Inequalities

Inequality Symbols

| Symbol | Meaning | |--------|---------| | $<$ | Less than | | $>$ | Greater than | | $\leq$ | Less than or equal to | | $\geq$ | Greater than or equal to |

Graphing Inequalities

On a number line:

• $<$ or $>$: Open circle (value not included)
• $\leq$ or $\geq$: Closed circle (value included)
• Shade in the direction of the solutions

Writing Inequalities from Words

| Words | Inequality | |-------|-----------| | "at least 5" | $x \geq 5$ | | "no more than 10" | $x \leq 10$ | | "fewer than 8" | $x < 8$ | | "more than 3" | $x > 3$ |

Solving Inequalities

Same as equations, EXCEPT: flip the inequality sign when you multiply or divide by a negative number.

$-4x \geq 20$ $x \leq -5$ (sign flipped!)

Compound Inequalities

$-3 < x \leq 7$

This means $x$ is between $-3$ (exclusive) and $7$ (inclusive).

Checking Solutions

Is $x = 4$ a solution to $2x + 3 > 9$? $2(4) + 3 = 11 > 9$ ✓ Yes!

Is $x = 2$ a solution? $2(2) + 3 = 7 > 9$ ✗ No!

Key difference from equations: Inequalities have infinitely many solutions, not just one!