The Donald Reid was a special submarine craft that could also fly. While floating on the water, the submarine's highest point is at most 4 meters above sea level. To understand the range of depths it can reach, we can use the mathematical inequality a\leq4, where a is the altitude of the submarine compared to sea level.

When we say "a is less than or equal to 4 meters ", we're not just talking about one specific number. We're talking about a whole set of altitudes, including a=4 meters, a=2 meters, a=0 meters (sea level), a=-1 meter (below sea level) and a=-1000 meters (deep underwater). All of these altitudes are less than or equal to 4.

If we graph all of the altitudes that are less than or equal to 4 on a number line, we get something that looks like this:

Our number line represents all the whole numbers that are shown and less than or equal to 4 meters, but what about fractions like \dfrac{1}{2} or decimals like -2.5? These numbers are also less than or equal to 4 meters, so we need to include them in our graph.

Rather than trying to graph all of the individual points, which would get very messy, we can draw a shaded line that includes all of the points, since they all are numbers that make the inequality true.

What if there is a different submarine that will start to malfunction if it reaches an altitude of 4 meters? We could use the inequality a<4 to show that a cannot be equal to 4. In this case the graph should not include the point where a=4.

We can show this on our number line by placing an unfilled point (open circle) at 4.

To graph inequalities that are greater than or greater than or equal to, we use the same method but the arrow will be facing to the right.

When the Donald Reid is above land and flying, it has a cruising altitude of above 2 meters.

The inequality that would match this scenario is a>2, where a is the altitude of the Donald Reid. The Donald Reid can not fly at 2 meters, but any altitude above 2 meters is fine so, there must be an unfilled endpoint (open circle) on the number line at 2 meters and shading to the right.

The graph of a>2 looks like this:

Examples

Example 1

Consider the following number line.

Which values are included on the number line?

-5.5

-\dfrac{2}{3}

7 \dfrac{1}{2}

Worked Solution

Create a strategy

Identify each point on the number line and see if it is within the shaded region of our graph.

Apply the idea

Looking at the points on the graph, we can see that 5, and 7 \dfrac{1}{2}are within our shaded region, so they are included in the number line.

We can see that -5.5,\,-\dfrac{2}{3},\, 0, and 1 are not within our shaded region, so they are not included in our number line.

1 is not within our shaded region because it is where the open (unfilled) circle is on the line. This means that 1 is not included in the set.

So, the correct options are A and F.

Write two inequalities that are represented by the number line.

Worked Solution

Create a strategy

To represent the same number line with two inequalities, one inequality should have the variable on the left side, and the other should have the variable on the right side.

Use the type of endpoint and the direction of shading to choose the correct inequality symbols.

Apply the idea

The graph starts at the point 1 and stretches out to the right, which means the line covers all values greater than 1. Since the endpoint at 1 is open/unfilled, it means that it is not included in the inequality.

We can say all of the solutions are greater than 1 or 1 is less than all of the solutions.

The inequalities that represent this are x>1 and 1<x.

Example 2

Graph the following inequalities and identify three values that are in the solution set for each:

x \geq 1

Worked Solution

Create a strategy

The greater than or equal to \left(\geq \right) symbol tells us all of the solutions are larger than 1 or equal to 1. So 1 and any larger number will be shaded on the number line.

Apply the idea

Start with a filled (closed) point at 1 and shade all values greater than (to the right of) 1.

1,4, and 10 are part of the solution set, along with many other values.

x \lt 7

Worked Solution

Create a strategy

The less than \left(\lt \right) symbol tells us all of the solutions are smaller than 7. So 7 will not be shaded but every number smaller than it will be.

Apply the idea

Start with an unfilled (open) point at 7 and shade all values less than (to the left of) 7.

6.9,\,\dfrac{1}{2}, and 0 are part of the solution set, along with many other values.

Example 3

Identify which values are in the solution set for each:

x \leq 5

-3

7.3

\dfrac{5}{6}

Worked Solution

Create a strategy

Substitute each value for x to see it makes the inequality true. If the statement is true, then the value is part of the solution set. If the statement is false, then the value is not part of the solution set.

Apply the idea

2 \leq 5 True
-3 \leq 5 True
5 \leq 5 True
7.3 \leq 5 False
\dfrac{5}{6} \leq 5 True

The values that are in the solution set are: 2,\,-3,\,5 and \dfrac{5}{6}.

So the correct choices are A, B, C, and E.

Reflect and check

We can check this by graphing the inequality on a number line and plotting each point. Any point in the shaded region is in the solution set.

We can see that 7.3 is the only value not in the shaded region.

x \gt -2

-2

-5

5\dfrac{3}{4}

-0.5

Worked Solution

Create a strategy

Apply the idea

-2 \gt -2 False
-5 \gt -2 False
5\dfrac{3}{4} \gt -2 True
-0.5 \gt -2 True
2 \gt -2 True

The values that are in the solution set are: 5\dfrac{3}{4},\,-0.5,\, and 2.

So the correct choices are C, D, and E.

Reflect and check

We can check this by graphing the inequality on a number line and plotting each point. Any point in the shaded region is in the solution set.

We can see that the values -2 and -5 are not in the solution set.

Example 4

Create an inequality to represent the following scenario:

A local pool has adult swim times, and requires swimmers to be at least 16 years old to swim during these times.

Worked Solution

Create a strategy

To determine which inequality to use, think about the ages of swimmers who would be allowed to swim during this time. Can someone who is older than 16 swim at this time? What about someone younger than 16? Would someone who is exactly 16 years old be allowed to swim during adult swim?

Apply the idea

To be at least 16 years old, a swimmer can be 16 or older than 16. The inequality needs to represent ages that are greater than or equal to 16. Let a represent the age of the swimmer who can swim during adult swim.

a \geq 16

Reflect and check

The inequality a \geq 16 includes very large ages such as 800 years old. Sometimes this can happen with real world scenarios, where we have a correct inequality to represent the statement, but some of the solutions might not make sense in the context.

Example 5

Create a scenario that could fit the following inequality:

x \lt 48

Worked Solution

Create a strategy

Think about a scenario where something has to be under 48.

Apply the idea

A local carnival has rides based off of heights. Some rides are designed for children under the height of 48 inches. So, you must be under 48 inches to ride the kids train ride at the carnival.

Idea summary

To graph an inequality, start by determining which direction the line will be shaded, right or left. This can be determined by making sure that the shaded line covers all the values that make the inequality true.

The end point of the line will be an open circle circle if the inequality has a < or >.

The end point of the line will be a closed circle if the inequality has a \leq or \geq.

The solution set of an inequality is made up of all values that make the inequality true. Solutions lie in the shaded region on a number line.

Outcomes

6.PFA.4

The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.

6.PFA.4a

Given the graph of a linear inequality in one variable on a number line, represent the inequality in two equivalent ways (e.g., x < -5 or -5 > x) using symbols. Symbols include <, >, â‰¤, â‰¥.

6.PFA.4b

Write a linear inequality in one variable to represent a given constraint or condition in context or given a graph on a number line.

6.PFA.4c

Given a linear inequality in one variable, create a corresponding contextual situation or create a number line graph.

6.PFA.4d

Use substitution or a number line graph to justify whether a given number in a specified set makes a linear inequality in one variable true.

6.PFA.4e

Identify a numerical value(s) that is part of the solution set of a given inequality in one variable.

6.06 Solutions to inequalities

Ideas

Solutions to inequalities

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Outcomes

6.PFA.4

6.PFA.4a

6.PFA.4b

6.PFA.4c

6.PFA.4d

6.PFA.4e

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