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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.

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.

Boat-on-river.jpg

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

a Let 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 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 or There are several ways each inequality can be phrased.

Inequality Symbol Key Phrases
$\bm{\lt}$
$\bm{\gt}$
$\bm{\leq}$
$\bm{\geq}$

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

Symbol Example Meaning
The variable is less than
The algebraic expression is less than or equal to
The expression is greater than
The variable is greater than or equal to the expression
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 is a number.

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.
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.
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 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 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.
Let 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 Then, the inequality can be written as follows.
Therefore, the inequality is
b Like in Part A, start by describing the situation in one sentence.
Let 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 inequality is
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.
Notice that if is substituted for in the inequality, the inequality holds true. Therefore, it can be said that is a solution to the given inequality.
However, this is not the only value that makes the inequality true. There are other values like and 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 or 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.
The inequality is read as is greater than or equal to It is a non-strict inequality, so 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 is used.

For the given inequality, a closed circle is placed at because it is a solution.

Placing a closed circle at -6

Every value of greater than 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.

A closed circle at -6 and an arrow pointing to the right
This is the graph of the inequality. Examine the graphs of solutions sets of different inequalities.
Graph of some 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 hours.

Kevin and his dad are fishing
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:
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 hours for Kevin to catch as many fish as he wanted.
Let 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 inequality represents the situation.
b Now the number line graph for the inquality from Part A will be drawn.
The inequality is read as is less than This is a strict inequality, so is not a solution. Since is not a solution, an open circle is used at that point.

Every value of less than 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 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 This means that is a solution of the inequality. This also means that the graph represents a non-strict inequality.
The arrow pointing toward the left indicates that all values less than are also part of the solution set. Therefore, the inequality would be is less than or equal to This can be expressed algebraically as follows.
In the context of the question, this inequality means that Kevin can throw a stone at most yards.
b This time, let the variable be the distances that Kevin's father can throw a stone. Consider the given graph.
An open circle is placed at so is not a solution of the inequality. This means that the graph represents a strict inequality.
The arrow pointing toward the left of the line indicates that all values less than are part of the solution set. Therefore, the inequality would be is less than This can be expressed algebraically as follows.
This inequality means that Kevin's father can throw a stone no farther than 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 pounds and hold up to five people.

a Let 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 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:
Graph:
b Inequality:
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 and each boat can carry up to pounds. This means that the weight of a group that can be held in the boat is represented as the following inequality.
The inequality represents the situation. It is a non-strict inequality, so is a solution. Since is a solution, use a closed circle at thsi point on the number line.

Every value of less than 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 that can fit on each boat is at most people. Then the number of people that the boat can hold is represented as the following inequality.
The inequality represents this situation. This is also non-strict inequality, so is a solution. Since is a solution, a closed circle is used on the graph.

Every value of less than 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.
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.


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