Clear, Real-World Examples of Solving Inequalities

Everyday examples of solving inequalities

Let’s start with situations that feel familiar. These examples of examples of solving inequalities show up in day-to-day life: money, time, and limits.

Paycheck example of a one-step inequality

You earn \(15 per hour. You want to make at least \)120 this weekend. How many hours should you work?

Let \(h\) be the number of hours.

You earn \(15 each hour, so your pay is \(15h\). “At least \)120” means greater than or equal to 120:

[
15h \ge 120
]

Solve by isolating \(h\):

[
\frac{15h}{15} \ge \frac{120}{15} \Rightarrow h \ge 8
]

So you need to work 8 hours or more.

This is one of the simplest examples of solving inequalities: one operation, one step. But the idea—model the situation, solve, then interpret—never changes, even for harder problems.

Budget example of a two-step inequality

You have \(50 saved. You’re buying movie tickets that cost \)8 each and you want to keep at least $10 in your account afterward. How many tickets can you buy?

Let \(t\) be the number of tickets.

Total spent is \(8t\). Money left is \(50 - 8t\). You want that to be at least 10:

[
50 - 8t \ge 10
]

Now solve step by step.

Subtract 50 from both sides:

[

-8t \ge 10 - 50
]
[

-8t \ge -40
]

Divide both sides by -8. Important: dividing by a negative flips the inequality sign.

[
\frac{-8t}{-8} \le \frac{-40}{-8}
]
[
t \le 5
]

You can buy 5 tickets or fewer.

This is a classic example of a two-step inequality where you have to remember the sign flip. It’s one of the best examples to practice that rule.

Classroom-style examples of solving inequalities

Now let’s move into the kinds of examples of solving inequalities you’ll see in algebra classes and on standardized tests.

Example of solving a multi-step inequality

Solve and graph: \(3(2x - 1) - 4 \le 2x + 5\)

First, distribute the 3:

[
3(2x - 1) - 4 \le 2x + 5
]
[
6x - 3 - 4 \le 2x + 5
]
[
6x - 7 \le 2x + 5
]

Move the \(2x\) to the left by subtracting:

[
6x - 2x - 7 \le 5
]
[
4x - 7 \le 5
]

Add 7 to both sides:

[
4x \le 12
]

Divide by 4:

[
x \le 3
]

Graph idea (on a number line): open vs closed circle matters. Because we have \(\le\), we use a closed circle at 3 and shade to the left.

This is one of those examples include distribution, combining like terms, and solving the inequality—three skills in one.

Example of a compound inequality (in-between values)

Solve: \(2 < 3x - 1 \le 11\)

This is a compound inequality: two inequalities stuck together.

[
2 < 3x - 1 \le 11
]

Think of it as a three-part sentence. Whatever you do to the middle, you must do to both ends.

Add 1 to all three parts:

[
2 + 1 < 3x \le 11 + 1
]
[
3 < 3x \le 12
]

Now divide everything by 3:

[
\frac{3}{3} < x \le \frac{12}{3}
]
[
1 < x \le 4
]

So \(x\) is greater than 1 and less than or equal to 4.

On a number line, that’s an open circle at 1, closed circle at 4, and shading between them.

This is one of the best examples of examples of solving inequalities with a “sandwich” structure—very common in textbooks and tests.

Real examples: inequalities in temperature, health, and safety

Inequalities aren’t just abstract symbols; scientists and health experts use them constantly. Here are some real examples of solving inequalities that connect to 2024–2025 data and guidelines.

Temperature range example of an inequality

A lab keeps samples safe between 35°F and 39°F, inclusive. Write and interpret an inequality for the allowed temperature \(T\).

“Between 35°F and 39°F, inclusive” means 35 and 39 are allowed.

[
35 \le T \le 39
]

If the thermostat shows 41°F, you can check it quickly:

[
35 \le 41 \le 39
]

The right side, \(41 \le 39\), is false, so 41°F is not in the safe range.

Labs, hospitals, and even your home refrigerator use inequality ranges like this all the time. For more on safe storage temperatures for food, the U.S. Department of Agriculture gives current guidance here: https://www.fsis.usda.gov/food-safety.

Health guideline example of solving inequalities

Public health recommendations are often written as inequalities. For instance, physical activity guidelines from organizations like the CDC give minimum amounts of exercise.

Suppose adults are advised to get at least 150 minutes of moderate exercise per week. You walk 30 minutes per day on weekdays. How many additional weekend minutes do you need?

Let \(w\) be weekend minutes.

Weekday walking: 30 minutes × 5 days = 150 minutes.

Total weekly minutes: \(150 + w\).

Guideline: at least 150 minutes:

[
150 + w \ge 150
]

Subtract 150 from both sides:

[
w \ge 0
]

So you technically don’t need extra weekend minutes to meet that minimum. But if you want to go beyond the minimum, you might set a personal goal like \(150 + w \ge 200\) and solve that:

[
150 + w \ge 200
]
[
w \ge 50
]

This is a nice, simple example of how inequalities capture “at least” and “no more than” in health recommendations.

Best examples of linear inequalities with graphs

Let’s look at some best examples that combine algebra, graphs, and interpretation. These are the kinds of examples of solving inequalities that prepare you for both schoolwork and standardized tests.

Graphing a simple inequality: \(x > -2\)

Solve and describe the graph of \(x > -2\).

The inequality is already solved: any number greater than -2 works.

• On a number line, you’d put an open circle at -2.
• Shade everything to the right of -2.

To check, pick a test value like 0:

[
0 > -2 \quad \text{(true)}
]

Pick a value that should not work, like -3:

[

-3 > -2 \quad \text{(false)}
]

This is one of the simplest examples of graphing solutions to inequalities, and it’s a good warm-up before more complicated ones.

Real examples of double inequalities: test score ranges

A teacher gives a test scored from 0 to 100. A passing score is at least 70 but less than 90. Write an inequality for a passing-but-not-excellent score \(s\).

“At least 70” means \(s \ge 70\), and “less than 90” means \(s < 90\).

Together:

[
70 \le s < 90
]

This is another example of a compound inequality. If a student scored 88, plug it in:

[
70 \le 88 < 90
]

Both parts are true, so 88 is in the range.

If a student scored 92:

[
70 \le 92 < 90
]

The right side is false, so 92 is not in this particular category.

Education data often uses ranges like this to describe performance levels. For example, state testing reports in the U.S. typically define “proficient” and “advanced” with inequalities on score ranges.

More examples of solving inequalities with negatives

Working with negative numbers is where students most often slip. These examples of examples of solving inequalities will help that rule about flipping the sign stick.

Example of dividing by a negative

Solve: \(-5x > 20\)

Divide both sides by -5. Because -5 is negative, flip the inequality sign:

[
\frac{-5x}{-5} < \frac{20}{-5}
]
[
x < -4
]

Check with a number that should work, like \(x = -5\):

[

-5(-5) > 20
]
[
25 > 20 \quad \text{(true)}
]

Check with a number that shouldn’t work, like \(x = -3\):

[

-5(-3) > 20
]
[
15 > 20 \quad \text{(false)}
]

This is one of the best examples to remember: divide or multiply both sides by a negative → flip the sign.

Real-world inequality with a negative rate

A freezer is warming at a rate of 2°F per hour because the power went out. It started at -10°F. After how many hours will the temperature be above 0°F?

Let \(h\) be hours after the power went out.

Temperature after \(h\) hours:

[
T = -10 + 2h
]

We want \(T > 0\):

[

-10 + 2h > 0
]

Add 10 to both sides:

[
2h > 10
]

Divide by 2:

[
h > 5
]

So after more than 5 hours, the temperature will be above freezing. This kind of inequality shows up in climate and environmental modeling; for broader climate information, you can explore resources from NASA and NOAA.

Examples include word problems with inequalities

Let’s pull everything together with word problems. These real examples of solving inequalities look a lot like standardized test questions in 2024–2025.

Cell phone data plan example of an inequality

A phone plan includes 5 GB of high-speed data per month. After that, your speed slows. You’ve already used 2.3 GB. Write and solve an inequality to find how many more gigabytes \(g\) you can use before slowing.

Total data used: \(2.3 + g\).

You want this to stay at most 5 GB:

[
2.3 + g \le 5
]

Subtract 2.3 from both sides:

[
g \le 5 - 2.3
]
[
g \le 2.7
]

You can use up to 2.7 GB more before your speed slows.

Overtime pay example of a system with an inequality

A worker earns \(18 per hour for up to 40 hours per week, and \)27 per hour for overtime (hours beyond 40). They want to earn at least $900 in one week. What inequality describes the total pay \(P\) in terms of overtime hours \(o\)?

Regular hours: 40 at \(18 → \)720.

Overtime hours: \(o\) at $27 → \(27o\).

Total pay:

[
P = 720 + 27o
]

They want at least $900:

[
720 + 27o \ge 900
]

Subtract 720:

[
27o \ge 180
]

Divide by 27:

[
o \ge \frac{180}{27} \approx 6.67
]

So they need at least about 6.67 hours of overtime. Since you can’t work a fraction of an hour in many jobs, they’d likely need 7 hours of overtime.

This is a realistic example of how inequalities help answer “How much is enough?” questions about money and work.

FAQ: short answers with more examples of inequalities

What are some quick examples of solving inequalities?

Some quick ones:

• \(x + 4 > 10\) → \(x > 6\)
• \(7 - y \le 3\) → \(-y \le -4\) → \(y \ge 4\)
• \(\frac{x}{3} \ge -2\) → \(x \ge -6\)

These are simple examples of one-step inequalities.

Can you give an example of an inequality from health or science?

Yes. A safe fever guideline for adults might say: “Call a doctor if your temperature is above 104°F.” Mathematically, if \(T\) is temperature in °F:

[
T > 104
]

Medical sites like Mayo Clinic often use phrases like “higher than” and “less than,” which are inequality symbols in disguise.

How do I know when to flip the inequality sign?

You flip the sign only when you multiply or divide both sides by a negative number. Adding or subtracting doesn’t flip it. The earlier examples of examples of solving inequalities with -5x and -8t are good practice problems for this.

What is an example of a compound inequality in real life?

A classic example of a compound inequality is a safe blood pressure range, or a “healthy” BMI range. For instance, if a healthy range for some measurement \(M\) is between 50 and 80, inclusive, that’s:

[
50 \le M \le 80
]

This is a real-world example of an “in-between” inequality.

Why are there so many different examples of solving inequalities?

Because inequalities model limits: budgets, speed limits, safe ranges, minimum goals, and more. That’s why textbooks, exams, and real-world sources give so many examples of solving inequalities—they show how one idea stretches across many different situations.

The more you work through these examples of examples of solving inequalities, the more you’ll see the same patterns repeating: isolate the variable, watch the sign when negatives are involved, and always check whether your answer makes sense in the original story. Once those habits stick, inequalities stop feeling mysterious and start feeling like everyday tools.