Solving and Graphing Inequalities

Inequality Symbols

$<$ less than · $>$ greater than · $\leq$ less than or equal · $\geq$ greater than or equal

Solving Inequalities

Same as equations, with ONE key difference:

When you multiply or divide by a negative number, FLIP the inequality sign!

Examples

$x + 4 > 10 \implies x > 6$

$3x \leq 15 \implies x \leq 5$

$-2x > 8 \implies x < -4$ (sign flipped!)

$\frac{x}{-3} \geq 4 \implies x \leq -12$ (sign flipped!)

Graphing on a Number Line

• $x > 3$: Open circle at 3, shade right
• $x \geq 3$: Closed circle at 3, shade right
• $x < -1$: Open circle at -1, shade left
• $x \leq -1$: Closed circle at -1, shade left

Two-Step Inequalities

$2x - 5 > 7$ $2x > 12$ $x > 6$

Checking Solutions

Is $x = 8$ a solution to $2x - 5 > 7$? $2(8) - 5 = 11 > 7$ ✓ Yes!

Is $x = 5$ a solution? $2(5) - 5 = 5 > 7$? No! ✗

Key difference from equations: An inequality has infinitely many solutions, shown as a range on the number line.