Problem 23 - Thinking Mathematically

Question

In Exercises 23-38, graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}3 x+6 y \leq 6 \\ 2 x+y \leq 8\end{array}\right.\)

Step-by-Step Solution

Answer

The intersected or overlapped region on the graph represents the solution set to the system of inequalities.

Step 1: Graph the first inequality

Rewrite the inequality into slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. So the first inequality \(3x + 6y \leq 6\) can be rewritten as \(y \leq -0.5x + 1\). Now graph this line. Because the inequality is 'less than or equal to', shade the region that is below the line.

Step 2: Graph the second inequality

Similarly, rewrite the second inequality into slope-intercept form. Thus, \(2x + y \leq 8\) becomes \(y \leq -2x + 8\). Graph this line on the same coordinate plane. Because this inequality is also 'less than or equal to', again shade the area below the line.

Step 3: Find the solution set

The solution set to the system of inequalities is the overlapped region shaded by both inequalities. Identify this region on the graph.

Key Concepts

Slope-Intercept Form

To effectively solve systems of linear inequalities, we often rewrite equations in the slope-intercept form, which makes graphing much easier. The slope-intercept form is denoted as \( y = mx + c \). Here, \( m \) represents the slope of the line, which shows how steep the line is. Meanwhile, \( c \) is the y-intercept, where the line meets the y-axis.

By expressing inequalities like this, students can easily determine how to draw the line on a graph. Let's take an example from the exercise: with the inequality \( 3x + 6y \leq 6 \), by rewriting it into slope-intercept form as \( y \leq -0.5x + 1 \), it's clear that the line meets the y-axis at \( 1 \) and has a slope of \( -0.5 \).

Having a clear understanding of this form allows you to quickly identify the characteristics of a line, thereby aiding the process of sketching it on a graph.

Solution Set

In mathematics, the solution set is essentially the answer to a problem involving inequalities. For systems of inequalities, the solution set is the region on the graph where the shaded areas from all inequalities overlap or intersect. Each point in this overlapping region is a potential solution to the inequalities given.

For instance, in the exercise problem, both inequalities \( y \leq -0.5x + 1 \) and \( y \leq -2x + 8 \) need to be satisfied. When these inequalities are graphed, you'll shade the areas below each respective line.

The solution set is found at the intersection of these shaded areas. This is the region that meets the conditions of both inequalities, which means any point within this region satisfies both conditions. Identifying this region accurately is crucial, especially when dealing with multiple inequalities.

Shading Regions

Shading regions is a visual method used to represent linear inequalities graphically. Once a line is drawn to represent an equation or inequality, shading helps identify which side or part of the graph satisfies the inequality.

It's important to pay attention to the inequality sign to shade the correct region. For \( \leq \) or \( \geq \), the boundary line is solid, indicating that the points on the line are included in the solution set. In contrast, \( < \) or \( > \) will have a dashed line, excluding points on the line from the solution set.

In our example, since both inequalities use '\( \leq \)', solid lines are drawn for both. To find the correct shaded area, if the inequality is 'less than or equal to,' shade below the line. In the graph from the exercise, identifying the proper region to shade ensures that the solution set—the overlap of the shaded areas—depicts the correct solution visually.