Problem

Determine the solution region for the following system of linear inqeualities by inputting a point in that region. The graph, without shaded solution, is shown on the right. (Enter an ordered pair $(x, y)$ with integer values, that is, no decimals or fractions.) (Do not enter a point that is actually on one of the lines; keep it in the interior of the solution region.) \[ \begin{array}{r} -3 x-3 y<24 \\ -3 x+y<12 \end{array} \] Point in the Solution Region:

Solution

Step 1 :To determine a point in the solution region, we need to check if it satisfies both inequalities.

Step 2 :First inequality: \(-3x - 3y < 24\) becomes \(-3(0) - 3(0) < 24\), which simplifies to \(0 < 24\). This is true.

Step 3 :Second inequality: \(-3x + y < 12\) becomes \(-3(0) + 0 < 12\), which simplifies to \(0 < 12\). This is also true.

Step 4 :Since the origin satisfies both inequalities, it is part of the solution set.

Step 5 :To graph the solution, we first graph the two inequalities as lines and then shade the appropriate regions.

Step 6 :First inequality: \(y = -x + 8\) is a line with slope \(-1\) and y-intercept \(8\).

Step 7 :Second inequality: \(y = 3x + 12\) is a line with slope \(3\) and y-intercept \(12\).

Step 8 :Since both inequalities are strict inequalities \(<\), we use a dashed line for both lines.

Step 9 :We shade the region below the first line and above the second line.

Step 10 :One point that lies in the solution set is \((0,0)\), as it is in the shaded region.

Step 11 :\(\boxed{\text{Solution point: (0,0)}}\)

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Source: https://solvelyapp.com/problems/46027/

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