO ≥

• GREATER THAN OR EQUAL TO BECOMES LESS THAN OR EQUAL TO

≥ SWITCHES TO ≤

Division Property for Inequalities

Caution! Dividing by a
negative number

Notice: Sign
CHANGED

Same if multiplying?

Draw

Solve: -3x ≥ 15

Solving One-Step

Inequalities #1

Did you get:

x ≤ -5

Multiple Choice

Solve: g - 21 $$>$$ 17

• g < 4

• g > 4

• g < 38

• g > 38

Multiple Choice

7r $$\le$$ 35

• r $$\ge$$ 5

• r $$\le$$ 5

• r < 5

• r > 5

Multiple Choice

$$\frac{j}{-4}$$ < -6

• j < 10

• j > 24

• j < 24

• j $$\le$$ 10

Multiple Choice

w < 3

• w is greater than 3

• w is less than 3

• w is greater than or equal to 3

• w is less than or equal to 3

Multiple Choice

256 ≥ p

• 256 is less than p

• 256 is less than or equal to p

• 256 is greater than p

• 256 is greater than or equal to p

Multiple Choice

88 < 11b

• b $$\le$$ 99

• b $$\ge$$ 8

• b < 8

• b > 8

Graphing
Inequalities

<

<

<
>

<

>
<
<

Graphing Inequalities

• When we graph an inequality on a number

line we use open and closed circles to
represent the number.

<

<

Plot a closed circle

≤

≥

Plot an open circle

Graphing Inequalities

*Remember

When creating a number line, you can create it the way that you would like! Be sure that you include the number that is being graphed, 3 numbers before it, and 3 numbers after it.

Some text here about the topic of discussion

Multiple Choice

h $$\ge$$ 265

• open circle

• closed circle

Multiple Choice

w < 8

• open circle

• closed circle

x < 5

means the value of x must be

less than 5.

Think of three numbers that

are less than 5!

Numbers less than 5 are to the left

of 5 on the number line.

0

5

10

15

-20 -15 -10

-5

-25

20

25

• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
• There are also numbers in between the integers, like
2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
though, because 5 is not less than 5.

x ≥ 2

means the value of x must be

greater than or equal to 2.

Think of three numbers that are

greater than or equal to 2.

Numbers greater than 2 are to the

right of 2 on the number line.

0

5

10

15

-20 -15 -10

-5

-25

20

25

• If you said 3, 4, 5.2, 70 ½, 100, or 999 etc., you are
correct.
•The number 2 is also be a correct answer, because of
the phrase, “or equal to”.

2

Open Ended

Provide 3 solutions for c :

c > 9

Open Ended

Provide 3 solutions for k :

k $$\le$$ 49

Solving an Inequality

(Follow the same rules and steps that we

used to solve an equation.)

•Use the inverse operation to isolate the
variable.
•Remember whatever is done to one side
of the inequality must be done to the
other side. The goal is to get the variable
by itself.

Solve and Graph an Inequality

w + 5 < 8

- 5 -5

w < 3

All numbers less
than 3 are
solutions to this
problem!

0

3

6

9

-12

-9

-6

-3

-15

12

15

More Examples

r + 8 ≥ 18

-8 -8

r 10
All numbers greater than 10
(including 10)
≥

0

5

10

15

-20 -15 -10

-5

-25

20

25

More Examples

-2x <-2

-2 -2

x > 1

All numbers
greater than -1
make this problem
true!

0

1

2

3

-4

-3

-2

-1

-5

4

5

Practice # 1

Solve and graph the following inequality:

v - 18 > 32

Some text here about the topic of discussion

Practice # 2

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Practice # 3

Some text here about the topic of discussion