Linear Equations 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.
Solved Examples
Jump directly to a worked example by inequality type.
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. | $|u|\lt a$ means $-a\lt u\lt a$. |
| Absolute value greater than | Split into two outside cases. | $|u|\gt a$ means $u\lt -a$ or $u\gt a$. |
| Polynomial or rational inequality | Use a sign chart. | Critical numbers split the number line into test intervals. |
Content Formulas
Flip Rule
$$-2x\lt 8\quad \Longrightarrow \quad x\gt -4$$
Interval Notation
$$a\lt x\lt b\quad \Longrightarrow \quad (a,b)$$
Absolute Value Inside
$$|u|\lt a\quad \Longrightarrow \quad -a\lt u\lt a$$
Absolute Value Outside
$$|u|\gt a\quad \Longrightarrow \quad u\lt -a\ \text{or}\ u\gt a$$
Open circles match $<$ and $>$. Closed circles match $\le$ and $\ge$.
Classic Examples
Linear Inequality
Solve $-3x+5\le 17$.
Solution Steps
- Subtract $5$ from both sides.
- Divide by $-3$ and reverse the inequality sign.
- State the solution in interval notation: $[-4,\infty)$.
$$-3x\le 12$$$$x\ge -4$$$$[-4,\infty)$$
Compound Inequality
Solve $-2\lt 3x+1\le 10$.
Solution Steps
- Subtract $1$ from all three parts.
- Divide all three parts by $3$.
- State the solution in interval notation: $(-1,3]$.
$$-3\lt 3x\le 9$$$$-1\lt x\le 3$$$$(-1,3]$$
Absolute Value Less Than
Solve $|2x-1|\lt 7$.
Solution Steps
- Rewrite the absolute-value inequality as a compound inequality.
- Add $1$ to all three parts.
- Divide all three parts by $2$ and state the interval.
$$-7\lt 2x-1\lt 7$$$$-6\lt 2x\lt 8$$$$-3\lt x\lt 4$$
Absolute Value Greater Than
Solve $|x+2|\ge 5$.
Solution Steps
- Rewrite the absolute-value inequality as two outside cases.
- Solve both linear inequalities.
- Join the two regions with union notation.
$$x+2\le -5\quad \text{or}\quad x+2\ge 5$$$$x\le -7\quad \text{or}\quad x\ge 3$$$$(-\infty,-7]\cup[3,\infty)$$
Quadratic Sign Chart
Solve $(x-2)(x+3)\gt 0$.
Solution Steps
- Find the critical numbers by setting each factor equal to zero.
- Test the sign on the interval to the left of $-3$.
- Test the sign on the interval between $-3$ and $2$.
- Test the sign on the interval to the right of $2$.
- Select the positive intervals and state the solution.
$$x=-3,\quad x=2$$$$(-\infty,-3):\ (+)$$$$(-3,2):\ (-)$$$$(2,\infty):\ (+)$$$$(-\infty,-3)\cup(2,\infty)$$