CONFIDENTIAL 1 Algebra1 Graphing and Writing Inequalities.

CONFIDENTIAL 1

Algebra1Algebra1

Graphing and Graphing and WritingWriting

InequalitiesInequalities

CONFIDENTIAL 2

Warm UpWarm Up

1) 2b - 6 = b + 3

3) 2 (y + 1) = 2y + 1

4) x + (x + 1) + (x + 2)

5) 5 + (x + 3) + 5 + 2 (x + 3)

Solve each equation.

2) -3 (2 - x) = 5x + 2

Simplify each expression.

1) b = 9 2) b = 9

3) contradictory statement

4) 3x + 3

5) 3x + 19

CONFIDENTIAL 3

Inequalities

An inequality is a statement that two quantities are not equal. The quantities are compared by using one

of the following signs:

A <  B

A >  B

A ≤  B

A ≥  B

A ≠ B

Where, A and B are some integer value.

CONFIDENTIAL 4

Any number that makes an inequality true is a solution of the inequality.

For example: -4 is a solution of y ≥ -5because -4 ≥ -5.

Inequalities, like equations, can be true, false or open.

15 > 11

6 < 3

a < 20

This sentence is true.

This sentence is false .This sentence is open. It is neither true nor false until “a" replaced with a number.

CONFIDENTIAL 5

< > ≤ ≥ ≠ Less than Fewer than

Greater than More than Exceeds

Less than or equal to No more than At most

Greater than or equal to No less than At least

A is not equal to B.

Many situations in real life can be described using inequalities. The table below shows some of the

common phrases and corresponding inequalities.

CONFIDENTIAL 6

Describe the solutions of 3 + x < 9 in words.

Identifying Solutions of Inequalities

Test values of x that are positive, negative, and 0.

When the value of x is a number less than 6, the value of 3 + x is less than 9.

When the value of x is 6, the value of 3 + x is equal to 9.

When the value of x is a number greater than 6, the value of 3 + x is greater than 9.

x -2.75 0 5.99 6 6.01

3 + x -5.75 3 8.99 9 9.01

3 + x < 9 -2.75 < 9 3 < 9 8.99 < 9 9<9 9.01< 9

Solution? Yes Yes Yes No No

CONFIDENTIAL 7

1. Describe the solutions of 2p > 8 in words.

Now you try!

1) p > 4. When the value of p is a number greater than 4, the value of 2p > 8 is greater than 8.

CONFIDENTIAL 8

An inequality like 3 + x < 9 has too many solutions to list. You can use a graph on a number line to

show all the solutions.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < 6

The solutions are shaded and an arrow shows that the solutions continue past those shown on the graph. To show that an endpoint is a solution, draw a solid circle

at the number. To show that an endpoint is not a solution, draw an empty circle.

CONFIDENTIAL 9

Graphing Inequalities

WORDS ALGEBRA

All real numbers less than 5 x < 5

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < 5

All real numbers greater than -1 x > -1

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x > -1

CONFIDENTIAL 10

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x ≥ 0

WORDSALGEBRA

All real numbers greater than or equal to 0 x ≥ 0

All real numbers less than or equal to 1 x ≤ 1 2 2

0 1 2 3 4 5 6-5 -4 -3 -2 -1

x ≥ 1 2

CONFIDENTIAL 11

Graphing Inequalities

The solution of inequality x < 9 represented on the number line is

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < 9

A) Solve: Graph the solution of he inequality, x < 9 on the number line.

An open dot shows that 9 is not a solution. Draw an arrow pointing to the left.

Shade all points to the left of 9.

CONFIDENTIAL 12

The solution of inequality x ≥ -3 represented on the number line is

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x ≥ -3

B) Solve: Graph the solution of the inequality, x ≥ -3 on the number line.

An solid circle shows that -3 is a solution. Draw an arrow pointing to the right.

Shade all points to the right of -3.

CONFIDENTIAL 13

Graph each inequality.

Now you try!

1) c > 2.5 2) 22 - 4 ≥ w

2.5

0 1 2 3 4 5 6-5 -4 -3 -2 -1

2.5

1)

2)2.5

0 1 2 3 4 5 6-5 -4 -3 -2 -1

0

CONFIDENTIAL 14

Writing an Inequality from a Graph

Write the inequality shown by each graph.

2.5

0 1 2 3 4 5 6-5 -4 -3 -2 -1

Use any variable. The arrow points to the right, so use either > or ≥. The empty circle at 4.5 means that 4.5 is

not a solution, so use >.

x > 2.5

A)

2.5

x > 2.5

CONFIDENTIAL 15

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

B)

Use any variable. The arrow points to the left, so use either < or ≤. The solid circle at -5 means that -5 is a

solution, so use ≤.

x ≤ -5

- 5

CONFIDENTIAL 16

1. Write the inequality shown by the graph.

Now you try!

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

- 0.5

The open dot and the left arrow from point -0.5 shows the value x < -0.5

CONFIDENTIAL 17

Solution:

A) The members of a lightweight crew team can weigh no more than 165 pounds each. Define a variable and write

an inequality for the acceptable weights of the team members. Graph the solutions.

Sports Application

Let w represent the weights that are allowed.

i.e., w ≤ 165

Stop the graph at 0 because a person’s weight must be a positive number.

105 120 135 150 1650 15 30 45 60 75 90

CONFIDENTIAL 18

1. A store’s employees earn at least $8.25 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions.

Now you try!

1) x ≥ 8.25

2 3 4 5 6 7 8 9 10-1 0 1

8.25

CONFIDENTIAL 19

Assessment

Describe the solutions of each inequality in words.

1) g - 5 ≥ 6 2) -2 < h + 1

3) 20 > 5t

1) g ≥ 11. When the value of g is a number greater than 11, the value of g - 5 is greater than 9.

2) h > -3. When the value of h is a number greater than -3, the value of h + 1 is greater than -2.

3) t < 4. When the value of t is a number less than 4, the value of 5t is less than 20.

CONFIDENTIAL 20

Graph each inequality.

4) x < -5 5) (4 - 2)3 > m

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

4)

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

5)

-5-5

8

CONFIDENTIAL 21

Write the inequality shown by each graph.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

6) x ≤ - 5

6)

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

7) x > 2.5

7)

CONFIDENTIAL 22

Define a variable and write an inequality for each situation. Graph the solutions.

8) There must be at least 20 club members present in order to hold a meeting.

9) A trainer advises an athlete to keep his heart rate under 140 beats per minute.

10) The maximum speed allowed on Main Street is 25 miles per hour.

8) x ≥ 20

9) x < 140

10) x ≤ 25

CONFIDENTIAL 23

Inequalities

An inequality is a statement that two quantities are not equal. The quantities are compared by using one

of the following signs:

A <  B

A >  B

A ≤  B

A ≥  B

A ≠ B

Where, A and B are some integer value.

Let’s review

CONFIDENTIAL 24

< > ≤ ≥ ≠ Less than Fewer than

Greater than More than Exceeds

Less than or equal to No more than At most

Greater than or equal to No less than At least

A is not equal to B.

Many situations in real life can be described using inequalities. The table below shows some of the

common phrases and corresponding inequalities.

CONFIDENTIAL 25

Describe the solutions of 3 + x < 9 in words.

Identifying Solutions of Inequalities

Test values of x that are positive, negative, and 0.

When the value of x is a number less than 6, the value of 3 + x is less than 9.

When the value of x is 6, the value of 3 + x is equal to 9.

When the value of x is a number greater than 6, the value of 3 + x is greater than 9.

x -2.75 0 5.99 6 6.01

3 + x -5.75 3 8.99 9 9.01

3 + x < 9 -2.75 < 9 3 < 9 8.99 < 9 9<9 9.01< 9

Solution? Yes Yes Yes No No

CONFIDENTIAL 26

An inequality like 3 + x < 9 has too many solutions to list. You can use a graph on a number line to

show all the solutions.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < 6

The solutions are shaded and an arrow shows that the solutions continue past those shown on the graph. To show that an endpoint is a solution, draw a solid circle

at the number. To show that an endpoint is not a solution, draw an empty circle.

CONFIDENTIAL 27

Graphing Inequalities

The solution of inequality x < 9 represented on the number line is

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < 9

A) Solve: Graph the solution of he inequality, x < 9 on the number line.

An open dot shows that 9 is not a solution. Draw an arrow pointing to the left.

Shade all points to the left of 9.

CONFIDENTIAL 28

The solution of inequality x ≥ -3 represented on the number line is

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x ≥ -3

B) Solve: Graph the solution of the inequality, x ≥ -3 on the number line.

An solid circle shows that -3 is a solution. Draw an arrow pointing to the right.

Shade all points to the right of -3.

CONFIDENTIAL 29

Writing an Inequality from a Graph

Write the inequality shown by each graph.

2.5

0 1 2 3 4 5 6-5 -4 -3 -2 -1

Use any variable. The arrow points to the right, so use either > or ≥. The empty circle at 4.5 means that 4.5 is

not a solution, so use >.

x > 2.5

A)

2.5

x > 2.5

CONFIDENTIAL 30

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

B)

Use any variable. The arrow points to the left, so use either < or ≤. The solid circle at -5 means that -5 is a

solution, so use ≤.

x ≤ -5

- 5

CONFIDENTIAL 31

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