Overview

This topic focuses on solving, graphing, and interpreting linear equations and inequalities. You’ll learn techniques for isolating variables, understanding slope-intercept form, and handling compound and absolute value inequalities. Mastery of these fundamentals is essential for analyzing relationships and modeling real-world problems.

Key Concepts and Structures

• Linear Equation: An equation of the form ax + b = 0. The graph is a straight line.
• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
• Standard Form: Ax + By = C. Rearranged to find intercepts or slope.
• Solving Equations: Use inverse operations to isolate the variable. Always simplify both sides first.
• Graphing: Identify slope and intercept, plot the y-intercept, and apply slope to find other points.
• Linear Inequalities: Similar to equations but use <, >, ≤, ≥. Remember to reverse the inequality when multiplying or dividing by a negative.
• Compound Inequalities: Combine two inequalities with "and" (overlap) or "or" (union).
• Absolute Value Equations: Split into two cases: |x| = a becomes x = a or x = -a.
• Absolute Value Inequalities: For |x| < a, use a compound inequality. For |x| > a, use two separate inequalities with "or".

Step-by-Step Example

Problem: Solve and graph the inequality 2(x - 3) > 4x + 1

Step 1: Expand both sides:
2x - 6 > 4x + 1

Step 2: Move all terms to one side:
2x - 6 - 4x - 1 > 0 → -2x - 7 > 0

Step 3: Solve for x:
-2x > 7
Important: Divide by -2 → x < -3.5

Final Answer: x < -3.5. Graph as an open circle at -3.5 with shading to the left.

Quick Tip