Problem 40 - Algebra 1

Question

Use the following information. You own a bottle recycling center that receives bottles that are either sorted by color or unsorted. To sort and recycle all of the bottles, you can use up to 4200 hours of human labor and up to 2400 hours of machine time. The system below represents the number of hours your center spends sorting and recycling bottles where \(x\) is the number of tons of unsorted bottles and \(y\) is the number of tons of sorted bottles. \(4 x+y \leq 4200\) \(2 x+y \leq 2400\) \(x \geq 0, y \geq 0\) Graph the system of linear inequalities.

Step-by-Step Solution

Answer

The graph of the system of inequalities shows the feasible number of tons of unsorted and sorted bottles that can be processed within the given constraints. The overlaps of the shaded regions denote possible solutions. This visual representation should help to easier understand how these variables relate to each other and to make decisions based on these constraints.

Step 1: Set Up the Axes

Firstly, set up the axes where \(x\) stands for the tons of unsorted bottles and \(y\) for the tons of sorted bottles. Because \(x\) and \(y\) should be greater or equal to 0, only consider the first quadrant.

Step 2: Plot the First Inequality

Start by graphing the line \(4x + y = 4200\). To do that, find the x- and y-intercepts by making \(y\) equal 0 and solve for \(x\), and then make \(x\) equal 0 and solve for \(y\). Shade the area that satisfies the inequality \(4x + y \leq 4200\).

Step 3: Plot the Second Inequality

Next, graph the line \(2x + y = 2400\). Again, find the x- and y-intercepts and then shade the area that satisfies the inequality \(2x + y \leq 2400\).

Step 4: Identify the Feasible Region

Finally, identify the feasible region that satisfies all inequalities. This will be the region that is shaded both by the first and the second inequality and is in the first quadrant. This area represents all possible combinations of unsorted and sorted bottles which meet the constraints of available human and machine time.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting the lines represented by each inequality and shading the area of the graph that satisfies the inequality.
This process helps visualize all possible solutions that satisfy the given constraints. To graph an inequality:

• First, rewrite the inequality as an equation (replace the inequality sign with an equal sign) to find the line it represents.
• Find the intercepts of the line by setting one variable to zero and solving for the other.


• Next, graph the line using these intercepts.
• Determine which side of the line satisfies the inequality. You can do this by testing a point not on the line, typically the origin (if it’s not on the line already).
• Shade the region of the graph that makes the inequality true. This shaded area represents all solutions to the inequality.

For example, in the exercise, for the inequality \(4x + y \leq 4200\), the line \(4x + y = 4200\) is graphed, and the region below and including this line is shaded. By doing this for all inequalities, the overlapping shaded region shows where all conditions are met.

Feasible Region

The feasible region is the part of the graph that satisfies all given inequalities at once. It represents the set of possible solutions given the constraints in a problem.
For example, when you graph multiple inequalities, the feasible region is where the shaded parts of the graphs overlap.
This region is crucial because:

• It shows the values that both satisfy the constraints and are possible within the problem context.


• Only combinations of values found in the feasible region are practical, given the limitations imposed by the inequalities.
• In optimization problems, like determining maximum profit or minimum cost, the optimal solution often lies within the boundaries of this region.

In the context of our exercise, the feasible region is the area in the first quadrant where the inequalities \(4x + y \leq 4200\) and \(2x + y \leq 2400\) are satisfied simultaneously. This region defines the possible tonnage combinations of unsorted and sorted bottles that respect both human labor and machine time constraints.

Constraints

Constraints are the conditions or limitations defined by inequalities in a problem. They restrict the values that the variables involved can take.
In linear programming or optimization problems, constraints help to define the solution set.
To illustrate:

• Every constraint translates to a linear inequality when graphing.


• Constraints restrict the feasible region; only solutions fulfilling all stipulations contribute to the final feasible region.
• They usually originate from real-world limits, such as budget, time, or resource limitations.

In our exercise, the constraints are:- \(4x + y \leq 4200\) which reflects the human labor limit.- \(2x + y \leq 2400\) representing the machine time constraint.- \(x \geq 0\) and \(y \geq 0\), both implying that negative tonnages of bottles are impossible.

First Quadrant

The first quadrant in the coordinate system is the section where both the x and y values are positive. This is crucial, particularly in contexts involving quantities that cannot be negative.
In graphing, focusing on the first quadrant is essential when the variables represent quantities like distance, time, or number of items, since these cannot be negative.
Features of the first quadrant include:

• Both x and y axes have positive values; hence, expressions like \(x \geq 0\) and \(y \geq 0\) hold true here.


• It is one of the four sections created by dividing a graph into the x- and y- axes.
• When dealing with problems where direction matters, such as maximizing or minimizing a function, restricting to the first quadrant makes sense if negative values are not meaningful or possible.

In the provided exercise, the first quadrant is where the solution makes sense because you cannot have negative tons of bottles. As a result, all analysis remains within this positive value space.