Lesson 1: The Basics

Property

If $a > b$ and $b > c$, then $a > c$. This property also applies to other inequality symbols: if $a < b$ and $b < c$, then $a < c$; if $a \geq b$ and $b \geq c$, then $a \geq c$; if $a \leq b$ and $b \leq c$, then $a \leq c$.

Examples

Property

Subtraction Property of Inequality
For any numbers $a$, $b$, and $c$,
if $a < b$, then $a - c < b - c$.
if $a > b$, then $a - c > b - c$.

Addition Property of Inequality
For any numbers $a$, $b$, and $c$,
if $a < b$, then $a + c < b + c$.
if $a > b$, then $a + c > b + c$.

Examples

• To solve $x + 7 \leq 15$, subtract 7 from both sides. This gives $x \leq 8$. The solution is all numbers less than or equal to 8, or $(-\infty, 8]$.

Property

For any real numbers $a$, $b$, $c$
if $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$ and $ac < bc$.
if $a > b$ and $c > 0$, then $\frac{a}{c} > \frac{b}{c}$ and $ac > bc$.
if $a < b$ and $c < 0$, then $\frac{a}{c} > \frac{b}{c}$ and $ac > bc$.
if $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$ and $ac < bc$.
When we divide or multiply an inequality by a:
• positive number, the inequality stays the same.
• negative number, the inequality reverses.

Examples

• To solve $6x > 48$, divide both sides by 6. Since 6 is positive, the inequality stays the same: $x > 8$.

• To solve $-4y \geq 20$, divide both sides by -4. Since -4 is negative, the inequality reverses: $y \leq -5$.