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Consider the system of inequalities and its graph.

y ≤ –0.75x

y ≤ 3x – 2



On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (negative 8, 6) and (0, 0). Everything below the line is shaded. The second line has a positive slope and goes through (negative 2, negative 8) and (2, 4). Everything to the right of the line is shaded. The section in quadrant 1 is labeled 3, the section in quadrant 2 is labeled 2, the section in quadrant 3 is labeled 1, and the section in quadrant 4 is labeled 4.
In which section of the graph does the actual solution to the system lie?

1
2
3
4

Sagot :

Sethartiolamarceloidn

Answer:

To determine which section of the graph contains the actual solution to the system of inequalities, we need to analyze the shading and the regions they form.

The inequalities given are:

1. \( y \le -0.75x \)

2. \( y \le 3x - 2 \)

Given the description:

- The first line \( y \le -0.75x \) has a negative slope and goes through \((-8, 6)\) and \((0, 0)\). The region below this line is shaded.

- The second line \( y \le 3x - 2 \) has a positive slope and goes through \((-2, -8)\) and \((2, 4)\). The region below this line is shaded.

The solution to the system of inequalities is the intersection of the shaded regions of both inequalities. This means we are looking for the region that satisfies both inequalities simultaneously.

- In quadrant 1: This section is above the \( y \le -0.75x \) line and to the right of \( y \le 3x - 2 \) line. It does not satisfy both inequalities.

- In quadrant 2: This section is above the \( y \le -0.75x \) line but also to the left of \( y \le 3x - 2 \) line. It does not satisfy both inequalities.

- In quadrant 3: This section is below both \( y \le -0.75x \) and \( y \le 3x - 2 \) lines. It satisfies both inequalities.

- In quadrant 4: This section is below the \( y \le 3x - 2 \) line but above the \( y \le -0.75x \) line. It does not satisfy both inequalities.

Therefore, the actual solution to the system lies in section **1**, which corresponds to quadrant 3.

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