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Chapter 4: Problem 23

Graph. Label the solution region.a For exercises \(23-44\), graph. Label the solution region. $$ \begin{aligned} &x+3 y \leq 9 \\ &x \geq 0 \\ &y \geq 0 \end{aligned} $$

Short Answer

Expert verified
Graph the line \(x + 3y = 9\) and shade below it in the first quadrant.

Step by step solution

01

Understand the Inequalities

The inequalities given are: 1. \(x + 3y \leq 9\) 2. \(x \geq 0\) 3. \(y \geq 0\)
02

Graph the Equality Line

Convert the inequality \(x + 3y \leq 9\) to an equality, i.e., \(x + 3y = 9\). Graph this line by finding the intercepts. For the x-intercept (when y=0): \(x = 9\)For the y-intercept (when x=0): \(3y = 9\), which simplifies to \(y = 3\)Plot these intercepts (9,0) and (0,3) and draw the line.
03

Identify the Feasible Region for \(x + 3y \leq 9\)

Since the inequality is \(x + 3y \leq 9\), shade the region below the line \(x + 3y = 9\).
04

Add the Constraints \(x \geq 0\) and \(y \geq 0\)

These inequalities indicate that the solution must be in the first quadrant. Shade the area to the right of \(x = 0\) (the y-axis) and above \(y = 0\) (the x-axis).
05

Determine the Solution Region

The solution region is where all shaded areas overlap. This region is bounded by the line \(x + 3y = 9\), the y-axis, and the x-axis.
06

Label the Solution Region

Mark the overlapping shaded region clearly on the graph. Label this area as the solution region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear inequalities
Linear inequalities are similar to linear equations. They involve variables and produce straight lines when graphed. However, they differ because inequalities use symbols like \( \leq \), \( \geq \), \( < \), and \( > \) instead of an equal sign. Let's break down the concept using the given inequalities:
solution region
The solution region is the area on a graph where all inequalities are satisfied. For example, given the inequalities:
  • \(x + 3y \leq 9\)
  • \(x \geq 0\)
  • \(y \geq 0\)
The solution region is found by graphing each inequality and seeing where all shaded areas overlap. This overlapping area is where all inequalities hold true simultaneously.
graphing constraints
To graph constraints accurately, convert each inequality to its corresponding equality to plot the boundary line. For \(x + 3y \leq 9\), first plot the equality line \(x + 3y = 9\). Next, find the x- and y-intercepts by setting y to 0 and solving for x, and vice versa. Plot these intercepts and draw the line. Then, shade the region that satisfies the inequality. Repeat for \(x \geq 0\) and \(y \geq 0\), where you shade the regions to the right of the y-axis and above the x-axis, respectively.
feasible region
The feasible region is the overlapping area of all constraints on the graph. It's the region that satisfies all inequalities. Here, it is bounded by the line \(x + 3y = 9\), the x-axis, and the y-axis. Clearly mark and label this region on the graph as the solution region. This feasible region represents all possible solutions that meet the given constraints.

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