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\( > \)
\( < \)
\( \geq \)

Simple
Inequalities

Learn to write, interpret, and graph inequalities like \( x > c \) and \( x < c \) on number lines: one skill, infinitely many solutions.

6th Grade Math

What Is an Inequality?

An inequality is a mathematical statement that compares two values using symbols like \( > \), \( < \), \( \geq \), or \( \leq \). Unlike an equation, which says two things are equal, an inequality says one value is greater than or less than another.

The Four Inequality Symbols

\( > \) Greater Than
\( x > 5 \) means \( x \) is any number larger than 5
\( < \) Less Than
\( x < 5 \) means \( x \) is any number smaller than 5
\( \geq \) Greater or Equal
\( x \geq 5 \) means \( x \) is 5 or any number larger
\( \leq \) Less or Equal
\( x \leq 5 \) means \( x \) is 5 or any number smaller

Why Do Inequalities Matter?

Real life is full of constraints: speed limits, minimum ages, budget limits, and temperature ranges. Inequalities let us model these rules mathematically.

Speed Limit
"You must drive at most 65 mph" becomes \( s \leq 65 \)
Roller Coaster
"You must be taller than 48 inches" becomes \( h > 48 \)
Budget
"You can spend less than $20" becomes \( c < 20 \)

Infinitely Many Solutions

An equation like \( x = 3 \) has one solution. But an inequality like \( x > 3 \) has infinitely many solutions: 3.1, 4, 10, 100, and every number beyond 3.

1 2 3 4 5 6
\( x > 3 \)

Open vs. Closed Circles

\( x > 3 \) (open circle)
 1 2 3 4 5
\( x \geq 3 \) (closed circle)
 1 2 3 4 5
Open Circle
Used with \( > \) or \( < \): the boundary value is not included
Closed Circle
Used with \( \geq \) or \( \leq \): the boundary value is included

Equation vs. Inequality

Equation Inequality
Symbol \( = \) \( >, <, \geq, \leq \)
Example \( x = 5 \) \( x > 5 \)
Solutions Exactly one value Infinitely many values
Graph A single point A ray on a number line
Meaning "is equal to" "is greater/less than"

How to Write an Inequality

1
Identify the variable. What quantity is unknown? Assign it a letter like \( x \), \( t \), or \( h \).
2
Find the boundary value. What number is the constraint based on? This is your \( c \).
3
Choose the symbol. Is the variable greater or less? Is the boundary included (\( \geq, \leq \)) or excluded (\( >, < \))?
4
Write it out. Combine: variable, symbol, boundary. Example: \( x > 12 \).

How to Graph on a Number Line

1
Draw a number line and mark the boundary value \( c \).
2
Choose the circle: Open for \( > \) or \( < \). Closed for \( \geq \) or \( \leq \).
3
Shade the ray: Shade right for "greater than." Shade left for "less than."
4
Add an arrow to show the solutions continue forever.

Example 1: Roller Coaster Height

"You must be at least 48 inches tall to ride the roller coaster." Write an inequality and graph it.

44 46 48 50 52
\( h \geq 48 \)

Example 2: Temperature Warning

"The temperature must stay below 32 degrees for the pond to freeze." Write the inequality and graph it.

28 30 32 34 36
\( t < 32 \)

Example 3: Spending Budget

"Maya has $15 to spend. She can spend no more than $15." Write the inequality and graph it.

5 10 15 20
\( m \leq 15 \)

Translating Key Phrases

English Phrase Symbol Example
"more than," "above," "over" \( > \) \( x > 7 \)
"fewer than," "below," "under" \( < \) \( x < 7 \)
"at least," "no fewer than" \( \geq \) \( x \geq 7 \)
"at most," "no more than" \( \leq \) \( x \leq 7 \)

Reading a Number Line Graph

You can also work backwards: given a graph, write the inequality it represents.

-2 -1 0 1 2 3 What inequality is this?
1
Circle type: The circle at \( -1 \) is closed (filled in), so the boundary is included.
2
Direction: The ray goes left, meaning values less than or equal to the boundary.
3
Answer: \( x \leq -1 \)

Common Mistakes

✕ Wrong Circle Type
Using an open circle for \( \leq \) or \( \geq \). Remember: a line under the symbol means closed (filled) circle.
✕ Shading the Wrong Way
Shading left for "greater than" or right for "less than." Greater = right; Less = left.
✕ Flipping the Symbol
Writing \( x > 5 \) when you mean \( x < 5 \). Always read: "x is [symbol] 5" and check if it matches the word problem.

Question 1 of 5

"A student must score more than 70 to pass." Which inequality represents this?

Question 2 of 5

Which number line shows \( x \leq 2 \)?

A) 2 B) 2 C) 2 D) 2

Question 3 of 5

Which value is not a solution to \( x > -3 \)?

Question 4 of 5

"A bag can hold at most 50 pounds." Which inequality models this?

Question 5 of 5

A number line shows an open circle at 4 with shading to the right. Which inequality does this represent?