Question

use the drawing tools to graph the solution to this system of inequalities on the coordinate plane.

\\(y > 2x + 4\\)
\\(x + y \le 6\\)

Explanation:

🆕 New Concept Discovered: Graphing Systems of Linear Inequalities
Shading the overlapping region of two boundary lines

Step 1: Graph the first inequality

The first inequality is:

$$ y > 2x + 4 $$

• Boundary Line: The equation of the boundary line is \( y = 2x + 4 \).
• y-intercept: \((0, 4)\)
• Slope (\(m\)): \( 2 \) (move up 2 units and right 1 unit to find another point, such as \((1, 6)\))
• Line Type: Since the inequality symbol is strictly greater than (\(>\)), use a dashed line.
• Shading: Since it is \( y > \), shade the region above the dashed line.

Step 2: Graph the second inequality

The second inequality is:

$$ x + y \le 6 $$

First, rewrite it in slope-intercept form:

$$ y \le -x + 6 $$

• Boundary Line: The equation of the boundary line is \( y = -x + 6 \).
• y-intercept: \((0, 6)\)
• Slope (\(m\)): \( -1 \) (move down 1 unit and right 1 unit to find another point, such as \((1, 5)\) or \((6, 0)\))
• Line Type: Since the inequality symbol is less than or equal to (\(\le\)), use a solid line (labeled "Line" in your tool).
• Shading: Since it is \( y \le \), shade the region below the solid line.

Step 3: Identify the overlapping solution region

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

• This region is above the dashed line \( y = 2x + 4 \) and below the solid line \( y = -x + 6 \).
• The two boundary lines intersect where:

$$ 2x + 4 = -x + 6 $$

$$ 3x = 2 \implies x = \frac{2}{3}, \quad y = \frac{16}{3} \approx 5.33 $$

• Use the Shaded Region tool to click inside the section that lies above the dashed line and below the solid line (this will be the region containing points like \((-2, 2)\) or \((0, 5)\)).

Answer:

To graph the solution on your screen:

1. First Boundary (Dashed Line):

• Select the Dashed Line tool.
• Plot points at \((0, 4)\) and \((1, 6)\) to draw the dashed line \( y = 2x + 4 \).

1. Second Boundary (Solid Line):

• Select the Line tool.
• Plot points at \((0, 6)\) and \((6, 0)\) to draw the solid line \( x + y = 6 \).

1. Shading:

• Select the Shaded Region tool.
• Click in the region that is above the dashed line and below the solid line (for example, click near the point \((-2, 3)\)).