Linear Equations and Inequalities Cheat Sheet

This cheat sheet summarizes key concepts and techniques for solving linear equations, inequalities, and absolute value equations/inequalities, including graphing solutions and understanding different forms of linear equations.

Core Principles

Linear equations have a variable power of 1.
Solving inequalities is similar to solving equations, but multiplying/dividing by a negative number reverses the inequality sign.
Absolute value represents the distance from zero and requires solving two separate equations or inequalities.
Different forms of linear equations exist: point-slope, slope-intercept, and general form.
Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals.

Action Steps

For linear equations: Isolate the variable by performing inverse operations on both sides.
For linear inequalities: Solve like an equation, but reverse the inequality sign when multiplying or dividing by a negative number.
For absolute value equations: Isolate the absolute value, then set up two separate equations (positive and negative cases).
For absolute value inequalities: Isolate the absolute value, then set up two separate inequalities (greater than/less than cases).
For rational inequalities: Move all terms to one side, find a common denominator, find critical points (numerator=0, denominator=0), and test intervals.

Formulas

Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Point-Slope Form: $y - y_1 = m(x - x_1)$
Slope-Intercept Form: $y = mx + b$
General Form: $Ax + By + C = 0$
Parallel Lines: $m_1 = m_2$
Perpendicular Lines: $m_1 m_2 = -1$
Absolute Value Equation: $|x| = c \Rightarrow x = c \text{ or } x = -c$
Absolute Value Inequality (gt; $): $|x| > c \Rightarrow x > c \text{ or } x < -c$
Absolute Value Inequality (lt; $): $|x| < c \Rightarrow -c < x < c$

Key Terms

Slope: The measure of the steepness of a line, calculated as the ratio of the change in y-values to the change in x-values.
Point-Slope Form: An equation of a line given a point $(x_1, y_1)$ and the slope $m$: $y - y_1 = m(x - x_1)$.
Slope-Intercept Form: An equation of a line given the slope $m$ and the y-intercept $b$: $y = mx + b$.
Absolute Value: The distance of a number from zero on the number line, always non-negative.
Extraneous Solution: A solution obtained through the solving process that does not satisfy the original equation or inequality.
Interval Notation: A way to represent a set of numbers using parentheses for open intervals and brackets for closed intervals.

Pro Tips

Always check your solutions for radical equations and rational equations/inequalities to eliminate extraneous solutions.
When solving absolute value inequalities, remember that '$|x| > c#39; results in two separate intervals, while '$|x| < c#39; results in a single interval.
For inequalities, pay close attention to whether the inequality symbol includes 'equal to' (solid dot, bracket) or not (open dot, parenthesis).
When dealing with rational inequalities, the values that make the denominator zero are never included in the solution set.

Pitfalls to Avoid

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Including values that make the denominator zero in the solution set for rational inequalities.
Not checking for extraneous solutions in radical equations.
Confusing the 'and' (overlap) for '$|x| < c#39; with the 'or' (union) for '$|x| > c#39;.
Incorrectly handling the 'equal to' part of inequality symbols (< vs. ≤).