Chapter 5 — Linear Inequalities

Linear inequalities are mathematical statements relating two real numbers or algebraic expressions using symbols <, >, ≤, or ≥. This chapter teaches how to solve one-variable and two-variable linear inequalities algebraically and graphically.

Chapter 5 covers linear inequalities in one and two variables, essential mathematical concepts used in science, statistics, economics, and psychology. Students learn the formal definition of inequalities and how they differ from equations. The chapter focuses on algebraic solutions using two key rules: adding/subtracting equal numbers from both sides preserves the inequality sign, while multiplying/dividing by negative numbers reverses it. Real-world applications include determining maximum purchases with a budget, mixing solutions to specific concentrations, and finding ranges for temperature conversions. Solutions can be expressed as sets, intervals, or graphs on a number line. The chapter includes double inequalities (e.g., 3 < x < 5) and systems of simultaneous inequalities.

1. 01An inequality consists of two real numbers or algebraic expressions related by <, >, ≤, or ≥; unlike equations, inequalities represent ranges of values, not exact solutions.
2. 02The solution of an inequality is any value of the variable that makes the statement true; solutions are often represented as intervals (e.g., x ∈ (−∞, 2)) or graphically on a number line.
3. 03Rule 1: Equal numbers may be added to or subtracted from both sides of an inequality without changing the inequality sign; Rule 2: Multiplying or dividing both sides by a positive number preserves the sign, but by a negative number reverses it (e.g., 3 > 2 becomes −3 < −2).
4. 04Linear inequalities in one variable have the form ax + b < 0 (or >, ≤, ≥); in two variables, ax + by < c (or >, ≤, ≥); higher-degree inequalities like ax² + bx + c ≤ 0 are quadratic, not linear.
5. 05Double inequalities (e.g., −8 ≤ 5x − 3 < 7) are solved by treating them as two simultaneous inequalities; graphically, the solution is the region where both conditions overlap on the number line.
6. 06Practical applications include solving real-world problems with constraints: finding minimum test scores for target averages, determining ranges for temperature conversions (Celsius to Fahrenheit), calculating mixture ratios for acid solutions, and identifying pairs of numbers satisfying multiple conditions.