Linear Equations and Inequalities Cheat Sheet

This cheat sheet covers solving and graphing linear inequalities, providing examples and key steps for various inequality types. It emphasizes algebraic manipulation and graphical representation of solutions.

Core Principles

Isolate the variable to find the solution set.
Maintain inequality direction when adding/subtracting the same value on both sides.
Reverse inequality direction when multiplying or dividing by a negative number.
Use open circles for strict inequalities (<, >) and closed circles for inclusive inequalities (<=, >=).
Graph solutions on a number line, shading the appropriate region.
Check solutions by substituting values into the original inequality.

Action Steps

Identify the inequality type and the variable to solve for.
Perform algebraic operations to isolate the variable.
Pay close attention to the sign of the number you multiply or divide by.
If dividing or multiplying by a negative, reverse the inequality sign.
Represent the solution on a number line.
Use an open circle for '<' or '>' and a closed circle for '≤' or '≥'.
Shade the number line to the left or right of the circle, according to the inequality.

Formulas

If $ax > b$, then $x < b/a$ (if $a < 0$)
If $ax < b$, then $x > b/a$ (if $a < 0$)
If $ax \ge b$, then $x \le b/a$ (if $a < 0$)
If $ax \le b$, then $x \ge b/a$ (if $a < 0$)

Key Terms

Inequality: A mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥.
Solution Set: The collection of all values that satisfy an inequality.
Number Line: A visual representation of numbers, used to graph inequality solutions.
Open Circle: Indicates that the endpoint is not included in the solution set (for < or >).
Closed Circle: Indicates that the endpoint is included in the solution set (for ≤ or ≥).

Real World Examples

Budgeting: Spending less than or equal to a certain amount: $x \le \text{budget}$.
Speed Limits: Driving at or below the speed limit: $v \le \text{limit}$.
Age Restrictions: Being older than a certain age to enter: $age > \text{restriction}$.