Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{r}x+2 y \leq 8 \\ 0 \leq x \leq 4 \\ 0 \leq y \leq 3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution region is the area on a graph bounded by the line \( x + 2y = 8 \), \( x = 0 \), \( x = 4 \), \( y = 0 \), and \( y = 3 \).

Step 1: Identify the System of Inequalities

The exercise presents a system of inequalities that need to be solved graphically or analytically. The system is:1. \( x + 2y \leq 8 \)2. \( 0 \leq x \leq 4 \)3. \( 0 \leq y \leq 3 \).

Step 2: Graph the Boundary Line

Start by graphing the boundary line for the inequality \( x + 2y = 8 \). To do so, find the x-intercept by setting \( y = 0 \), which gives \( x = 8 \), and the y-intercept by setting \( x = 0 \), which gives \( y = 4 \). Draw this line on a coordinate plane.

Step 3: Determine the Solution Area

Using a test point, such as \( (0,0) \), plug it into the inequality \( x + 2y \leq 8 \). Since \( 0 + 2(0) \leq 8 \) is true, the solution region is the area below the line, including the boundary.

Step 4: Apply Additional Constraints

Apply the constraints \( 0 \leq x \leq 4 \) and \( 0 \leq y \leq 3 \). Draw vertical lines at \( x = 0 \) and \( x = 4 \), and horizontal lines at \( y = 0 \) and \( y = 3 \). The solution region is now constrained to the intersection of these lines and the area below \( x + 2y = 8 \).

Step 5: Shade the Feasible Region

Shade the region that satisfies all inequalities. This is the region below the line, bounded on the right by \( x = 4 \), on the left by \( x = 0 \), on the top by \( y = 3 \), and above by \( y = 0 \). All points in this region are solutions to the system.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting the boundary lines on a coordinate plane to determine the set of all solutions that satisfy the inequality. To graph the inequality \( x + 2y \leq 8 \), start by sketching the line \( x + 2y = 8 \). This line acts as a boundary that separates the solutions that satisfy the inequality from those that don't.

Here's how to do it effectively:

• Find the intercepts, which are the easy points where the graph crosses the axes.
• Determine if a line should be solid or dashed; use a solid line when the inequality sign is either \( \leq \) or \( \geq \), and a dashed line when it is \( < \) or \( > \).
• Pick a test point, like \( (0,0) \), which is often the simplest choice if it is not on the line. Substitute it into the inequality to check which side of the line represents the solution set.

When the test point satisfies the inequality, shade the region that includes it. The shaded area represents all the possible solutions to the inequality.

Solution Region

The solution region is the area on the graph where all the inequalities in a system overlap. It represents all the potential solutions that satisfy the set of inequalities. The exercise provided involves multiple inequalities: \( x + 2y \leq 8 \), \( 0 \leq x \leq 4 \), and \( 0 \leq y \leq 3 \).

To find the solution region:

• Graph each inequality one by one, marking their respective solution sets.
• For \( x + 2y \leq 8 \), the solution set is below the boundary line.
• The inequalities \( 0 \leq x \leq 4 \) and \( 0 \leq y \leq 3 \) are simple vertical and horizontal lines, creating a rectangular boundary.
• The solution region is the common area bounded by these two types of lines.

This constrained area within these lines includes all solutions that meet all the given conditions. Shading this region effectively on a graph highlights the feasible solutions.

X-Intercept and Y-Intercept

Intercepts are crucial in graphing linear equations as they provide easy-to-identify points for sketching lines quickly. The x-intercept of a line is the point where it crosses the x-axis, and the y-intercept is where it crosses the y-axis. For the boundary line \( x + 2y = 8 \), intercepts are solved by setting one variable to zero to find the value of the other.

Here's how:

• To find the x-intercept, set \( y = 0 \). Substituting into the equation gives \( x = 8 \), so the x-intercept is \( (8, 0) \).
• To determine the y-intercept, set \( x = 0 \). This leads to \( 2y = 8 \), or \( y = 4 \). Thus, the y-intercept is \( (0, 4) \).

The intercepts \( (8, 0) \) and \( (0, 4) \) help to plot the boundary line accurately, forming a guide to easily graph the inequality \( x + 2y \leq 8 \). Knowing these intercepts simplifies understanding the relationship between x and y in the context of this inequality.