Solving Word Problems with Inequalities - 7th Grade Math

Section 1

Where Do Inequalities Come From?

People have been comparing sizes and amounts for thousands of years. Long before anyone wrote the symbols "<" or ">", traders needed to know whether they had enough goods, and builders needed to check whether a bridge was strong enough to hold weight. The math idea behind all of these questions is the inequality—a statement that one quantity is bigger or smaller than another.

~1700 BCE

Ancient Babylonian tablets show problems about amounts being "more than" or "less than" certain limits. They didn't have symbols yet, but they understood the idea.

1631 CE

English mathematician Thomas Harriot introduced the symbols < (less than) and > (greater than) in a book published after his death. These are the exact symbols you use today!

1734 CE

French mathematician Pierre Bouguer first used the symbols ≤ (less than or equal to) and ≥ (greater than or equal to), completing the set of inequality signs.

1900s

Inequalities became a central tool in optimization—the math of finding the best option. Everything from airplane design to phone apps uses inequalities to set limits and constraints.

Today, any time you ask "Do I have enough money?", "Can I finish in time?", or "Is this safe?", you're thinking about an inequality. In this lesson, you'll learn to take questions like those, write them as math statements, solve them, and show the answers on a number line.

Section 2

Core Ideas You Need

Before we dive into word problems, let's make sure you're comfortable with four building blocks. If you know how equations work (like 2x + 3 = 11), you're already most of the way there—inequalities just swap the "=" for a comparison symbol.

What Is an Inequality?

An inequality is a math sentence that uses <, >, ≤, or ≥ instead of =. It says one side is bigger or smaller than the other. For example, x > 5 means "x is more than 5."

The Form px + q > r (or < r)

In this lesson, every inequality you build will look like a number times x, plus (or minus) another number, compared to a third number. The letters p, q, and r stand for specific rational numbers (fractions, decimals, or whole numbers).

Solution Set

An equation usually has one answer. An inequality has many answers—a whole range of numbers that make it true. This range is called the solution set.

Graphing on a Number Line

We show the solution set by shading part of a number line. An open circle (○) means "not including this point" (used with < or >). A closed circle (●) means "including this point" (used with ≤ or ≥).

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KEY TAKEAWAY

Think of an inequality like a speed limit sign. A sign that says "Speed Limit 55" doesn't mean you must drive exactly 55 mph. It means your speed must be less than or equal to 55. There's a whole range of legal speeds (0, 10, 30, 55…). Inequalities work the same way—instead of one answer, you get a range of answers.

Section 3

Seeing Inequalities on a Number Line

One of the best things about inequalities is that you can see the answer. Below is a diagram showing how to graph two common types of solutions. Pay attention to the open circle versus the shading direction—those two details tell you everything.

Number line diagrams showing x > 3 (open circle, shading right) and x < −1 (open circle, shading left)

In Example A, every number to the right of 3 makes x > 3 true. Numbers like 3.5, 4, 10, and 100 all work. But 3 itself does not work (that's why the circle is open). In Example B, every number to the left of −1 makes x < −1 true. Numbers like −2, −5, and −100 all work, but −1 itself does not.

Here's a quick memory trick: the arrow points the same direction as the inequality symbol. If the symbol opens to the right (>), you shade to the right. If it opens to the left (<), you shade to the left.

Section 4

How to Solve Inequalities (Step by Step)

Solving an inequality of the form px + q > r or px + q < r is almost the same as solving an equation. You use inverse operations to get x by itself. There's just one extra rule you need to remember.

General Form

px + q > r or px + q < r

p, q, and r are rational numbers (fractions, decimals, or integers); x is the unknown.

Step 1 — Subtract q from both sides

px > r − q

This removes the constant term from the left side.

Step 2 — Divide both sides by p

x > (r − q) ÷ p

This isolates x. If p is positive, the inequality sign stays the same. ⚠ If p is negative, FLIP the inequality sign!

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KEY TAKEAWAY

Imagine you and a friend are standing on a number line. You're at 4, your friend is at 8, so you're "less than" your friend (4 < 8). Now imagine a mirror flips both of you to the negatives: you land at −4, your friend at −8. Suddenly you're bigger! (−4 > −8.) That's why multiplying or dividing both sides by a negative number flips the inequality sign.

Section 5

Translating Words into Inequalities

The trickiest part of word problems is figuring out which math symbol to use. Below is a handy table of keyword translations. Keep this in mind every time you read a word problem!

Words You'll See	Symbol	Example Phrase → Inequality
more than, greater than, above, over, exceeds	>	"more than 20" → `x > 20`
less than, fewer than, below, under	<	"fewer than 15" → `x < 15`
at least, no less than, minimum	≥	"at least 50" → `x ≥ 50`
at most, no more than, maximum	≤	"at most $30" → `x ≤ 30`

Now let's look at a full process diagram. This flowchart shows the steps from reading a word problem all the way to graphing the answer.

Flowchart showing four steps: Read and identify, Write inequality, Solve algebraically, Graph on number line

Let's walk through each of those steps with a real word problem in the next section.

Section 6

Worked Example

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Problem

Maya is saving up for a skateboard that costs $75. She already has $18 saved and earns $4.50 per hour doing yard work. How many hours does she need to work so that her total savings are more than $75?

Maya's Skateboard Savings

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Step 1 — Read & Identify

Let's figure out the pieces. Maya's rate of earning is $4.50 per hour. Her starting savings are $18. The target is $75. The comparison word is "more than," which means >. The unknown is the number of hours, which we'll call h.
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Step 2 — Write the Inequality

Total savings = (hourly rate × hours) + starting amount. We need this to be more than $75: 4.50h + 18 > 75. This matches the form px + q > r, where p = 4.50, q = 18, and r = 75.
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Step 3 — Solve

Subtract 18 from both sides to undo the + 18: 4.50h + 18 − 18 > 75 − 18 → 4.50h > 57. Divide both sides by 4.50 to isolate h. Since 4.50 is positive, the sign stays the same: h > 57 ÷ 4.50 → h > 12.666…

As a fraction, that's h > 12⅔ (or h > 38/3).
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Step 4 — Graph

On a number line, place an open circle at 12⅔ (because the sign is >, not ≥) and shade to the right.
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Step 5 — Interpret

The solution h > 12⅔ tells us Maya needs to work more than 12⅔ hours. Since she probably gets paid in whole hours, that means she needs to work at least 13 hours to have more than $75 saved. After 13 hours she'll have 4.50 × 13 + 18 = $76.50, which is indeed more than $75. ✓

Number line showing h > 12⅔, open circle at 12⅔, shading to the right

Section 7

Tips, Pitfalls, and Comparisons

Inequalities are close cousins of equations, but they have some key differences. Here's a side-by-side look so you can keep them straight.

Feature	Equation (px + q = r)	Inequality (px + q > r or < r)
Symbol	`=`	`<, >, ≤, ≥`
Number of solutions	Usually one answer	Infinitely many (a range)
Solving steps	Inverse operations	Same inverse operations + flip rule
Graphing the answer	A single point on the number line	A ray (with open or closed circle)
Multiplying/dividing by a negative	No extra rule	FLIP the inequality sign!

Common Mistakes to Avoid

Mistake 1: Forgetting to flip the sign. When you divide (or multiply) both sides by a negative number, the inequality direction reverses. For example, if −2x > 10, dividing both sides by −2 gives x < −5 (not x > −5).

Mistake 2: Mixing up open and closed circles. Strict inequalities (< and >) get an open circle. "Or equal to" inequalities (≤ and ≥) get a closed (filled) circle.

Mistake 3: Shading the wrong direction. Always check your answer by plugging in a number from the shaded part. If it makes the original inequality true, you shaded correctly!

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KEY TAKEAWAY

Think of solving an inequality like a balance scale with a "greater than" or "less than" sign in the middle instead of an equals sign. You can add or remove the same weight from both sides without changing which side is heavier. But if you flip both sides upside down (multiply by a negative), the heavier side becomes the lighter side — so you have to flip the sign too.

Section 8

Where This Leads Next

You've just learned to solve one-variable linear inequalities. That's an important building block! Here's a sneak peek at where this math takes you in the years ahead.

What You Learned Now	What Comes Later
One inequality, one variable (px + q > r)	Compound inequalities — two inequalities joined by "and" or "or" (8th grade)
Graphing on a number line (1-D)	Graphing on a coordinate plane — shading regions above or below a line (Algebra 1)
One constraint	Systems of inequalities — multiple constraints at once, used in real-world optimization (Algebra 2)
Rational number coefficients	Absolute value inequalities — dealing with distance from zero (Algebra 1)

The skills you're building right now — translating words to math, solving step by step, and interpreting answers — will carry forward into every one of those future topics. You're laying a strong foundation!

Section 9

Practice Problems

Try these five problems on your own before revealing the answers. They start easy and get more challenging. You've got this!

Question 1: Problem 1 — Conceptual

Look at this inequality: 3x + 5 > 20. Without solving, is x = 4 part of the solution set? How about x = 6? Explain how you checked.

Show answer

Check x = 4: Plug it in. 3(4) + 5 = 12 + 5 = 17. Is 17 > 20? No, so x = 4 is not in the solution set. Check x = 6: 3(6) + 5 = 18 + 5 = 23. Is 23 > 20? Yes! So x = 6 is in the solution set. To check any value, substitute it for x and see if the resulting statement is true.

Question 2: Problem 2 — Basic Calculation

Solve the inequality: 2x + 7 < 19. Then graph the solution on a number line.

Show answer

Step 1: Subtract 7 from both sides. 2x + 7 − 7 < 19 − 7 → 2x < 12. Step 2: Divide both sides by 2. x < 12 ÷ 2 → x < 6. Graph: Draw a number line. Place an open circle at 6 (since it's <, not ≤). Shade everything to the left of 6. Any number less than 6 (like 5, 0, −10) makes the original inequality true.

Question 3: Problem 3 — Intermediate

Solve: ½x + 3 > 8. Write your answer as an inequality and describe the solution set in words.

Show answer

Step 1: Subtract 3 from both sides. ½x + 3 − 3 > 8 − 3 → ½x > 5. Step 2: Multiply both sides by 2 (or divide by ½). x > 5 × 2 → x > 10. In words: The solution set is all numbers greater than 10. On a number line, you'd draw an open circle at 10 and shade to the right. Any number bigger than 10 (like 10.5, 11, 100) makes the inequality true.

Question 4: Problem 4 — Applied Word Problem

A movie streaming service costs $8.50 per month plus a one-time signup fee of $12. Jaden's parents said he can spend less than $100 total on the service. Write an inequality and solve it to find how many months Jaden can keep the subscription. Graph and interpret your answer.

Show answer

Set up: Let m = number of months. Total cost = 8.50m + 12. This must be less than 100: 8.50m + 12 < 100. Step 1: Subtract 12. → 8.50m < 88. Step 2: Divide by 8.50. → m < 88 ÷ 8.50 → m < 10.35… (approximately 10.35). Graph: Open circle at about 10.35, shade left. Interpret: Since months must be whole numbers, Jaden can subscribe for at most 10 months. At 10 months: 8.50(10) + 12 = $97, which is under $100. ✓ At 11 months: 8.50(11) + 12 = $105.50, which is over $100. ✗

Question 5: Problem 5 — Challenge

A store sells custom T-shirts. It costs the store $6.25 to make each shirt, plus a fixed machine cost of $40 per day. The store sells each shirt for $15. How many shirts must the store sell in a day so that its revenue (money it earns) is more than its costs? Set up and solve an inequality, graph the solution, and explain what your answer means for the store. Hint: Revenue = 15s. Cost = 6.25s + 40. You want Revenue > Cost.

Show answer

Set up: Let s = number of shirts sold. Revenue > Cost becomes: 15s > 6.25s + 40. Step 1: Subtract 6.25s from both sides. 15s − 6.25s > 40 → 8.75s > 40. Step 2: Divide by 8.75. s > 40 ÷ 8.75 → s > 4.571… As a fraction: s > 4 and 4/7 (since 40/8.75 = 32/7 ≈ 4.571). Graph: Open circle at about 4.57, shade right. Interpret: You can't sell a fraction of a shirt, so the store must sell at least 5 shirts to make more money than it spends. At 5 shirts: Revenue = $75, Cost = 6.25(5) + 40 = $71.25. Since 75 > 71.25, the store makes a profit! This is called the break-even point, and it's a real concept businesses use every day. 🎉

Summary

Lesson Summary

Summary

In this lesson, you learned how to take real-world word problems and turn them into linear inequalities of the form px + q > r or px + q < r, where p, q, and r are rational numbers (fractions, decimals, or integers). You practiced four key steps: reading the problem to identify the rate, starting value, and comparison word; writing the inequality using the correct symbol (<, >, ≤, or ≥); solving algebraically by subtracting and dividing (remembering to flip the sign when dividing by a negative); and graphing on a number line with the right circle type (open for strict, closed for "or equal to") and the correct shading direction.

The solution set of an inequality is not just one number—it's an entire range of values that make the statement true. On the number line, this shows up as a ray pointing left or right. When you interpret the solution, you connect the math back to the real world: "Maya needs more than 12⅔ hours" or "Jaden can subscribe for at most 10 months." Building these translation skills now gives you a strong foundation for compound inequalities, graphing in two dimensions, and optimization problems you'll meet in later courses.