Sketch the graph of the system [latex]\left\{\begin{aligned}x+y & \leq 4 \\2 x-y & \leq 4 \\x & \geq 0 \\y & \geq 0\end{aligned}\right.[/latex]

Question 8.3.4: Sketch the graph of the system {x+y ≤ 4 2 x-y ≤ 4 x ≥ 0 y ≥ ......

Precalculus Functions and Graphs [1893897]

Sketch the graph of the system

\left\{\begin{aligned}x+y & \leq 4 \\2 x-y & \leq 4 \\x & \geq 0 \\y & \geq 0\end{aligned}\right.

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Step 1:

The problem states that we have a system of inequalities. In this system, we have four inequalities.

Step 2:

The first two inequalities are the same as those considered in Example 3. This means that the points on the graph of the system must lie within the purple region shown in Figure 6.

Step 3:

Additionally, the third and fourth inequalities in the system tell us that the points must lie in the first quadrant or on its boundaries.

Step 4:

Combining all the information from the inequalities, we can determine the region in which the points must lie. This region is shown in Figure 7.

Step 5:

Therefore, the answer is that the points of the system must lie within the purple region shown in Figure 6 and within the first quadrant or on its boundaries, as shown in Figure 7.

Final Answer

The first two inequalities are the same as those considered in Example 3, and hence the points on the graph of the system must lie within the purple region shown in Figure 6. In addition, the third and fourth inequalities in the system tell us that the points must lie in the first quadrant or on its boundaries. This gives us the region shown in Figure 7.