📈 Linear Equations & Inequalities — One-Variable Systems
Fundamental building blocks for more complex algebraic problems.
🎯 Key Ideas
Linear Equations — Equations of the form $ax + b = 0$ where $a \neq 0$. The solution is $x = -\frac{b}{a}$.
Linear Inequalities — Inequalities using $<$, $>$, $\leq$, $\geq$. Solutions are intervals on the number line.
Interval Notation — Compact way to represent solution sets: $(a,b)$ for open intervals, $[a,b]$ for closed intervals.
📊 Essential Concepts
Linear Equations
| Type | Form | Solution | Example |
|---|---|---|---|
| Standard | $ax + b = 0$ | $x = -\frac{b}{a}$ | $2x + 3 = 0$ → $x = -\frac{3}{2}$ |
| With fractions | $\frac{x}{a} + \frac{b}{c} = 0$ | $x = -\frac{ab}{c}$ | $\frac{x}{2} + \frac{1}{3} = 0$ → $x = -\frac{2}{3}$ |
| With parentheses | $a(x + b) = c$ | $x = \frac{c}{a} - b$ | $3(x + 2) = 9$ → $x = 1$ |
Linear Inequalities
| Symbol | Meaning | Interval | Example |
|---|---|---|---|
| $<$ | Less than | $(a,b)$ | $x < 5$ → $(-\infty, 5)$ |
| $\leq$ | Less than or equal | $[a,b]$ | $x \leq 3$ → $(-\infty, 3]$ |
| $>$ | Greater than | $(a,b)$ | $x > -2$ → $(-2, \infty)$ |
| $\geq$ | Greater than or equal | $[a,b]$ | $x \geq 0$ → $[0, \infty)$ |
🎯 Micro-Examples
Example 1: Solve $3x - 7 = 2x + 1$
- Subtract $2x$: $x - 7 = 1$
- Add $7$: $x = 8$
- Answer: $x = 8$
Example 2: Solve $2(x + 3) = 4x - 2$
- Distribute: $2x + 6 = 4x - 2$
- Subtract $2x$: $6 = 2x - 2$
- Add $2$: $8 = 2x$
- Divide by $2$: $x = 4$
- Answer: $x = 4$
Example 3: Solve $3x + 2 < 8$
- Subtract $2$: $3x < 6$
- Divide by $3$: $x < 2$
- Answer: $x \in (-\infty, 2)$
⚠️ Common Traps & Fixes
Trap: Forgetting to flip inequality sign when multiplying/dividing by negative
- Fix: Always flip the sign when multiplying or dividing by a negative number
- Example: $-2x > 6$ becomes $x < -3$ (flipped from $>$ to $<$)
Trap: Incorrect interval notation
- Fix: Remember $(a,b)$ is open, $[a,b]$ is closed
- Example: $x < 5$ is $(-\infty, 5)$, not $(-\infty, 5]$
Trap: Losing solutions when clearing fractions
- Fix: Multiply by the LCD of all denominators
- Example: $\frac{x}{2} + \frac{1}{3} = \frac{x}{6}$ → multiply by $6$: $3x + 2 = x$
🎯 AMC-Style Worked Example
Problem: Find all real values of $x$ such that $2x - 1 \leq 3x + 2 < 5x - 1$.
Solution:
- Split compound inequality: $2x - 1 \leq 3x + 2$ and $3x + 2 < 5x - 1$
- Solve first inequality: $2x - 1 \leq 3x + 2$
- Subtract $2x$: $-1 \leq x + 2$
- Subtract $2$: $-3 \leq x$
- Solve second inequality: $3x + 2 < 5x - 1$
- Subtract $3x$: $2 < 2x - 1$
- Add $1$: $3 < 2x$
- Divide by $2$: $\frac{3}{2} < x$
- Combine: $-3 \leq x$ and $x > \frac{3}{2}$
- Answer: $x \in \left(\frac{3}{2}, \infty\right)$
Key insight: Compound inequalities require solving each part separately, then finding the intersection.
🔗 Related Topics
- Systems of Equations — Linear equations are components of systems
- Absolute Value — Often involves linear expressions
- Word Problems — Linear equations model many real-world situations
- Graphing — Linear equations have straight-line graphs
📝 Practice Checklist
- Master basic linear equation solving
- Practice with fractions and parentheses
- Learn interval notation
- Practice compound inequalities
- Understand graphical interpretation
- Work on speed and accuracy
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