Much like solving equations from real world problems, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.

Recall from the equations section that we have the following key terms to indicate operations and quantities:

Addition	Subtraction	Multiplication	Division	Parentheses
plus	minus	times	divided by	quantity
the sum of	the difference of	the product of	the quotient of	times the sum
increased by	decreased by	multiplied by	half	times the difference
total	fewer than	each/per	split
more than	less than	twice	equally shared
added to	subtracted from	double

In addition to operations, certain key terms and phrases will indicate which inequality will be needed for the given situation:

\gt	\lt	\geq	\leq
greater than	less than	greater than or equal to	less than or equal to
more than	is below	at least	at most
over	under	minimum	maximum
		no less than	no more than

Recall that a solution to an inequality is any value that makes the inequality statement true. Based on the context, some values might be calculated algebraically, but are not reasonable based on the restrictions of the scenario. For example, time and lengths generally cannot be negative, which can create restrictions on the possible values x and y can take on.

Solution set

The set of all values that make the inequality or equation true

Example:

3 \leq x

Viable solutions

A valid solution that makes sense within the context of the question or problem

Non-viable solution

An algebraically valid solution that does not make sense within the context of the question or problem

We can represent the solutions to inequalities on a number line. The inequality symbol will determine if the ray should have a filled (closed) or unfilled (open) endpoint and which direction the arrow extends.

The less than or equal to symbol \left( \leq \right) will be represented with a filled (closed) endpoint and an arrow pointing left. The inequality x \leq 2 will be graphed like this:

This graph tells us that any number that falls on the ray, including 2 is a part of the solution set.

The less than symbol \left(<\right) will be represented with an unfilled (open) endpoint and an arrow pointing left. The inequality x < 2 will be graphed like this:

This graph tells us that any number that falls on the ray, except for 2 is a part of the solution set.

The greater than or equal to symbol \left( \geq \right) will be represented with a filled (closed) circle and an arrow pointing right. The inequality x \geq 2 will be graphed like this:

This graph tells us that any number that falls on the ray, including 2 is a part of the solution set.

The greater than symbol \left(>\right) will be represented with an unfilled (open) circle and an arrow pointing right. The inequality x \gt 2 will be graphed like this:

This graph tells us that any number that falls on the ray, excluding 2 is a part of the solution set.

Examples

Example 1

Consider the inequality: k \geq -25

Represent the inequality on a number line.

Worked Solution

Create a strategy

To represent the solution to the inequality on a number line, first, identify the critical value, which is -25. Then, decide how to show that all values equal to and greater than this are included in the solution set.

Apply the idea

Draw a horizontal line to represent the number line. Mark the point -25 on this line.

Use a closed (filled) circle to indicate that -25 is included in the solution set (since the inequality includes equal to), and draw an arrow extending to the right from -25, showing that all values greater than -25 are included.

Name 3 values that are in the solution set.

Worked Solution

Create a strategy

Since the inequality k \geq -25 includes all values greater than or equal to -25, we can choose any three values greater than or equal to this. We can use the number line from part (a) to help us.

Apply the idea

Some of the values in the solution set are -25,\,0,\, and 10.

We can see that they are all included in the shaded section of the number line:

We can also see that when substituted for k in k \geq -25 they make the inequality true.

\displaystyle -25	\displaystyle \geq	\displaystyle -25	Reads '-25 is greater than or equal to -25'
\displaystyle 0	\displaystyle \geq	\displaystyle -25	Reads '0 is greater than or equal to -25'
\displaystyle 10	\displaystyle \geq	\displaystyle -25	Reads '10 is greater than or equal to -25'

Example 2

Sasha is drawing a pentagon-shaped house design and wants to make sure the combined length of two sides representing the roof are at least twice as long as the sum of the remaining sides.

A pentagon with sides of two 2x minus 5 units, two x units, and x plus 3 units.

Write two different inequalities that Sasha could use to represent this situation.

Worked Solution

Create a strategy

The two expressions representing the roof are each 2x-5, the word twice will multiply their sum by 2, the remaining three expressions will be added, and at least represents \geq.Possible inequalities can show different levels of simplification or rearranging of variables.

Apply the idea

Possible inequalities representing the situation could include

2(2x-5)+2(2x-5) \geq x + x + (x+3)
2(2x-5 + 2x-5) \geq 2x + x+3
2(4x - 10) \geq 3x +3
x + x + x + 3 \leq 2(4x - 10)

There are many other possibilities, and these represent just a few.

Is 4 a possible value for x?

Worked Solution

Create a strategy

Choose one of the inequalities written in part (a) and substitute x=4.

Apply the idea

\displaystyle 2(2x-5)+2(2x-5)	\displaystyle \geq	\displaystyle x + x + (x+3)	Write the inequality
\displaystyle 2((2\cdot4)-5)+2((2\cdot4)-5)	\displaystyle \geq	\displaystyle 4 + 4 + (4 + 3)	Substitute x=4
\displaystyle 12	\displaystyle \geq	\displaystyle 15	Evaluate

Since 12 is not greater than or equal to 15, x=4 does not make the inequality true. Thus, x=4 is not a possible value for x to satisfy the given inequality.

Example 3

Calandra charges \$ 37.72 to style hair, as well as an additional \$ 6 per foil. Pauline would like the total cost for her styling to be no more than \$ 95.86.

Write an inequality that represents the number of foils Pauline could get.

Worked Solution

Create a strategy

Pauline has no more than \$ 95.86 to spend. "No more than" means "less than or equal to."

Apply the idea

We can write an inequality in words that represents the cost to style Pauline's hair:

\text{Cost of styling}+ \text{Cost per foil} \cdot \text{Number of foils} \leq \text{Total Pauline can spend}

Translating that into an algebraic expression we get: 37.72+6N\leq95.86 where N represents the number of foils.

The solution for the inequality is N \leq 9.69. Determine whether N=-2 is a viable solution to the inequality in the context of the question.

Worked Solution

Create a strategy

Keep in mind it is not realistic to get part of a foil or a negative number of foils.

Apply the idea

Pauline can get a maximum of 9 foils and a minimum of 0 foils, so while -2 is mathematically part of the solution set for the inequality N \leq 9.69 it is not a viable solution in this context.

Reflect and check

Unlike a value that is not in a solution set of an inequality, this is an example of a solution that was mathematically valid and part of the original solution set but when considering the context we have found that it is non-viable.

Example 4

Given the inequality 3x + 5 - 6x > 8, create a verbal situation that can be represented by this inequality.

Worked Solution

Create a strategy

Think of a scenario where the result needs to exceed a certain value.

Apply the idea

Imagine you are saving money for a new video game. You start with \$5. Each week, you can save \$3 from your allowance, but you also spend \$6 on other expenses. To determine when you can afford the game, which costs more than \$8, you use the inequality 3x + 5 - 6x > 8, where x represents the number of weeks.

Idea summary

The following key terms can be used to indicate operations and quantities in real-world situations:

Addition	Subtraction	Multiplication	Division	Parentheses
plus	minus	times	divided by	quantity
the sum of	the difference of	the product of	the quotient of	times the sum
increased by	decreased by	multiplied by	half	times the difference
total	fewer than	each/per	split
more than	less than	twice	equally shared
added to	subtracted from	double

Phrases that describe inequalities include

\gt	\lt	\geq	\leq
greater than	less than	greater than or equal to	less than or equal to
more than	is below	at least	at most
over	under	minimum	maximum
		no less than	no more than

Because inequalities have infinite solutions, inequalities used to represent real-world situations often include solutions that are unreasonable in context and therefore non-viable.

Outcomes

8.PFA.5

The student will write and solve multistep linear inequalities in one variable, including problems in context that require the solution of a multistep linear inequality in one variable.

8.PFA.5b

Represent solutions to inequalities algebraically and graphically using a number line.

8.PFA.5c

Write multistep linear inequalities in one variable to represent a verbal situation, including those in context.

8.PFA.5d

Create a verbal situation in context given a multistep linear inequality in one variable.

8.PFA.5f

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

2.05 Write and represent multistep inequalities

Ideas

Write and represent multistep inequalities

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

8.PFA.5

8.PFA.5b

8.PFA.5c

8.PFA.5d

8.PFA.5f

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