Questions: Graph the solution set of the system of inequalities. 2x-3y ≤ 6 2x-2y > 4

Transcript text: Graph the solution set of the system of inequalities. \[ \begin{array}{l} 2 x-3 y \leq 6 \\ 2 x-2 y>4 \end{array} \]

Solution

Solution Steps

Step 1: Solve the first inequality

The first inequality is: \[ 2x - 3y \leq 6 \]

To find the boundary line, we set the inequality to equality: \[ 2x - 3y = 6 \]

Solving for $ y $: \[ 3y = 2x - 6 \] \[ y = \frac{2}{3}x - 2 \]

Step 2: Solve the second inequality

The second inequality is: \[ 2x - 2y > 4 \]

To find the boundary line, we set the inequality to equality: \[ 2x - 2y = 4 \]

Solving for $ y $: \[ 2y = 2x - 4 \] \[ y = x - 2 \]

Step 3: Determine the solution set

The solution set is the region that satisfies both inequalities:

$ y \leq \frac{2}{3}x - 2 $
$ y > x - 2 $

Final Answer

The solution set is the region below the line $ y = \frac{2}{3}x - 2 $ and above the line $ y = x - 2 $.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (2/3)x - 2", "y = x - 2"], "latex_expressions": ["$y = \\frac{2}{3}x - 2$", "$y = x - 2$"]}

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