01_Ex 5.1

Exercise Questions

Questions	Links
Q1. Graph the solution set of each of the following…	5.1 Q-1
Q2. Indicate the solution set of the following…	5.1 Q-2
Q3. Indicate the region of the following…	5.1 Q-3
Q4. Graph the solution region of the following systems…	5.1 Q-4
Q5. Graph the solution region of the following systems…	5.1 Q-5

The Questions provide a general method and might skip over specific parts, Refer to video appended at the bottom for a deeper understanding of each part

Overview

This exercise focuses on graphing the solution sets of linear inequalities in the $xy$ -plane. By solving inequalities and plotting them, students gain a deeper understanding of the relationship between algebraic expressions and their graphical representations. The concepts covered are critical for visualizing and solving systems of inequalities, a skill applicable in optimization problems and linear programming.

Key Concepts

Linear Inequalities: Expressions involving variables $x$ and $y$ with inequality signs (e.g., $\leq$ , $\geq$ ).
Solution Set: The set of points $(x, y)$ that satisfy the inequality.
Graphing: Plotting points and regions on the $xy$ -plane that satisfy the inequality. Boundary lines or curves are drawn for equalities and the shading represents solutions to the inequality.
Feasible Region: The shaded area that satisfies the inequality constraints in a system.

Important Formulas

Linear Inequality: $ax + by \leq c$ $a x + b y \leq c$ or $ax + by \geq c$ $a x + b y \geq c$
- Solution involves plotting the boundary line (where equality holds), and shading the region that satisfies the inequality.
Point Calculation for Intersection:
- Set $x = 0$ or $y = 0$ to find intercepts, then plot these points.
- Example: For $2x + y \leq 6$ , the boundary line is $2x + y = 6$ .

Tips and Tricks

Always check whether the inequality is strict (e.g., $< \text{ or } >$ ) or non-strict (e.g., $\leq \text{ or } \geq$ ) to determine if the boundary line is dashed or solid.
For systems of inequalities, graph each inequality separately and then find the intersection of the shaded regions.
For quick graphing, find two intercepts by setting $x = 0$ and $y = 0$ , then draw the line through those points.

Summary

This exercise demonstrates how to solve and graph linear inequalities in the $xy$ -plane. By identifying the solution set, students learn to visualize the relationship between algebraic equations and their geometric representation. Understanding how to graph these inequalities is foundational in topics like linear programming and optimization.

Reference

By Sir Shahzad Sair: