Introduction to Inequalities

While equations use $=$, inequalities compare values that are not equal.

Inequality Symbols

Reading Inequalities

Tip: The "hungry alligator" always eats the bigger number!

• $x < 5$ reads as "x is less than 5"
• $x > 5$ reads as "x is greater than 5"
• $x \leq 5$ reads as "x is less than or equal to 5" (x can be 5!)
• $x \geq 5$ reads as "x is greater than or equal to 5" (x can be 5!)

Solutions to Inequalities

Unlike equations with one solution, inequalities have many solutions.

For $x > 3$:

• $x = 4$ ✓
• $x = 5$ ✓
• $x = 100$ ✓
• $x = 3.001$ ✓
• $x = 3$ ✗ (not greater than 3)
• $x = 2$ ✗

Graphing on a Number Line

We show solutions on a number line:

• Open circle (○): Number is NOT included ($<$ or $>$)
• Closed circle (●): Number IS included ($\leq$ or $\geq$)
• Arrow: Shows direction of all solutions

Example: $x > 2$

Use an open circle at 2, arrow pointing right →

Example: $x \leq -1$

Use a closed circle at -1, arrow pointing left ←