Questions: Graph the solution set of the system. 9x + 2y ≤ 18 x - y ≥ 3 y ≥ -3

Transcript text: Graph the solution set of the system. \[ \left\{\begin{array}{l} 9 x+2 y \leq 18 \\ x-y \geq 3 \\ y \geq-3 \end{array}\right. \]

Solution

Solution Steps

Step 1: Rewrite the inequalities in slope-intercept form.

The first inequality $9x + 2y \le 18$ can be rewritten as $y \le -\frac{9}{2}x + 9$. The second inequality $x - y \ge 3$ can be rewritten as $y \le x - 3$. The third inequality is already in the desired form: $y \ge -3$.

Step 2: Graph the inequalities.

The first inequality $y \le -\frac{9}{2}x + 9$ is graphed by drawing the line $y = -\frac{9}{2}x + 9$ and shading the region below the line. The second inequality $y \le x - 3$ is graphed by drawing the line $y = x - 3$ and shading the region below the line. The third inequality $y \ge -3$ is graphed by drawing the horizontal line $y=-3$ and shading the region above the line.

Step 3: Identify the solution set.

The solution set is the region where all three shaded regions overlap. This is the triangular region bounded by the lines $y = -\frac{9}{2}x + 9$, $y = x - 3$, and $y = -3$.

Final Answer:

The solution set is the triangular region below the lines $y = -\frac{9}{2}x + 9$ and $y = x - 3$, and above the line $y=-3$. The vertices of this triangular region are $(0, -3)$, $(6,-3)$, and $(2, -1)$.

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