Linear Inequalities

Recall the following rules for working with inequalities (where $a$, $b$, and $c$ are real numbers):

1. Addition/Subtraction:
   1. If $a \leq b$, then $a \pm c \leq b \pm c$.
   
   
   2. If $a \geq b$, then $a \pm c \geq b \pm c$.
   
   
2. Multiplication/Division (if $c>0$):
   1. If $a \leq b$, then $ca \leq cb$ and $\displaystyle \frac{a}{c}\leq \frac{b}{c}$.
   
   
   2. If $a \geq b$, then $ca \geq cb$ and $\displaystyle \frac{a}{c}\geq \frac{b}{c}$.
   
   
3. Multiplication/Division (if $c< 0$):
   1. If $a \leq b$, then $ca \geq cb$ and $\displaystyle \frac{a}{c}\geq \frac{b}{c}$.
   
   
   2. If $a \geq b$, then $ca \leq cb$ and $\displaystyle \frac{a}{c}\leq \frac{b}{c}$.
   
   
[Note: Rules 3.i and 3.ii are often described “if you multiply or divide by a negative number you must reverse the inequality".]
4. Reciprocals:
   1. If $0 < a \leq b$, then $\displaystyle \frac{1}{a} \geq \frac{1}{b}$.
   
   
   2. If $a \geq b > 0$, then $\displaystyle \frac{1}{a} \leq \frac{1}{b}$.