One-Step Inequalities — I can solve one-step inequalities...

SECTION 1

Where Did Inequalities Come From?

Have you ever heard someone say, "You must be at least 48 inches tall to ride this roller coaster"? That's an inequality in everyday life. An inequality tells us that two things are not necessarily equal — one might be bigger or smaller than the other.

People have been comparing amounts for thousands of years. Ancient traders needed to know if they had enough goods. Builders checked if a wall was tall enough. Over time, mathematicians created symbols and rules to write these comparisons neatly.

~1700 BCE

Ancient Babylon

Babylonian scribes solved problems about amounts that were "more than" or "less than" a given value, but they wrote everything in words — no symbols yet.

~300 BCE

Greek Geometry

Euclid and other Greek mathematicians compared lengths and areas, stating when one was greater than another. They used sentences, not symbols.

1631

The < and > Symbols Are Born

English mathematician Thomas Harriot introduced the less-than (<) and greater-than (>) symbols. These are the same symbols you use today!

1734

≤ and ≥ Appear

French mathematician Pierre Bouguer added a line underneath to create ≤ (less than or equal to) and ≥ (greater than or equal to).

Today, inequalities show up everywhere — speed limits, budget constraints, game scores, and more. The big question is: how do you solve an inequality and show all the numbers that make it true?

SECTION 2

Core Principles & Definitions

Before we start solving, let's nail down the key vocabulary. An inequality is a math sentence that uses a comparison symbol instead of an equals sign. Instead of saying "x equals 5," we might say "x is greater than 5."

Inequality Symbols

There are four symbols: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Solution Set

An equation usually has one answer. An inequality has a whole solution set — many (often infinite) numbers that make it true.

Open vs. Closed Circles

On a number line, an open circle (○) means the number is NOT included (< or >). A closed circle (●) means it IS included (≤ or ≥).

The Flip Rule

When you multiply or divide both sides by a negative number, you must flip the inequality symbol. This is the biggest rule to remember!

✦ KEY TAKEAWAY

Think of an inequality like a video game high-score board. If the rule is "you need more than 100 points to level up," then 101, 102, 500, and 1 million all work — but 100 does not. An inequality doesn't pick just one winner; it picks every number that passes the test.

SECTION 3

Seeing Inequalities on a Number Line

A number line is the best way to show the solution to an inequality. It lets you see at a glance which numbers are included. The diagram below shows all four inequality types.

Notice the difference: open circles (top two) mean the boundary number is NOT part of the answer. Closed (filled) circles (bottom two) mean the boundary number IS included. The shaded ray shows which direction the solutions go.

A quick trick: if the symbol has a line underneath (≤ or ≥), the circle is filled in. Think of the line in the symbol as "filling" the circle. If there's no line (< or >), the circle stays open.

SECTION 4

Solving One-Step Inequalities

Solving an inequality is a lot like solving an equation. You use inverse operations (doing the opposite) to get the variable alone. There is one extra rule you must watch out for.

ADDITION / SUBTRACTION RULE

If x + a > b, then x > b − a

Add or subtract the same number on both sides. The inequality symbol stays the same.

MULTIPLICATION / DIVISION BY A POSITIVE NUMBER

If ax > b (a > 0), then x > b ÷ a

Multiply or divide both sides by a positive number. The inequality symbol stays the same.

MULTIPLICATION / DIVISION BY A NEGATIVE NUMBER

If −ax > b, then x < b ÷ (−a) ← FLIP!

When you multiply or divide by a negative number, you must flip the inequality symbol. This is the #1 rule students forget!

💡 Why does the symbol flip?

Think about it: 2 < 5 is true. But multiply both sides by −1: you get −2 and −5. On the number line, −2 is to the right of −5, so −2 > −5. The order reversed! That's why we flip the symbol.

SECTION 5

Types of One-Step Inequalities

One-step inequalities come in four flavors based on which operation is used. The table below shows each type, an example, and how to solve it.

Summary of one-step inequality types

Operation	Example	What You Do	Solution
Addition	x + 4 > 10	Subtract 4 from both sides	x > 6
Subtraction	x − 3 ≤ 7	Add 3 to both sides	x ≤ 10
Multiplication (positive)	3x ≥ 12	Divide both sides by 3	x ≥ 4
Division (positive)	x ÷ 5 < 2	Multiply both sides by 5	x < 10
Multiplication (negative)	−2x > 8	Divide by −2 and FLIP	x < −4

This diagram walks through solving x + 4 > 10. We subtract 4 from both sides to get x > 6, then graph it with an open circle at 6 and shade to the right.

SECTION 6

Worked Example: Solving and Graphing

Let's solve a complete problem from start to finish. Pay attention to each step — especially the graphing part at the end.

Solve −5x ≥ 20 and graph the solution.

Step 1 — Identify the OperationThe variable x is being multiplied by −5. To undo multiplication, we need to divide both sides by −5.

Step 2 — Divide Both Sides by −5We divide each side by −5: (−5x) ÷ (−5) and 20 ÷ (−5). Since we are dividing by a negative number, we must flip the inequality symbol from ≥ to ≤.

x ≤ −4

Step 3 — Check with a Test ValuePick a number that should work, like x = −5. Plug it in: −5 × (−5) = 25. Is 25 ≥ 20? Yes! Now try one that should NOT work, like x = 0. −5 × 0 = 0. Is 0 ≥ 20? No. Our solution checks out.

Step 4 — Graph on a Number LineDraw a number line. Place a closed circle at −4 (because the symbol is ≤, meaning −4 IS included). Shade everything to the left of −4, since x is less than or equal to −4.

Solution: x ≤ −4 — closed circle at −4, shade left.

✅ Always Check Your Work!

After solving, plug a number from your shaded region into the original inequality. If it makes the inequality true, you're on the right track. If not, double-check whether you needed to flip the symbol.

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students make mistakes with inequalities. Here are the most common errors and how to steer clear of them.

Avoid these common pitfalls!

Mistake	Why It's Wrong	How to Fix It
Forgetting to flip the symbol when dividing by a negative	Multiplying or dividing by a negative reverses the order of numbers.	Write "FLIP!" next to any step where you divide or multiply by a negative.
Using a closed circle for < or >	A closed circle means the endpoint is included, but < and > do NOT include it.	Remember: no line under the symbol = no fill in the circle.
Shading the wrong direction	Shading left when the answer should go right (or vice versa) gives the wrong solution set.	After solving, read the inequality aloud. "x is greater than 6" means shade toward bigger numbers (right).
Flipping the symbol when adding or subtracting	You only flip when multiplying or dividing by a negative — never when adding or subtracting.	The flip rule ONLY applies to multiplication and division by negatives.

🔑 REMEMBER THIS

Think of the inequality symbol as a little mouth that always wants to eat the bigger number. When you multiply or divide both sides by a negative, the numbers swap positions on the number line — so the mouth has to turn around to face the new bigger side. That's the flip rule!

SECTION 8

From One-Step to Multi-Step Inequalities

Once you've mastered one-step inequalities, the next level is multi-step inequalities. These require two or more operations to solve — just like multi-step equations. But the same core rules apply.

Comparing one-step and multi-step inequalities

Feature	One-Step Inequality	Multi-Step Inequality
Number of operations to undo	1	2 or more
Example	x + 5 > 12	2x + 5 > 12
Flip rule	May apply (one chance)	May apply (check each step)
Graphing	Same number line method	Same number line method
Compound inequalities?	No	Yes (e.g., 3 < x + 1 ≤ 8)

The good news? Everything you learned today — inverse operations, the flip rule, open vs. closed circles, and shading — carries forward. You're building a strong foundation for all future work with inequalities.

SECTION 9

Practice Problems

Try these five problems on your own. Each one builds on the skills from this lesson. Write your solution and sketch a number line for each.

PROBLEM 1 — CONCEPTUAL

What is the difference between an open circle and a closed circle on a number line? Give an example inequality for each.

PROBLEM 2 — BASIC CALCULATION

Solve x + 9 > 15 and describe how you would graph the solution on a number line.

PROBLEM 3 — INTERMEDIATE

Solve −3x ≤ 18 and graph the solution. Explain why the inequality symbol changes direction.

PROBLEM 4 — APPLIED

You are saving money for a skateboard that costs $75. You already have $28. You earn the same amount each week from chores. Write and solve an inequality to find the minimum amount you must earn per week to buy the skateboard in exactly one more week.

PROBLEM 5 — CRITICAL THINKING

A student solves the inequality x ÷ (−4) > 2 and writes the answer as x > −8. They graph it with an open circle at −8 and shade right. Identify the mistake, correct the solution, and explain what the correct graph looks like.

SUMMARY

Lesson Summary

An inequality compares two expressions using the symbols <, >, ≤, or ≥. To solve a one-step inequality, use inverse operations (the opposite of addition is subtraction; the opposite of multiplication is division) to isolate the variable. The most important rule is the flip rule: whenever you multiply or divide both sides by a negative number, the inequality symbol reverses direction.

To graph on a number line, use an open circle for < or > (boundary not included) and a closed circle for ≤ or ≥ (boundary included). Then shade in the direction of the solutions. Always check your answer by plugging a test value back into the original inequality.