# What Are Inequalities Used For? (9 Important Applications)

You will see inequalities all the time in mathematics.  However, you might be wondering when you will see them used in real life applications.

So, what are inequalities used for?  Inequalities are used in real life to express estimates, limits, and restrictions. They are used in fields including construction, finance, law, manufacturing, medicine, real estate, software, and staffing (human resources). A compound inequality gives an interval when we don’t know exact values.

Of course, the context matters.  An inequality might express a limit in one case (such as a speed limit) or it might express an estimate (such as an upper bound for cost of a project).

Let’s get started.

## What Are Inequalities Used For?

Inequalities are used in many applications, including the following areas:

• Age Restrictions
• Construction
• Finance
• Manufacturing (limits on labor, time, or materials)
• Medicine
• Real Estate (minimum lot size or home area)
• Software (Computer Programming)
• Speed Limits (Minimums or Maximums)
• Staffing

Let’s take a look at each one, starting with age restrictions.

### Age Restrictions

Age restrictions use inequalities to express an age limit in months or years.

For example, certain cities, towns, and states give tax exemptions for elderly citizens.  For example, Massachusetts offers a \$700 exemption for taxpayers over age 65 in that tax year.

If A is the age of the taxpayer, the inequality would be written as:

• A >= 65 years

This is an example of a minimum age.  Another would be minimum age requirement for getting a driver’s license.

For example, if you must be at least 16 to get a driver’s license, the inequality would be written as:

• A >= 16 years

There are also inequalities that involve maximum age.  For example, a park may offer free admission for kids under the age of 5.

This inequality would be expressed as:

• A <= 5 years

### Construction

Inequalities are also used in construction when coming up with estimates.

For example, let’s say that you think the cost of building a house will come out to at least \$600,000.  If the area of the house is 1,500 square feet, then the cost per square foot is at least:

• \$600,000 / 1,500 = \$400 per square foot

If the cost per square foot is C, then the inequality would be written as:

• C >= \$400 per square foot

You might also use an inequality when estimating the number of workers to assign to a project.

For example, let’s say that you estimate a project will take at most 1000 labor hours to complete.  If you want to get the project done in at most 5 weeks, then on average you would need:

• 1,000 / 5 = 200 labor hours per week

If each of your workers puts in 40 labor hours in a normal week, then you would need:

• 200 / 40 = 5 workers

To get the job done on time, you should assign 5 workers to the task.  All 5 of them would be working full-time on that job for the next 5 weeks.

You would need to make sure you didn’t assign anything else to them during that time (or pay them overtime if you did).

### Finance

Inequalities are used in finance when analyzing companies and investments.

For example, a financial analyst may come up with a range of estimates for the price of a company’s stock in 1 year.  If the range is [\$50, \$70], then it might make sense to buy the company stock today at \$40.

If the stock price in one year is S1, this inequality would be expressed as:

• \$50 <= S1 <= \$70

If the company stock costs \$80 today, it would not make sense to buy it (based on our estimates).

Since we have limited money to invest, we can also compare the potential return for two different companies we are considering.

Let’s say that company 1 has a stock price of \$40 right now.  We estimate that the price will be in the range [\$50, \$60] one year from now.

Also, company 2 has a stock price of \$100 right now.  We estimate that the price will be in the range [\$110, \$120] one year from now.

Even though we think company 2 will have the higher stock price next year, we want to look at return on investment as a percentage of money invested.

Here are the percentage returns for each company:

So, it looks like company 1 will return a higher percentage than company 2, so company 1 is probably the better investment to make.

### Manufacturing

Inequalities are also used in manufacturing, based on limits for the labor, time, or materials we have available.

For example, let’s say that a table requires 10 square feet of wood and 2 hours of labor to build.

If we only have 160 square feet of wood and 40 hours of labor available this week, we can express those inequalities as:

• W <= 160 square feet
• L <= 40 hours

We can also figure out a limit on how many tables we can make this week, based on what is available.

We have 160 square feet of wood, and a table requires 10 square feet of wood, so we can make at most:

• 160 / 10 = 16 tables (T <= 16)

We have 40 hours of labor, and a table needs requires 2 hours of labor, so we can make at most:

• 40 / 2 = 20 tables (T <= 20)

Combining both of the above inequalities into a compound inequality, we get T <= 16 and T <= 20, which means T <= 16.

So, we can only make 16 tables – we will run out of wood before we run out of labor.  If we get 40 more square feet of wood, then we can make an extra 4 tables and completely use up both the wood and the labor hours.

### Medicine

Inequalities are also used in medicine when tracking various health metrics.

For example, all of the following have “normal” or expected ranges:

• Blood pressure (both systolic and diastolic)
• Blood sugar (glucose)
• Cholesterol (both HDL and LDL)
• Iron
• Etc.

A doctor can also use inequalities to prescribe a maximum or minimum amount of something for a patient.

For instance, during recovery for a surgery, a doctor may suggest a maximum of 20 minutes of walking per day for the first week:

• W <= 20 minutes

A doctor may also suggest a minimum amount of food (perhaps 2,000 calories daily) for someone who is underweight:

• C >= 2,000 calories

Inequalities are also used in medical research to express ranges for the effects of a drug.  For example, a cholesterol medication may reduce blood cholesterol by 10 to 30 mg per dL”

• 10 <= R <= 30 mg/dL

### Real Estate

Inequalities are often used in real estate, in terms of zoning or lot restrictions.

For example, there may be a minimum lot size of 9,000 square feet for building a home:

• L >= 9,000 square feet

There may also be a minimum amount of frontage (space between house and road) of 40 feet:

• F >= 40 feet

Inequalities can also be used to make good decisions involving real estate investments.

For example, if ranch style houses in your city are selling in the range of \$400,000 to \$450,000, then it might be a good deal to buy one at \$410,000 if you can.

Of course, you should have an inspection done to make sure there is nothing wrong with a house before buying.

### Software (Computer Programming)

Inequalities are also use in software to tell a computer how many times to repeat an operation.

For example, the code below tells the computer to call a function 10 times:

``````for(i = 0, i < 10, i++){
}``````
• The expression i = 0 tells the computer to start the variable i at a value of 0.
• The inequality i < 10 tells the computer to keep calling the function until the variable i has a value of 10.
• The expression i++ tells the computer to increase the value of the variable i by one each time the loop is run.

### Speed Limits

A speed limit can be expressed using an inequality.  For example, if the speed limit on a road is 65 miles per hour, we can write:

• S <= 65 mph

Some roads also have a minimum speed, which would use the opposite inequality symbol.  For example, with a minimum speed of 40 miles per hour, we can write:

• S >= 40 mph

### Staffing

Inequalities can also be used in staffing to figure out pay offers for new employees.

If a software engineer position is paying \$80,000 to \$95,000 in your city, then your offer should be somewhere in that range.  It would be higher for more experienced hires, and lower for less experienced ones.

The inequality would be written as:

• \$80,000 <= P <= \$95,000

## Conclusion

Now you know what inequalities are used for and where they might show up in everyday life.  Can you think of some other applications for inequalities?