Problem 27 - Thinking Mathematically

Question

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x+y<4 \\ x-y>4\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of inequalities is the overlapping region in your graph where the pointed half of the line passing through points (0, 4) and (2, 0) and the non-pointed half of the line passing through points (0, -4) and (4, 0) intersect.

Step 1: Graph the Inequalities Individually

Start by treating each inequality as a two-variable linear equation and sketch the lines on the same graph. For the equation \(2x + y = 4\), the y-intercept is 4 and the slope is -2, so the graph is a line that passes through the points (0, 4) and (2, 0). For the equation \(x - y = 4\), the y-intercept is -4 and the slope is 1, so the graph is a line that passes through the points (0, -4) and (4, 0).

Step 2: Apply the Sign of Inequality

The less than '<' in the equation \(2x + y < 4\) means that the solutions lie below the line \(2x + y = 4\). Similarly, the greater than '>' sign in the equation \(x - y > 4\) means that the solutions lie above the line \(x - y = 4\). Fill these regions in the graph with different colors or patterns to indicate the solution set of each inequality individually.

Step 3: Identify Overlapping Region

The solution to the system of inequalities will be the region where the solution sets of both inequalities overlap. Find and highlight this region on your graph.

Key Concepts

Graphing Inequalities

Graphing inequalities can be a bit tricky if you're just getting started, but once you get the hang of it, it becomes a straightforward process. Begin by translating each inequality into a graphable line, known as its boundary line. Inequalities often come in two parts: a linear equation and a sign that tells us about the solution's position in relation to the line.

For example, for the inequality \(2x + y < 4\), imagine the boundary line being \(2x + y = 4\). This line forms the border of the region where solutions might be found. The type of line (solid or dashed) depends on the sign of the inequality:

• Use a dashed line for "<" or ">" since points on the line are not part of the solution.
• Use a solid line for "\(\leq\)" or "\(\geq\)" as points on the line are included in the solution.

After plotting the line, check which side of the line satisfies the inequality. A common technique is to substitute a test point, like (0,0), into the inequality. If it satisfies the inequality, shade the side containing the point; if not, shade the opposite side. Remember, each inequality will have its own shaded region.

Linear Equations

Linear equations are the backbone of systems of inequalities. They provide the boundary lines that help define where the solutions to the inequalities lie. A linear equation typically looks like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations graph as straight lines, and understanding their slope and intercepts is essential in graphing inequalities.

The slope provides the steepness of the line. A slope of 2, as in \(y = 2x + 3\), means the line rises 2 units for every 1 unit it runs horizontally. It's a crucial piece of information for sketching the line correctly. The intercepts, usually where the line crosses the axes, offer easy points to plot on our graph. The y-intercept is the point where the line crosses the y-axis, while the x-intercept is where it crosses the x-axis.

When graphing both \(2x + y = 4\) and \(x - y = 4\), these intercepts are handy starting points. Once the lines intersect, you can determine the solution sets' region's boundary, forming a convenient visual grid for locating solutions of the inequalities.

Solution Set of Inequalities

The solution set of a system of inequalities represents the common area on the graph that satisfies all inequalities in the system simultaneously. It is crucial to graph each inequality first, which allows you to find where their shaded regions intersect. This overlapping region is your solution set.

In our original exercise, the solution set is found by identifying the graph of \(2x + y < 4\) below its boundary line, and \(x - y > 4\) above its line. These inequalities together form a region on the graph if you shade them appropriately.

When the two regions overlap, that intersection becomes significant. It is the solution set where both inequalities hold true. If unsure, always double-check this area with test points from the region to ensure they satisfy both original inequalities.

Understanding this concept is vital, especially for systems with more than two inequalities. It gives you confidence in visualizing and determining solutions to complex inequality systems.