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Pre-Algebra View details

5. Writing and Graphing Inequalities

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Pre-Algebra /

Chapter 6

Writing and Graphing Inequalities

This educational lesson focuses on the concept of inequalities and how they can be represented both in writing and graphically. It teaches how to use a number line to visualize the solution set of an inequality. The lesson also emphasizes the importance of understanding the different types of inequalities and how they can be applied in various real-world scenarios. For example, inequalities can be used to determine the range of acceptable values for a certain variable, such as the amount of weight a bridge can hold or the speed limit on a particular road. Overall, the lesson aims to provide a thorough understanding of inequalities, making them accessible and useful in everyday life.

Lesson Settings & Tools

	12 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 12

It is natural to compare quantities or mathematical expressions. For example, a person might compare the amount of money in their pocket to the price of a bar of chocolate. Here, the price of the chocolate can be seen as a constraint. In this lesson, real-life problems that involve constraints will be modeled by inequalities.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Inverse Operations
Equation
Number Line

Challenge

Fishing Day With Dad

Kevin loves to go fishing with his father. They usually fish on a nearby lake. At the lake, small fishing boats are rented to groups of tourists for daily fishing trips.

For safety reasons, each boat can carry at most $600$ pounds of weight. Additionally, each boat can hold a maximum of five people.

a Let

w

represent the total weight of a group of people wishing to rent a boat. What mathematical sentence can be written to represent all possible total weights for a boat? Draw a number line that shows solutions to the mathematical sentence.

b Let

p

represent the total number of people going on the boat. What mathematical sentence can be written to represent the number of people that a boat can hold? Draw a number line that shows solutions to the mathematical sentence.

Discussion

Inequality

An inequality, like an equation, is a mathematical statement that compares two quantities. An inequality contains the symbols $<,$ $>,$ $\leq,$ or $\geq .$ There are several ways each inequality can be phrased.

Inequality Symbol	Key Phrases
$<$	$∙ is less than or equal toter ∙ is fewer than$
$>$	$∙ is greater than or equal to ∙ is more than$
$\leq$	$∙ is less than or equal to ter ∙ is at most ∙ is no more than$
$\geq$	$∙ is greater than or equal to ∙ is at least ∙ is no less than$

With an inequality, it is possible to compare any combination of two numbers, variables, numeric expressions, or algebraic expressions.

Symbol	Example	Meaning
$<$	$x < 1$	The variable $x$ is less than $1 .$
$\leq$	$x + 1 \leq - 3$	The algebraic expression $x + 1$ is less than or equal to $- 3 .$
$>$	$2 x - 5 > 5$	The expression $2 x - 5$ is greater than $5 .$
$\geq$	$x \geq 2 x + 1$	The variable $x$ is greater than or equal to the expression $2 x + 1 .$

As shown in the table, some inequalities indicate that the two quantities are not necessarily equal, whereas others indicate that they are strictly never equal.

This lesson will focus on inequalities of the following forms, where $a$ is a number.

x < a, x > a, x \leq a, and x \geq a

Discussion

Strict and Non-Strict Inequality

An inequality that compares two quantities that are strictly not equal is called a strict inequality. There are two types of strict inequalities.

Less Than : Greater Than : < >

The boundary values in strict inequalities are not included in the solution set. On the other hand, an inequality that compares two quantities that are not necessarily different is called a non-strict inequality. There are two types of non-strict inequalities.

Less Than or Equal To : Greater Than or Equal To : \leq \geq

The boundary values in non-strict inequalities are included in the solution set.

Example

Fishing Time

Kevin and his father manage to find a boat so that the two of them can go fishing.

Kevin and his dad are fishing in a boat. Kevin fishes while his dad rows. There are a few fish in the water.

a Kevin excitedly tells his father that he wants to catch at least

7

fish. Write an inequality that represents the number of fish Kevin want to catch.

b Kevin's father warns Kevin about the lengths of the fish. If they catch a gray mullet, its length must be longer than

11

inches. If it is not that long, they must return the fish to the lake. Write an inequality that represents the lengths of gray mullets they are allowed to keep.

Hint

a Which inequality symbol represents at least?

b Which inequality symbol represents more than?

Solution

a The situation can be described in one sentence as follows.

The number fish Kevin wants to catch is at least 7 .

Let

f

be the number of fish Kevin wants to catch. The key phrase in the sentence is is at least. This phrase represents the inequality symbol

\geq .

Then, the inequality can be written as follows.

The number fish Kevin wants to catch f is at least \geq 7 . 7

Therefore, the inequality is

f \geq 7 .

b Like in Part A, start by describing the situation in one sentence.

The length of gray mullets must be longer than 11 inches .

Let

g

be the length of the gray mullet they are allowed to keep. The key phrase in this case is longer than. This phrase can also be interpreted as more than. It follows that the inequality symbol to use is

> .

Then, the inequality could be written as follows.

The length of gray mullets must be longer than 11 inches . ⇓ g > 11

The inequality is

g > 11 .

Discussion

Solution Set of an Inequality

A solution of an inequality is any value of the variable that makes the inequality true. As an example, consider the following inequality.

2 x - 3 < 5

Notice that if

0

is substituted for

x

in the inequality, the inequality holds true. Therefore, it can be said that

0

is a solution to the given inequality.

2 (0) - 3 < ? 5 \Rightarrow - 3 < 5 ✓

However, this is not the only value that makes the inequality true. There are other

x -

values like

1

and

2

that make it true. The set of all possible values that satisfy an inequality is the solution set of an inequality.

Pop Quiz

Is the Given Number a Solution of the Inequality?

The applet shows an inequality of the form $x < a,$ $x > a,$ $x \leq a,$ or $x \geq a .$ Determine if the given number is a solution to the inequality shown.

Discussion

Graphing an Inequality on a Number Line

Inequalities can have an unlimited number of solutions — in other words, they might have infinitely many solutions. In such cases, number lines can be useful for showing all the values that make the inequality true. Consider graphing the solution set of the following inequality.

x \geq - 6

The inequality is read as

x

is greater than or equal to

- 6 .

It is a non-strict inequality, so

x = - 6

is a solution. There are two possible cases when representing a number on a number line.

If a number is a solution, a closed circle $(∙)$ is used.
If a number is not a solution, an open circle $(\circ)$ is used.

For the given inequality, a closed circle $(∙)$ is placed at $- 6$ because it is a solution.

Every value of $x$ greater than $- 6$ has to be included in the graph. Since greater numbers lie to the right on the number line, this is graphed as an arrow pointing to the right.

This is the graph of the inequality. Examine the graphs of solutions sets of different inequalities.

As can be seen, an open circle is used if the inequality is strict. A closed circle is used if the inequality is non-strict.

Example

How Long Did It Take to Fish?

Kevin caught as many fish as he wanted in less than $5$ hours.

a Write an inequality to describe the amount of time it took him to catch the fish.

b Graph the inequality on a number line.

Answer

a Inequality:

t < 5

b Graph:

Hint

a Which inequality symbol represents is at least?

b Is it a strict or non-strict inequality?

Solution

a It took less than

5

hours for Kevin to catch as many fish as he wanted.

The amount of time spent fishing is less than 5 hours .

Let

t

be the amount of time that Kevin spent fishing. The key phrase in the sentence, is less than, is represented by the symbol

< .

This means that the inequality can be written as follows.

The amount of time spent fishing t is less than < 5 hours . 5

The inequality

t < 5

represents the situation.

b Now the number line graph for the inquality from Part A will be drawn.

t < 5

The inequality is read as

t

is less than

5 .

This is a strict inequality, so

x = 5

is not a solution. Since

5

is not a solution, an open circle

\circ

is used at that point.

Every value of $t$ less than $5$ has to be included in the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.

This is the graph of the inequality.

Example

How Far Can Kevin Throw a Stone?

After sitting in the boat for so long, Kevin wants to go for a walk by the lake. He is curious about how far he and his father can throw stones.

Kevin and his father throw stones into the lake.

a The graph shows the distances in yards that Kevin can throw a stone. Write the inequality that the graph represents.

b The graph shows the distances in yards that Kevin's father can throw a stone. Write the inequality that the graph represents.

Hint

a Determine the type of circle used in the graph. Does this give any information about the inequality symbol?

b Determine the type of circle used in the graph. Does this give any information about the inequality symbol?

Solution

a Use the variable

k

to represent the distances that Kevin can throw a stone. Take a look at the given graph.

Notice that there is a closed circle on the graph at

17 .

This means that

k = 17

is a solution of the inequality. This also means that the graph represents a non-strict inequality.

A closed circle (∙) is used . ⇓ Inequality symbol is either \leq or \geq .

The arrow pointing toward the left indicates that all values less than

17

are also part of the solution set. Therefore, the inequality would be

k

is less than or equal to

17 .

This can be expressed algebraically as follows.

k is less than or equal to 17 . ⇓ k \leq 17

In the context of the question, this inequality means that Kevin can throw a stone at most

17

yards.

b This time, let the variable

f

be the distances that Kevin's father can throw a stone. Consider the given graph.

An open circle is placed at

35,

f = 35

is not a solution of the inequality. This means that the graph represents a strict inequality.

An open circle (\circ) is used . ⇓ Inequality symbol is either < or > .

The arrow pointing toward the left of the line indicates that all values less than

35

are part of the solution set. Therefore, the inequality would be

f

is less than

35 .

This can be expressed algebraically as follows.

f is less than 35 . ⇓ f < 35

This inequality means that Kevin's father can throw a stone no farther than

35

yards.

Pop Quiz

Writing an Inequality From Its Graph

Examine the given graph and determine its inequality.

Graph of an inequality and four possible inequalities

Closure

Father and Son Fishing Adventure

Similar to equations, inequalities are mathematical expressions. Inequalities are useful for modeling a constraint or condition in a real-world problem. Consider the situation presented at the beginning of the lesson. Boats are rented to groups of tourists on the lake where Kevin and his father go fishing.

Two facts are known about the boats. Each boat can carry up to $600$ pounds and hold up to five people.

a Let

w

represent the total weight of a group of people wishing to rent a boat. Write and graph an inequality to represent all possible total weights for the boat.

b Let

p

represent the total number of people that are getting on a boat. Write and graph an inequality to represent the number of people that the boat can hold.

Answer

a Inequality:

w \leq 600

Graph:

b Inequality:

p \leq 5

Graph:

Hint

a Determine the inequality symbol that would mean up to. Is the given number a solution to the inequality?

b If a number is a solution, use a closed point on the graph. If a number is not a solution, use an open point.

Solution

a The total weight of a group of people is

w

and each boat can carry up to

600

pounds. This means that the weight of a group that can be held in the boat is represented as the following inequality.

The total weight of a group of people is at most 600 pounds . ⇓ w \leq 600

The inequality

w \leq 600

represents the situation. It is a non-strict inequality, so

x = 600

is a solution. Since

600

is a solution, use a closed circle

(∙)

at thsi point on the number line.

Every value of $w$ less than $600$ has to be included in the solution set on the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.

This is the graph of the inequality.

b The number of people

p

that can fit on each boat is at most

5

people. Then the number of people that the boat can hold is represented as the following inequality.

The number of people is at most 5 . ⇓ p \leq 5

The inequality

p \leq 5

represents this situation. This is also non-strict inequality, so

x = 5

is a solution. Since

5

is a solution, a closed circle

(∙)

is used on the graph.

Every value of $p$ less than $5$ has to be included in the solution set. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.

In the examples solved throughout this lesson, the variables are considered as real numbers and conditions other than those specified in the examples are ignored. For example, a negative number of people or any partial number of people would not make sense in Part B of this exercise. These constraints can be added to the solution set.

p \leq 5, where p is a non - negative integer

Now see what would happen to the graph if these additional constraints were added to the solution set.

Now go through the examples again and determine in which examples negative values are meaningless or only integers make sense for the solution sets.

Level 1

Level 2

Level 3

Writing and Graphing Inequalities

Exercise 1.1

Write the given sentence as an inequality.

A number q is no less than - 12 .

A number z is less than 13 .

Let's recall some key phrases that we can replace with inequality symbols.

Key Phrases	Symbol
& ∙ is less than & ∙ is fewer than	<
& ∙ is greater than & ∙ is more than	>
& ∙ is less than or equal to & ∙ is at most & ∙ is no more than	≤
& ∙ is greater than or equal to & ∙ is at least & ∙ is no less than	≥

The key phrase in the given sentence is is no less than. We can replace this phrase with the inequality symbol ≥. Let's write the sentence as an inequality. A numberq & is no less than & - 12. q & ≥ & - 12 The inequality is q ≥ - 12.

Let's look for the key phrases and determine where we need to place the inequality symbol. A numberz is less than 13. We can see that the sentence contains the key phrase is less than. We can use the inequality symbol < for ths phrase. A numberz & is less than & 13. z & < & 13

Use substitution to determine which value(s) from the given set makes the given inequality true. Select all that apply.

Inequality : Set : t < 10 {9.1, 10.1, 11}

Inequality : Set : y \geq 5 {0, 5, 10}

Let's start by reading the given inequality. Inequality: t < 10 It is read as t is less than 10. This means that all values of t less than 10 make the inequality true. When we look at the numbers in the given set, we can see that only 9.1 is less than 10. The other numbers are greater than 10. rc Inequality: & t < 10 [0.5em] Set: & { 9.1,10.1,11 } We can also use the substitution to determine the answer.

Value of t	t < 10	Is the Inequality True?
9.1	9.1 < 10	Yes
10.1	10.1 ≮ 10	No
11	11 ≮ 10	No

From the given set, only 9.1 makes the inequality true.

Let's examine the inequality. Inequality: y ≥ 5 It is read as y is greater than or equal to 5. This means that all values of y greater than or equal to 5 make the inequality true. When we look at the numbers in the given set, we can see that 5 is equal to 5 and 10 is greater than 5. rc Inequality: & y ≥ 5 [0.5em] Set: & {0, 5, 10 } The answers are 5 and 10. Let's use substitution to check our answer.

Value of y	y ≥ 5	Is the Inequality True?
0	0 ≱ 5	No
5	5 ≥ 5	Yes
10	10 ≥ 5	Yes

From the given set, 5 and 10 make the inequality true.

Identify the graph of the given inequality.

x \leq 52

- 7 \leq c

Let's graph the given inequality on a number line. x ≤ 52 We can read the inequality as x is less than or equal to 52. From this sentence, we can understand that 52 is a solution of the inequality. This means that we use a closed circle to represent the number on the number line.

The solution set of the inequality also includes the numbers less than 52. Since smaller numbers lie to the left on the number line, we can graph these solutions as an arrow pointing to the left.

This graph matches with the graph in option B.

We want to graph the inequality on a number line. - 7 ≤ c We read the inequality as - 7 is less than or equal to c. We can conclude that - 7 is a solution to the inequality because the inequality is true when we substitute - 7 for c. - 7 ≤ - 7 ✓ We use a closed circle to represent it on the number line.

The solution set of the inequality also includes the numbers greater than - 7. Since greater numbers lie to the right on the number line, we can graph these solutions as an arrow pointing to the right.

This graph matches with the graph in option A.

Write an inequality with the variable $x$ that represents the graph.

We want to write an inequality for the given graph using the variable x.

From the graph, we can see that there is an open circle at - 1. Since an open circle is used to indicate a number that is not a solution to an inequality, the number - 1 is not a solution. This means that we need to write a strict inequality. An open circle (∘) is used. ⇓ Inequality symbol is either < or > . We also see that the arrow is pointing to the left. This indicates that all values less than - 1 form the solution set. We can express the inequality as x is less than - 1. xis less than -1. ⇓ x < - 1

We will write an inequality that represents the following graph using the variable x.

We can see that there is an open circle at - 5. Since open circles are used for numbers that are not solutions of inequalities, the number - 5 is not a solution and our inequality is a strict inequality. An open circle (∘) is used. ⇓ Inequality symbol is either < or >. Since the arrow is pointing to the right, all values greater than - 5 form the solution set. As a result, the graph represents the values x is greater than - 5. xis greater than -5. ⇓ x > - 5

Writing and Graphing Inequalities

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