Graphing Systems of Inequalities: Step-by-Step Visual Guide

Welcome to our lesson on graphing linear inequalities!We'll start with a simple inequality: y is greater than two x plus one.To graph an inequality, we first treat it as an equation and draw the line y equals two x plus one.Since our inequality uses the greater than symbol, we use a dashed line to show that points exactly on the line are not part of the solution.To determine which side of the line to shade, we'll test a point. Let's use the origin, point zero comma zero.Let's substitute zero comma zero into our inequality to see if it's a solution.Since zero is not greater than one, the point zero comma zero is not in our solution region. Therefore, we shade the region above the line.All points in this shaded region satisfy our inequality y greater than two x plus one.Remember these key points when graphing inequalities.Now that we understand how to graph a single inequality, let's explore what happens when we have two inequalities that need to be satisfied at the same time.Let's start with our first inequality: y is greater than 2x plus 1.For our second inequality, we'll add y is less than negative x plus 5.The solution to our system of inequalities is the region where both conditions are true simultaneously. This is where our shaded regions overlap.One important point in our solution region is where the two lines intersect. This occurs at the point four-thirds comma eleven-thirds.Notice that these two inequalities create what we call a bounded region - meaning our solution has a finite area with clear boundaries on all sides.The purple region represents all points that satisfy both inequalities. Any point inside this region is a valid solution to our system.Now that we have our system of inequalities graphed, let's verify our solution by testing specific points.Let's test three points: one inside our solution region, and two outside.This point satisfies both inequalities, confirming it's in our solution region.This point fails at least one inequality, confirming it's not a solution.This point also fails our inequalities, confirming it's outside our solution region.Now let's see how systems of inequalities apply to real business problems, like production planning.A company needs to decide how many widgets and gadgets to produce, subject to various constraints.Let's visualize these constraints on a coordinate plane, where x represents widgets and y represents gadgets.The blue line represents our budget constraint, the red line shows our storage limit, and the green line indicates our minimum production requirement.The shaded region shows all possible combinations of widgets and gadgets that satisfy all our constraints.For example, producing 100 widgets and 100 gadgets would be a feasible solution that satisfies all constraints.