Problem 39 - Elementary Algebra Within Reach

Question

Explain how to sketch the graph of a system of inequalities in two variables.

Step-by-Step Solution

Answer

To sketch the graph of a system of inequalities in two variables, graph each inequality individually, shading the region that satisfies each inequality according to whether the inequality sign is '> / <' or '≥ / ≤'. The solution to the system of inequalities will be the overlapping region of these individual solutions. Remember to use dashed lines for '>' and '<' and solid lines for '≥' and '≤'.

Step 1: Graphing the First Inequality

First, decide one inequality to plot. Change the inequality symbol to an equal sign and graph the corresponding line. If the inequality is 'greater than' or 'greater than or equal to', you will use a dashed line. If it's 'less than' or 'less than or equal to', you will use a solid line.

Step 2: Shading the Correct Region for the First Inequality

Next, choose a test point not on the line (usually, (0,0) is the easiest, unless the line passes through the origin). Substitute it into the original inequality. If it makes the inequality true, shade the side of line containing the test point. If it doesn't, shade the other side.

Step 3: Graphing the Second Inequality

Repeat steps 1 and 2 for the second inequality - graph the line and then shade the appropriate region.

Step 4: Find the Overlapping Region

The solution to the system of inequalities is the overlapping region that satisfies both inequalities. Highlight this section of the graph to clearly illustrate the solution region.

Key Concepts

Inequality Symbols

Understanding inequality symbols is fundamental when graphing systems of inequalities. Inequalities express the relationship between two expressions, where one is not necessarily equal to the other. The symbols used are '<' for 'less than', '<=' for 'less than or equal to', '>' for 'greater than', and '>=' for 'greater than or equal to'. Each of these symbols determines the nature of the line and the area to be shaded on a graph. For instance, a '<' symbol suggests shading below the line for a standard y inequality, and a '>' symbol suggests shading above the line. Remember, using a solid line indicates that points on the line are part of the solution (with '='), whereas a dashed line means they are not (without '=').

Test Point Method

The test point method is a practical approach used to determine which side of the boundary line to shade when graphing an inequality. After graphing the line, select any point not on that line as your test point. The origin, \( (0,0) \), is the most common choice unless the line passes through it. Plug this point into the inequality: if the statement remains true, then the region containing the test point is the solution area. If the statement is false, shade the opposite region. This method simplifies the process of graphing by eliminating guesswork and ensuring accuracy in representing the inequality's solutions.

Shading Regions

Shading represents the infinite set of points that satisfy an inequality. When graphing an inequality on a two-dimensional plane, the part of the plane that represents all possible solutions is shaded. Shading is not arbitrary; it depends on the inequality symbol and the test point method. Once you determine which side of the line to shade, use horizontal or vertical pencil strokes to fill in the region consistently. This visual distinction is crucial for identifying the set of points that are solutions to the inequality. It is especially important when graphing systems of inequalities, as you will need to determine where multiple shaded regions overlap.

Solution Region

The solution region of a system of inequalities is where the shaded regions of all individual inequalities intersect. This region represents the set of points that satisfy all inequalities simultaneously. To find it, graph each inequality separately with its appropriate boundary line and shaded area. The common overlapping area is the solution region, which is the focus of the entire graph. Make sure to clearly outline or highlight this region for visibility. In most cases, this region will be a segment, a polygon, or an irregular shape bounded by the lines you've drawn. Its shape and size directly depend on the constraints imposed by each inequality in the system.