We can also write inequality statements in two orders, but we need to be careful and switch the inequality sign being used as well. For example, x>10 means the same thing as 10<x. That is, "x is greater than ten" is the same as "ten is less than x".

When graphing inequalities we have to consider which inequality symbol is used.

The \gt and \lt both use an open (unfilled) circle as their endpoint. The \geq and \leq use a closed (filled) circle for their endpoint.

Let's see an example of each case.

The graph of x \gt 3 or "x is greater than 3" is:

Notice the open circle to show that 3 is not part of the solution set and the shading to the right to show that all numbers larger than 3 are in the solution set.

The graph of x \geq 3 or "x is greater than or equal to 3" is:

Notice the closed circle to show that 3 is included in the solution set and the shading to the right to show that all numbers larger than 3 are also in the solution set.

The graph of x \lt 3 or "x is less than 3" is:

Notice the open circle to show that 3 is not part of the solution set and the shading to the left to show that all numbers smaller than 3 are in the solution set.

The graph of x \leq 3 or "x is less than or equal to 3" is:

Notice the closed circle to show that 3 is included in the solution set and the shading to the left to show that all numbers smaller than 3 are also in the solution set.

Inequalities are used in many real life scenarios.

Consider the situation where Alex is at a trampoline park. Jumping on a trampoline costs \$ 12 per hour and there is a one-time entrance fee of \$ 8. He has a budget of \$ 52.

In this situation, we know Alex only has \$ 52 to spend. He can spend all of his money or just some of it, but it is not possible for him to spend more money than he has. This means the amount he spends at the trampoline park has to be less than or equal to \$ 52.

The amount he spends at the park is the cost of the entrance fee, \$ 8, plus the cost for jumping, \$ 12 per hour. We can represent the cost of jumping as \$ 12 times the number of hours, h or 12h. The total cost can be represented by 8+12h.

Putting this all together we get the inequality 8+12h \leq 52.

Assume Alex solves this inequality and gets the solution h \leq 4. This means that to stay within his budget, Alex can spend no more than 4 hours jumping on trampolines.

We can graph the inequality, h \leq 4.

Notice, that the inequality and the numberline include values that are not possible in the scenario, i.e. -2. It is impossible to go to the trampoline park for -2 hours. These negative values are mathematical solutions to the inequality but they do not make sense for this context. We always need to think about the meaning of each value in a solution set to determine if it makes sense for the context.

Examples

Example 1

Write and graph an inequality to represent the statement " k is less than or equal to seven."

Worked Solution

Create a strategy

The words "less than or equal to" refer to the symbol \leq and the graph will have a closed (filled) circle at 7.

Apply the idea

The inequality is:k\leq7

To graph it on a numberline we will start with a closed (filled) point at 7 and shade all values that are less than (to the left of) 7.

Reflect and check

There are a whole range of values of k which make this inequality true, including k=7,\,k=6.999,\,k=0 and k=-15.5.

Example 2

Write an inequality to represent the statement "The sum of 3 and 5 groups of x is at least 23. "

Worked Solution

Create a strategy

The word "sum" refers to addition, "groups of" refers to multiplication. "At least" means the amount needs to be that much or more which is represented by \geq.

Apply the idea

3+5x \geq 23

Example 3

Write an inequality to represent the following situations.

A painter is buying cans of paint. Each can costs \$ 40 and the painter has a budget of \$ 320.

Worked Solution

Create a strategy

To find the total cost, we need to multiply the cost of a can by the number of cans bought. Having a budget means you have a maximum amount that you can spend. The painter can spend all of the money, or some of the money, but not more than \$320.

Apply the idea

Let n be the number of cans the painter buys.

The total cost is 40n.

The inequality symbol that represents being on a budget is \leq because you can spend less than your total budget or all of it but not more.

The inequality to represent this situation is: 40n \leq 320

A gardener is planning to plant trees in a park. Each tree costs \$ 250 to plant, and there are additional setup costs totaling \$ 1500. The gardener's budget for planting trees is \$ 5500.

Worked Solution

Create a strategy

Just like in part (a) being on a budget means you can spend all of the money or some of the money which is represented by \leq. To find the total cost we need to multiply the cost of planting a tree by the number of trees planted and add that to the setup costs.

Apply the idea

Let t represent the number of trees planted. Then we can write the cost of planting trees as 250t and the total cost of planting all of the trees as 250t+1500.

Using the \leq symbol to show the total cost cannot be more than \$ 5500 we get the inequality:

250t + 1500 \leq 5500

Reflect and check

We can also represent this inequality where the total cost and the budget are flipped to the other side of the inequality.

5500 \geq 250t + 1500

This is tell us that the budget has to be greater than or equal to the total cost of planting the trees.

A contractor is bidding on a job to install flooring in several rooms of a building. Each room costs \$ 400 to install, and there are additional setup costs amounting to \$ 1000. The contractor needs to make at least \$ 3000 to cover their expenses and profit.

Worked Solution

Create a strategy

We need to set up an inequality to represent the total earnings from installing the rooms, including setup costs, and ensure it is greater than or equal to the the amount to cover the expenses.

Apply the idea

Let r be the number of rooms.

400r + 1000 \geq 3000

Example 4

Write a real-world scenario for each inequality.

50x \leq 1000

Worked Solution

Create a strategy

We can think of x as an amount that can change, and each x has to be 50 of something. All together this must be no more than 1000.

Apply the idea

A sample scenario would be "A worker is loading 50 lb bags of cement onto a truck. The total truck load weight can not exceed 1000 lbs. Let x represent the number of bags that can fit on the truck."

100+25p \geq 600

Worked Solution

Create a strategy

We can think of 100 as a fixed starting amount and p as an amount that can change. Each p must be 25 of something and the total amount cannot be less than 600.

Apply the idea

A sample scenario would be "A graphic designer is creating posters for a client. She charges a one time consultation fee of \$ 100 and each poster costs \$ 25 to produce. The designer needs to make at least \$600. Let p represent the number of posters being created."

\dfrac{b}{5}\lt 20

Worked Solution

Create a strategy

We can think of b as an amount that can change, and when divided into 5 equal groups the size of each group is less than 20.

Apply the idea

A sample scenario would be "A baker is baking cookies for some family members. She has 5 boxes to fill with the same number of cookies in each. Each box must contain fewer than 20 cookies. Let b represent the total number of cookies she bakes."

Idea summary

The smaller side of the inequality symbol matches the side with the smaller number. That is, the inequality symbol "points to" the smaller number.

The greater than symbol is >. This is an open circle on the number line and is shaded to the right.

The less than symbol is <. This is an open circle on the number line and is shaded to the left.

The greater than or equal to symbol is \geq. This is an closed circle on the number line and is shaded to the right.

The less than or equal to symbol is \leq. This is an closed circle on the number line and is shaded to the left.

Outcomes

7.PFA.4

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

7.PFA.4c

Represent solutions to one- or two-step linear inequalities in one variable algebraically and graphically using a number line.

7.PFA.4d

Write one- or two-step linear inequalities in one variable to represent a verbal situation, including those in context.

7.PFA.4e

Create a verbal situation in context given a one or two-step linear inequality in one variable.

4.04 Write and represent inequalities

Ideas

Write and represent inequalities

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

7.PFA.4

7.PFA.4c

7.PFA.4d

7.PFA.4e

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