1.8: Inequalities
Inequalities are just equations with the $=$ replaced with $<, >, \leq$ or $\geq$.
For example: \[3x + 5 \leq 17\] is an inequality.
We are interested in solving these, similar to solving equations in Section 1.5.
To manipulate inequalities, use these properties:
Properties of Inequalities
- If $A \leq B$, then $A \pm C \leq B \pm C$.
English Adding the same number on both sides does not flip the inequality.
- If $C > 0$, then given $A \leq B$, we know $A\cdot C \leq B\cdot C$.
English Multiplying a positive number on both sides does not flip the inequality.
- If $C < 0$, then:
- Given $A \leq B$, we know $A\cdot C \geq B \cdot C$
- Given $A < B$, we know $A\cdot C > B \cdot C$
English Multiplying a negative number on both sides flips the inequality.
Solving Linear Inequalities
To solve an inequality, isolate the variable using the previous properties.
Solve the inequality $3x < 9x + 4$.
Solve the inequality $4 \leq 3x - 2 < 13$.