1. Linear Equations
An equation is a balanced statement. Think of it like a scale: whatever you do to one side, you must do to the other.
Goal: Isolate the variable $x$.
Standard Procedure:
- Expand/Simplify: Use distributive properties to remove parentheses.
- Collect Variables: Move all terms containing $x$ to one side and constants to the other.
- Divide: Divide by the coefficient of $x$ to solve.
2. Linear Inequalities
Unlike equations, inequalities represent ranges of values rather than a single point.
One Variable (Number Line)
Example: $2x + 4 > 10 \implies 2x > 6 \implies x > 3$
Two Variables (Coordinate Plane)
Example: $y > mx + c$
Solid Line ($\leq, \geq$): Used when the boundary is included.
Dashed Line ($<, >$): Used when the boundary is excluded.
Shading: Shade above for $y >$ and below for $y <$.
⚠️ The Golden Rule: If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the inequality sign (e.g., $<$ becomes $>$).