Question

systems of inequalities: mastery test

use the drawing tool(s) to form the correct answer on the provided graph.

graph the solution to this system of inequalities in the coordinate plane.

\\3y > 2x + 12\\
\\2x + y \le -5\\

Explanation:

🆕 New Concept Discovered: Solving Systems of Linear Inequalities by Graphing
Finding where shaded regions overlap on a coordinate plane

Step 1: Graph the first inequality

The first inequality is:

$$ 3y > 2x + 12 $$

To graph this, we first rewrite it in slope-intercept form \( y > mx + b \) by dividing both sides by \( 3 \):

$$ y > \frac{2}{3}x + 4 $$

• Boundary Line: The boundary line is \( y = \frac{2}{3}x + 4 \).
• The \( y \)-intercept is \( (0, 4) \).
• The slope is \( \frac{2}{3} \), meaning from \( (0, 4) \), we go up \( 2 \) units and right \( 3 \) units to reach \( (3, 6) \).
• Since the inequality symbol is strictly greater than (\( > \)), use a dashed line to show that points on the line are not included in the solution.
• Shading: Since it is \( y > \), we shade above this dashed line.

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Step 2: Graph the second inequality

The second inequality is:

$$ 2x + y \le -5 $$

We rewrite this in slope-intercept form by subtracting \( 2x \) from both sides:

$$ y \le -2x - 5 $$

• Boundary Line: The boundary line is \( y = -2x - 5 \).
• The \( y \)-intercept is \( (0, -5) \).
• The slope is \( -2 \), meaning from \( (0, -5) \), we go down \( 2 \) units and right \( 1 \) unit to reach \( (1, -7) \), or up \( 2 \) units and left \( 1 \) unit to reach \( (-1, -3) \).
• Since the inequality symbol is less than or equal to (\( \le \)), use a solid line to show that points on the line are included in the solution.
• Shading: Since it is \( y \le \), we shade below this solid line.

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Step 3: Identify the solution region

The solution to the system is the region where the shaded areas of both inequalities overlap.

To find this region, we can test a point in the overlapping area, such as \( (-6, 2) \):

• For the first inequality:

$$ 3(2) > 2(-6) + 12 \implies 6 > 0 \quad \text{(True)} $$

• For the second inequality:

$$ 2(-6) + 2 \le -5 \implies -10 \le -5 \quad \text{(True)} $$

Thus, the region containing \( (-6, 2) \) (to the left of the solid line and above the dashed line) is the correct shaded region.

Answer:

To graph the solution on your screen, use the drawing tools to plot the following:

1. Dashed Line passing through the points:

• \( (0, 4) \) and \( (3, 6) \)

1. Solid Line passing through the points:

• \( (0, -5) \) and \( (-1, -3) \)

1. Shaded Region:

• Click the Shaded Region tool and click inside the region that is above the dashed line and below/to the left of the solid line (containing the point \( (-6, 2) \)).