An equation says two things are exactly equal: x = 5. An inequality says one thing is bigger or smaller than another — and it can have many possible answers.
For example, x > 3 means x could be 4, 5, 10, 100, or any number greater than 3. That's an infinite number of solutions — and we show them all on a number line.
Key idea: Solving inequalities uses the same steps as solving equations — with one extra rule you'll learn below.
🔣 The Four Inequality Symbols
Memorize these four symbols — they each have a name, a real-world meaning, and a circle type on number lines.
Symbol
Meaning
Also means…
Circle on graph
>
Greater than
more than, above
Open
<
Less than
fewer than, below
Open
≥
Greater than or equal to
at least, minimum
Closed
≤
Less than or equal to
at most, maximum
Closed
Memory tip: If the symbol has a line under it (≥ or ≤), the value is included — use a closed (filled) circle. No line = open circle.
📏 Reading Number Lines
Every inequality can be shown on a number line. Two things matter: the circle type and the direction of shading.
Open Circle
The value is NOT included
Used with > and < Example: x > 3 → open circle at 3
Closed Circle
The value IS included
Used with ≥ and ≤ Example: x ≥ 3 → closed circle at 3
x > 3 — open circle, shade right
x ≤ 2 — closed circle, shade left
The symbol acts like an arrow tip. x > 3 — the > symbol points right, so shade right. x < 5 — the < symbol points left, so shade left.
🔄 Always Put the Variable First
When the variable is on the right side (like 4 < x), it's easy to graph in the wrong direction. Always rewrite so the variable comes first before you graph.
The Rewriting Rule
Swap both sides → the symbol flips direction
4 < x → x > 4 (open circle at 4, shade right)
9 ≥ x → x ≤ 9 (closed circle at 9, shade left)
−2 > x → x < −2 (open circle at −2, shade left)
⚠️ The One Extra Rule
Solving inequalities works almost exactly like solving equations. There is one extra rule — and it is the most commonly missed concept on the GED.
⚠️ The Flip Rule
When you multiply or divide both sides by a negative number, you must flip the inequality sign.
Positive → no flip. Negative → flip.
Example: −2x > 10
Divide both sides by −2 (negative!) → flip > to < → x < −5
Why does this happen? Think about it: 2 < 5 is true. Multiply both sides by −1: −2 and −5. Now −2 > −5 — the direction reversed! Multiplying by a negative flips the number line.
❌ Forgot to Flip
−3x > 12 → x > −4
Wrong! Divided by −3 but kept >
✓ Flipped Correctly
−3x > 12 → x < −4
Correct! Divided by −3 and flipped > to <
🔢 Worked Examples
Example 1 — Reading an Inequality
Graph x > 4
Symbol: > means strictly greater than → open circle (4 is not included)
Direction: > points right → shade right
Graph: x > 4
Example 2 — Closed Circle
Graph x ≤ 2
Symbol: ≤ means less than or equal to → closed circle (2 is included)
Direction: ≤ points left → shade left
Graph: x ≤ 2
Example 3 — One-Step Inequality
Solve: x + 7 < 12
Subtract 7 from both sides: x < 12 − 7
No negative division → no flip needed
Answer: x < 5
Graph: x < 5
Example 4 — Two-Step Inequality
Solve: 3x − 4 ≥ 11
Step 1: Add 4 to both sides → 3x ≥ 15
Step 2: Divide by 3 (positive → no flip) → x ≥ 5
Answer: x ≥ 5
Graph: x ≥ 5
Example 5 — The Flip Rule ⚠️
Solve: −2x > 8
Divide both sides by −2
−2 is negative → ⚠️ FLIP THE SIGN: > becomes <
Answer: x < −4
Graph: x < −4
🗺️ What Comes Next
This lesson covers the core inequality skills. Later in the sequence you will learn:
Coming up: Inequality word problems · Translating verbal statements into inequalities · Multi-step inequalities
✏️ Practice Questions
Guided Practice
Questions are randomized — answer as many as you like
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GED Level Questions
Questions are randomized — answer as many as you like