Writing and Graphing Inequalities

Grade 6 · Math · Study Guide

Learn to write, solve, and graph one-variable inequalities on a number line.

An inequality is like an equation, but instead of saying two things are equal, it says one is greater or less than the other. The four inequality symbols are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). An example: x > 3 means 'x can be any number greater than 3.'

Solving an inequality works almost exactly like solving an equation — you use the same inverse operations. To solve x + 5 > 12, subtract 5 from both sides: x > 7. There is one important exception: if you multiply or divide both sides by a negative number, you must FLIP the inequality sign. For example: −2x < 8 → divide by −2 → x > −4 (sign flipped!).

Graphing an inequality on a number line shows all possible solutions. Use an open circle (○) for strict inequalities (< or >) because the boundary number is NOT included. Use a closed circle (●) for ≤ or ≥ because the boundary IS included. Then shade the line in the direction of the solutions.

Inequalities describe real-world limits all the time. 'You must be at least 48 inches tall' means height ≥ 48. 'Speed limit 65 mph' means speed ≤ 65. Whenever you see 'at least,' 'at most,' 'no more than,' 'more than,' or 'less than,' an inequality is the right tool.

Inequality Symbols and Meanings
Symbol	Meaning	Example	Circle Type
<	Less than	x < 5 (x is less than 5)	Open ○
>	Greater than	x > 2 (x is greater than 2)	Open ○
≤	Less than or equal to	x ≤ 8 (x is 8 or less)	Closed ●
≥	Greater than or equal to	x ≥ 0 (x is 0 or more)	Closed ●

Key terms

Inequality: A mathematical statement using <, >, ≤, or ≥ to compare two expressions
Open Circle: On a number line graph, shows that the boundary value is NOT included (used for < and >)
Closed Circle: On a number line graph, shows that the boundary value IS included (used for ≤ and ≥)
Solution Set: All values that make an inequality true

Flashcards

< symbol: Less than — the open end points toward the smaller number
> symbol: Greater than — the open end points toward the larger number
≤ and ≥: Less/greater than OR EQUAL to — boundary value is included
Open circle on number line: Boundary is NOT included — used for strict < or >
Closed circle on number line: Boundary IS included — used for ≤ or ≥
Flip the sign rule: Multiply or divide by a negative number → flip the inequality sign

Practice questions

Which inequality means 'x is at least 10'?
1. x ≥ 10
2. x > 10
3. x ≤ 10
4. x < 10
Show answer
Answer: x ≥ 10
'At least 10' means 10 or more — that's the ≥ (greater than or equal to) symbol.
Solve: x + 4 < 11. What is the solution?
1. x < 7
2. x < 15
3. x > 7
4. x > 15
Show answer
Answer: x < 7
Subtract 4 from both sides: x + 4 − 4 < 11 − 4 → x < 7.
When graphing x > 5, you use a closed (filled) circle at 5.
Show answer
Answer: False
x > 5 uses an OPEN circle because 5 itself is NOT a solution (x must be strictly greater than 5).
When you divide both sides of an inequality by a negative number, you must ___ the inequality sign.
Show answer
Answer: flip
Multiplying or dividing by a negative reverses the order of the inequality — e.g. −2x < 8 → x > −4.
Which value is a solution to x ≤ −2?
1. −5
2. 0
3. 2
4. −1
Show answer
Answer: −5
x ≤ −2 means x can equal −2 or any number less. −5 < −2, so −5 is a solution.

Fun facts

The inequality symbols < and > were introduced by English mathematician Thomas Harriot in 1631.
Every speed limit sign is an inequality in disguise — 'Speed Limit 55' really means your speed ≤ 55 mph.

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