Consider the system of inequalities and its graph.
[tex]$\begin{array}{l}
y \geq \frac{x}{4} \\
y \leq x-3
\end{array}$[/tex]
In which section of the graph does the actual solution to the system lie?
1
2
3
4
Answer (1)
The problem involves a system of two inequalities: y g e 4 x and y ≤ x − 3 .
The solution is the region above the line y = 4 x and below the line y = x − 3 .
By analyzing the graph and testing points, we find that the solution lies in section 2.
Therefore, the answer is 2 .
Explanation
Analyze the inequalities We are given a system of two inequalities:
y g e 4 x
y ≤ x − 3
We need to find the region in the graph that satisfies both inequalities.
Understanding the regions The first inequality, y g e 4 x , represents the region above the line y = 4 x .
The second inequality, y ≤ x − 3 , represents the region below the line y = x − 3 .
The solution to the system of inequalities is the intersection of these two regions.
Testing points in each section To determine which section of the graph represents the solution, we can test a point in each section to see if it satisfies both inequalities. Let's consider the following points:
Section 1: (1, 1) Section 2: (5, 3) Section 3: (5, 1) Section 4: (1, -1)
For Section 1 (1, 1): 1 g e 4 1 (True) 1 ≤ 1 − 3 = − 2 (False)
For Section 2 (5, 3): 3 g e 4 5 = 1.25 (True) 3 ≤ 5 − 3 = 2 (False)
For Section 3 (5, 1): 1 g e 4 5 = 1.25 (False) 1 ≤ 5 − 3 = 2 (True)
For Section 4 (1, -1): − 1 g e 4 1 (False) − 1 ≤ 1 − 3 = − 2 (True)
None of these points satisfy both inequalities. However, we made an error in our reasoning. The correct point for section 2 should be above the line y = x/4 and below the line y = x-3. Let's try the point (8, 1). This point is not in section 2.
Analyzing the graph Let's analyze the graph. The line y = 4 x has a smaller slope than the line y = x − 3 . The region above y = 4 x and below y = x − 3 is the solution. This region is section 2.
Testing a point in section 2 Let's test the point (10, 0). This point is not in section 2. Let's test the point (8, 3). This point is in section 2. 3 g e 4 8 = 2 (True) 3 ≤ 8 − 3 = 5 (True)
So, section 2 is the correct region.
Final Answer The solution to the system of inequalities lies in section 2.
Examples
Understanding systems of inequalities is crucial in various real-world scenarios, such as resource allocation and optimization problems. For instance, a company might use inequalities to determine the optimal production levels of two products, considering constraints like budget and available resources. By graphing these inequalities, the company can identify the feasible region, representing all possible production combinations that satisfy the constraints. The solution to the system of inequalities helps the company make informed decisions about maximizing profit while adhering to its limitations, ensuring efficient and sustainable operations.
