Graphing and WritingInequalities

with One VariableDAY 9

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When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.

Important Words

Sample Sentence

Equivalent Translation

is more than Trenton is more than 10 miles away.

t > 10

is greater than A is greater than B.

A > B

must exceed The speed must exceed 25 mph.

The speed is greater than 25 mph.

s > 25

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.

Important Words

Sample Sentence

Equivalent Translation

cannot exceed Time cannot exceed 60 minutes.

Time must be less than or equal to 60 minutes.

t < 60

is at most At most, 7 students were late for class.

Seven or fewer students were late for class.

n < 7

is at least Bob is at least 14 years old.

Bob's age is greater than or equal to 14.

B > 14

How are these inequalities read?

2 + 2 > 3 Two plus two is greater than 3

2 + 2 ≥ 4 Two plus two is greater than or equal to 4

2 + 2 < 5 Two plus two is less than 5

2 + 2 ≤ 5 Two plus two is less than or equal to 5

2 + 2 ≤ 4 Two plus two is less than or equal to 4

2 + 2 > 3 Two plus two is greater than or equal to 3

Writing inequalities

Let's translate each statement into an inequality.

x is less than 10

20 is greater than or equal to y

x < 10

words

inequality statement

translate to

20 > y

You try a few:

1. 14 is greater than a

2. b is less than or equal to 8

3. 6 is less than the product of f and 20

4. The sum of t and 9 is greater than or equal to 36

5. 7 more than w is less than or equal to 10

6. 19 decreased by p is greater than or equal to 2

7. Fewer than 12 items

8. No more than 50 students

9. At least 275 people attended the play

Do you speak math?

Change the following expressions from English into math.

Double a number is at most four.

Three plus a number is at least six.

2x ≤ 4

3 + x ≥ 6

Answer

Answer

Five less than a number is less than twice that number.

The sum of two consecutive numbers is at least thirteen.

Three times a number plus seven is at least nine.

x - 5 < 2x

x + (x + 1) ≥ 13

3x + 7 > 9

Answer

Answer

Answer

0 1 2 3 4 5 6 7 8 9 10

7.5

$7.50

7.5

at least

>

An employee earns

e

A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour.

Let e represent an employee's wages.

Try this:

The speed limit on a road is 55 miles per hour. Define a variable and write an inequality.

118 You have $200 to spend on clothes. You already spent $140 and shirts cost $12. Which equation shows this scenario?

A 200 < 12x + 140B 200 12x + 140

C 200 > 12x + 140

D 200 12x + 140

≤

≥

119 A sea turtle can live up to 125 years. If one is already 37 years old, which scenario shows how many more years could it live?

125 < 37 + x

125 37 + x≤

A

BC 125 > 37 + x

D 125 37 + x≥

120 The width of a rectangle is 3 in longer than the length. The perimeter is no less than 25 inches.

A 4a + 6 < 25B 4a + 6 25

C 4a + 6 > 25

D 4a + 6 ≥ 25

≤

121 The absolute value of the sum of two numbers is less than or equal to the sum of the absolute values of the same two numbers.

ABCD

A solution to an inequality is NOT a single number. It will have more than one value.

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

This would be read as the solution set is all numbers greater than or equal to negative 5.

Solution Sets

Let's name the numbers that are solutions of the given inequality.

r > 10 Which of the following are solutions? {5, 10, 15, 20}

5 > 10 is not trueSo, not a solution

10 > 10 is not trueSo, not a solution

15 > 10 is trueSo, 15 is a solution

20 > 10 is trueSo, 20 is a solution

Answer:{15, 20} are solutions of the inequality r > 10

Let's try another one.

30 ≥ 4d; {3, 4, 5, 6, 7, 8}

30 ≥ 4d30 ≥ (4)330 ≥ 12

30 ≥ 4d30 ≥ (4)430 ≥ 16

30 ≥ 4d30 ≥ (4)530 ≥ 20

30 ≥ 4d30 ≥ (4) 630 ≥ 24

30 ≥ 4d30 ≥ (4)730 ≥ 28

30 ≥ 4d30 ≥ (4)830 ≥ 32

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Graphing Inequalities - The Circle

An open circle on a number shows that the number is not part of the solution. It is used with "greater than" and "less than". The word equal is not included.< >

A closed circle on a number shows that the number is part of the solution. It is used with "greater than or equal to" and "less than or equal to". < >

Graphing Inequalities - The Arrow

The arrow should always point in the direction of those numbers who satisfy the inequality.

*If the variable is on the left side of the inequality, then < and ≤ will show an arrow pointing left.

*If the variable is on the left side of the inequality, then > and ≥ will show an arrow pointing right.

Notice that < and ≤ look like an arrow pointing left and that > and ≥ look like an arrow pointing right.

But what if the variable isn't on the left? Do the opposite of where the inequality symbol points.

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

INEQUALITY SYMBOLS

DOT AND DIRECTION

<>

What is the number in the inequality?

What kind of circle should be used?

In what direction does the line go?

Graphing Inequalities

Step 1: Rewrite this as x < 5.

Step 2: What kind of circle? Because it is less than, it does not include the number 5 and so it is an open circle.

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

Graphing Inequalitiesx is less than 5

Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.

Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left.

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

x < 5

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

Step 1: Rewrite this as x ≤ 5.

Step 2: What kind of circle? Because it is less than or equal to, it does include the number 5 and so it is a closed circle.

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

Graphing Inequalitiesx is less than or equal to 5

Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.

Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left.

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

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

x ≤ 5

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

You tryGraph the inequalityx > 2

Graph the inequality -3 > x

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

click 2 on the number line for answer

click -3 on the number line for answer

.05.05

Try these.Graph the inequalities.

1. x > -3

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

2. x < 4

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

.05.

Try these.State the inequality shown.1.

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

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

2.

122 This solution set would be x > -4.

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

A TrueB False

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

123

A x > 3

B x < 3

C x < 3

D x > 3

State the inequality shown.

5 6 7 8 9 10 11 12 13 14 15

124

A 13 < x

B 13 > x

C 13 > x

D 13 < x

State the inequality shown.

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

125

A x > -1

B x < -1

C x < -1

D x > -1

State the inequality shown.

-1-5 1 50-2-3-4 2 3 4

126

A -4 < x

B -4 > x

C -4 < x

D -4 > x

State the inequality shown.