Foundations Reference
Inequalities
An inequality describes a range of values instead of one exact value. The main skill is preserving the direction of the inequality while rewriting it.
Fact Table
| Type | Method | Reminder |
|---|---|---|
| Linear inequality | Solve like an equation. | Flip the sign when multiplying or dividing by a negative. |
| Compound "and" | Both conditions must be true. | The graph is the overlap. |
| Compound "or" | At least one condition must be true. | The graph is the union. |
| Absolute value less than | Rewrite as a bounded interval. | $|u| |
| Absolute value greater than | Split into two outside cases. | $|u|>a$ means $u<-a$ or $u>a$. |
| Polynomial or rational inequality | Use a sign chart. | Critical numbers split the number line into test intervals. |
Content Formulas
Flip Rule
$$-2x<8\quad \Longrightarrow \quad x>-4$$
Interval Notation
$$a
Absolute Value Inside
$$|u|
Absolute Value Outside
$$|u|>a\quad \Longrightarrow \quad u<-a\ \text{or}\ u>a$$
Open circles match $<$ and $>$. Closed circles match $\le$ and $\ge$.
Classic Examples
Linear Inequality
Solve $-3x+5\le 17$.
Inequality MoveWhen dividing by a negative, reverse the inequality sign.
$$-3x\le 12$$$$x\ge -4$$$$[-4,\infty)$$
Compound Inequality
Solve $-2<3x+1\le 10$.
Inequality MoveUndo operations to all three parts at the same time.
$$-3<3x\le 9$$$$-1$$(-1,3]$$
Absolute Value Less Than
Solve $|2x-1|<7$.
Inequality MoveLess than means the inside expression stays between two bounds.
$$-7<2x-1<7$$$$-6<2x<8$$$$-3
Absolute Value Greater Than
Solve $|x+2|\ge 5$.
Inequality MoveGreater than splits into two outside regions.
$$x+2\le -5\quad \text{or}\quad x+2\ge 5$$$$x\le -7\quad \text{or}\quad x\ge 3$$$$(-\infty,-7]\cup[3,\infty)$$
Quadratic Sign Chart
Solve $(x-2)(x+3)>0$.
Inequality MoveZeros split the line; test one value in each interval.
$$x=-3,\quad x=2$$$$(-\infty,-3):\ (+)$$$$(-3,2):\ (-)$$$$(2,\infty):\ (+)$$$$(-\infty,-3)\cup(2,\infty)$$