Write and Represent Simple Inequalities - 6th Grade Math

SECTION 1

Historical Context & Motivation

Math isn't just about finding one exact answer. Sometimes, you need to describe a whole range of possibilities. Think about it: a roller coaster might say "you must be at least 48 inches tall to ride." That rule doesn't name one height — it describes every height that works. That's exactly the kind of problem that inequalities (math statements using symbols like > and <) were invented to solve.

~1700 BCE • Ancient Babylonians

Babylonian scribes solved problems about boundaries and limits on clay tablets, thinking about "more than" and "less than" long before special symbols existed.

~300 BCE • Euclid's Geometry

The Greek mathematician Euclid compared lengths and areas, writing things like "line A is greater than line B." He used words instead of symbols.

1631 • Inequality Symbols Invented

English mathematician Thomas Harriot introduced the > (greater than) and < (less than) symbols in his book, giving math a quick shorthand.

1800s • Number Lines Appear

Mathematicians started drawing number lines to picture solutions. This made it much easier to see that an inequality like x > 3 includes infinitely many numbers.

Today • Inequalities Everywhere

Speed limits, age requirements, budgets, and temperature ranges all use inequality thinking. It's one of the most useful ideas in everyday math.

So here's the big question: how do we write a math statement that captures all the numbers that fit a rule — not just one? That's where inequalities come in. Let's learn how to write them and show their solutions on a number line.

SECTION 2

Core Principles & Definitions

Before we dive in, let's nail down the key ideas you'll need. An inequality is a math sentence that compares two values using a symbol like > or <. Unlike an equation (which uses the = sign and usually has one answer), an inequality can have many, many answers.

Greater Than ( > )

The symbol > means the value on the left is larger than the value on the right. Example: x > 5 means x can be 6, 7, 100, or even 5.01.

Less Than ( < )

The symbol < means the value on the left is smaller than the value on the right. Example: x < 10 means x can be 9, 3, 0, or even −4.

Variable as Unknown

The letter x (or any variable) stands for an unknown number. In an inequality, x represents all numbers that make the statement true.

Infinitely Many Solutions

Between any two numbers, there are always more numbers (like decimals and fractions). That's why x > 3 has infinitely many solutions — they never stop.

Open Circle on Number Lines

When graphing x > c or x < c, we use an open circle at c to show that c itself is NOT included, then shade in the direction of the solutions.

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KEY TAKEAWAY

Think of an inequality like a velvet rope at a concert. An equation says "only person #7 gets in." An inequality says "everyone with a ticket number greater than 7 gets in." That's a LOT of people — in fact, infinitely many!

SECTION 3

Visualizing Inequalities on Number Lines

A number line diagram is the perfect way to show the solutions of an inequality. It lets you see at a glance which numbers work and which don't. Let's look at two examples side by side.

For x > 3 (top), the open circle at 3 shows that 3 itself is not a solution. The shading goes to the right, toward bigger numbers. For x < 3 (bottom), the open circle is at 3 again, but the shading goes to the left, toward smaller numbers. Both arrows show the solutions keep going forever.

Notice a few important things. First, the open circle means the boundary number is not a solution. The number 3 does not satisfy x > 3 or x < 3. Second, the arrow at the end of the shaded region shows that solutions continue forever. There is no biggest number that satisfies x > 3 — you can always find a bigger one!

SECTION 4

Writing Inequalities from Words

A big part of this skill is translating real-world words into math symbols. Here are the two main inequality forms you'll use.

GREATER THAN

x > c

x is the unknown value, and c is a constant (a specific number). This reads: "x is greater than c." The solutions are all numbers to the right of c on a number line.

LESS THAN

x < c

This reads: "x is less than c." The solutions are all numbers to the left of c on a number line.

Key Phrases to Watch For

English Phrase	Inequality Symbol	Example
more than, above, over, exceeds	>	"more than 5" → x > 5
less than, below, under, fewer than	<	"below 20" → x < 20
must be greater than	>	"must be greater than 0" → x > 0
cannot exceed, must stay under	<	"must stay under 100" → x < 100

Common phrases and their inequality translations

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Helpful Tip

The inequality symbol is like an alligator's mouth — it always opens toward the bigger value. In 8 > 3, the mouth opens toward 8 because 8 is bigger.

SECTION 5

Understanding Infinitely Many Solutions

One of the most important ideas about inequalities is that they have infinitely many solutions. Let's explore why this is true using a concrete example.

Consider x > 2. Is 3 a solution? Yes, because 3 > 2 is true. Is 2.5 a solution? Yes! Is 2.1 a solution? Yes! Is 2.001 a solution? Absolutely! No matter how close you get to 2, you can always find another number that is still greater than 2. This goes on forever, which is what infinitely many means.

The green-shaded region on the number line covers every number greater than 2. Whole numbers like 3 and 4 are solutions, but so are decimals like 2.5 and 2.001. Numbers equal to 2 or less than 2 (shown in red) are NOT solutions.

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Important

The boundary number itself is never a solution for x > c or x < c. That's why we use an open circle on the number line — to show that point is not included.

SECTION 6

Worked Example: From Words to Number Line

Let's walk through a complete example. Here's the problem:

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Problem

A movie theater requires that you be over 12 years old to watch a PG-13 movie without an adult. Write an inequality for the ages that can watch the movie alone, and graph it on a number line.

Full Solution

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Step 1 — Identify the Variable

The unknown is the person's age. Let's call it a (you can use any letter, but 'a' for age makes sense).

Variable: a = age in years
2
Step 2 — Find the Key Phrase

The phrase "over 12 years old" means the person's age must be greater than 12. The word "over" tells us to use the > symbol.

Key phrase: "over" → greater than → >
3
Step 3 — Write the Inequality

Put the variable on the left, the > symbol in the middle, and the boundary number on the right.

a > 12
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Step 4 — Draw the Number Line

Draw a number line with 12 clearly labeled. Place an open circle at 12 (because 12 itself is not "over 12"). Then shade everything to the right of 12, because greater-than solutions go to the right.

Open circle at 12, shade right with an arrow
5
Step 5 — Check with a Test Value

Pick a number in the shaded region, like 15. Is 15 > 12? Yes! Now pick a number outside, like 10. Is 10 > 12? No! Our graph is correct.

✓ 15 > 12 is TRUE ✗ 10 > 12 is FALSE

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STRATEGY RECAP

Follow these steps every time: (1) pick a variable, (2) find the key comparison word, (3) write the inequality, (4) graph it with an open circle and shading, and (5) test a value to make sure it works.

SECTION 7

Equations vs. Inequalities — What's the Difference?

You already know how to work with equations. Inequalities are closely related, but there are some important differences. Let's compare them side by side.

Feature	Equation (x = c)	Inequality (x > c or x < c)
Symbol	=	> or <
Number of solutions	Exactly one	Infinitely many
Number line graph	A single filled-in dot	Open circle + shaded ray
Example	x = 5 → only the number 5	x > 5 → 5.1, 6, 100, …
Real-world use	"The answer IS 42."	"You must be OVER 42 inches tall."

Key differences between equations and inequalities

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KEY TAKEAWAY

An equation is like a lock with one key — only one number opens it. An inequality is like a lock that accepts any key above a certain size (or below it). That's why we need a whole shaded region on the number line, not just a single dot.

SECTION 8

Connection to Future Math: ≤, ≥, and Compound Inequalities

Right now, you're working with strict inequalities (that's the official name for > and <). But in later grades, you'll meet some close relatives. Here's a sneak peek.

What You Know Now	What's Coming Next
x > c (greater than) — open circle	x ≥ c (greater than or equal to) — closed (filled) circle
x < c (less than) — open circle	x ≤ c (less than or equal to) — closed (filled) circle
One inequality at a time	Compound inequalities like 2 < x < 7 (a range between two values)
Inequality with one variable	Inequalities with two variables graphed on a coordinate plane

How today's lesson connects to future algebra skills

The difference between > and ≥ is small but important. With ≥ (greater than or equal to), the boundary number is a solution. On a number line, you'd use a filled-in (closed) circle instead of an open one. But don't worry about that for now — master the open-circle version first, and the rest will feel easy later.

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Fun Fact

In high school, you'll solve inequalities by doing algebra, just like you solve equations. For example, if 2x > 10, you divide both sides by 2 to get x > 5. The skills you're learning today are the foundation for all of that!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own, then check the answer. The problems start easy and get harder.

Question 1: CONCEPTUAL

True or false: The number 7 is a solution to the inequality x > 7. Explain your reasoning.

Show answer

False. The inequality x > 7 means "x is greater than 7." The number 7 is not greater than 7 — it is equal to 7. For a number to be a solution of x > 7, it must be even a tiny bit larger than 7, like 7.001 or 8. That's also why we use an open circle at 7 on the number line — to show that 7 itself is not included.

Question 2: BASIC CALCULATION

Write an inequality for this situation: A store sells only items that cost less than $25. Use the variable p for the price. Then name three numbers that are solutions and one that is not.

Show answer

The inequality is p < 25. The phrase "less than $25" tells us to use the < symbol with 25 as the boundary. Three solutions could be p = 10, p = 24.99, and p = 0 (since all of these are less than 25). A non-solution is p = 25, because 25 is not less than 25, and p = 30 wouldn't work either.

Question 3: INTERMEDIATE

The temperature outside must be above 0°C for ice to melt. Write an inequality using the variable t. Then describe what the number line graph would look like.

Show answer

The inequality is t > 0. "Above 0" means greater than 0. On a number line, you would place an open circle at 0 (because 0 itself is not above 0 — ice doesn't melt at exactly 0°C in this problem). Then you would shade everything to the right of 0 with an arrow to show the solutions go on forever. Numbers like 0.5, 1, 10, and 100 are all solutions. Numbers like −3, −1, and 0 are not solutions.

Question 4: APPLIED

Marcus has a gift card with $50 on it. He wants to buy a video game, but he needs to have more than $15 left on the card afterward for a future purchase. If the video game costs g dollars, write an inequality for the amount of money left on the card (50 − g) and explain what values of g would work.

Show answer

The money left after buying the game is 50 − g. Marcus needs this leftover amount to be more than 15. So the inequality for the leftover money is 50 − g > 15. This means 50 minus the cost of the game must be greater than 15. If the game costs $30, then 50 − 30 = 20, and 20 > 15 is true, so that works. If the game costs $35, then 50 − 35 = 15, and 15 > 15 is false, so $35 does not work. The game must cost less than $35 for Marcus to have more than $15 left.

Question 5: CRITICAL THINKING

Jenna says that x > 3 and 3 < x are two different inequalities with different solutions. Is she correct? Explain why or why not, and describe what would happen if you graphed both on the same number line.

Show answer

Jenna is incorrect. The inequalities x > 3 and 3 < x mean exactly the same thing. Both say that x is greater than 3. In x > 3, the variable is on the left. In 3 < x, the variable is on the right, but the < symbol still opens toward x, meaning x is the bigger value. If you graphed both on the same number line, the graphs would be identical: an open circle at 3 with shading to the right. This is an important lesson — you can read an inequality from either direction, as long as the "mouth" of the symbol points to the larger value.

SUMMARY

Lesson Summary

Summary

In this lesson, you learned how to write simple inequalities of the form x > c or x < c to represent real-world constraints. The greater-than symbol ( > ) means the variable is larger than the number, while the less-than symbol ( < ) means it is smaller. Key English phrases like "more than," "above," "below," and "under" are your clues for choosing the right symbol.

You also discovered that these inequalities have infinitely many solutions — between any two solutions, you can always find another one. To show these solutions, you graph them on a number line using an open circle at the boundary number (to show it's NOT included) and a shaded ray with an arrow pointing toward all the solutions. Shade right for >, shade left for <. Always test a value to check your work!