Answer :
To solve the system of inequalities, we'll graph each inequality and find the region of the coordinate plane where both inequalities are satisfied.
The system of inequalities is:
1. [tex]\( x + y \leq -3 \)[/tex]
2. [tex]\( y < \frac{x}{2} \)[/tex]
### Graphing the First Inequality: [tex]\( x + y \leq -3 \)[/tex]
1. Rewrite the inequality: We can rewrite it in slope-intercept form, which is [tex]\( y \leq -x - 3 \)[/tex].
2. Graph the boundary line: The boundary line for this inequality is [tex]\( y = -x - 3 \)[/tex]. This is a straight line with a slope of [tex]\(-1\)[/tex] and a y-intercept of [tex]\(-3\)[/tex].
- When [tex]\( x = 0 \)[/tex], then [tex]\( y = -3 \)[/tex]. This gives the point [tex]\((0, -3)\)[/tex].
- When [tex]\( y = 0 \)[/tex], then [tex]\( x = -3 \)[/tex]. This gives the point [tex]\((-3, 0)\)[/tex].
Draw the line through these points.
3. Determine the shading: Since the inequality is [tex]\( y \leq -x - 3 \)[/tex], we shade below the line.
### Graphing the Second Inequality: [tex]\( y < \frac{x}{2} \)[/tex]
1. Rewrite the inequality: The inequality is already in slope-intercept form as [tex]\( y < \frac{x}{2} \)[/tex].
2. Graph the boundary line: The boundary line for this inequality is [tex]\( y = \frac{x}{2} \)[/tex]. This is a straight line with a slope of [tex]\(\frac{1}{2}\)[/tex] and a y-intercept of [tex]\(0\)[/tex].
- When [tex]\( x = 0 \)[/tex], then [tex]\( y = 0 \)[/tex]. This gives the point [tex]\((0, 0)\)[/tex].
- When [tex]\( x = 2 \)[/tex], then [tex]\( y = 1 \)[/tex]. This gives the point [tex]\((2, 1)\)[/tex].
Draw the line through these points but use a dashed line, indicating that the points on the line are not included (since it's a strict inequality).
3. Determine the shading: Since the inequality is [tex]\( y < \frac{x}{2} \)[/tex], we shade below the line.
### Finding the Intersection of the Two Regions
To find the solution to the system, we need to find where the shaded regions of both inequalities overlap.
1. Shade the region below the line [tex]\( y = -x - 3 \)[/tex].
2. Shade the region below the line [tex]\( y < \frac{x}{2} \)[/tex].
The overlapping region between these two shaded areas will be the solution to the system of inequalities.
### Sketch of the Graph
1. Graph the line [tex]\( y = -x - 3 \)[/tex]:
- Passes through [tex]\((0, -3)\)[/tex] and [tex]\((-3, 0)\)[/tex].
- Use a solid line and shade below it.
2. Graph the line [tex]\( y = \frac{x}{2} \)[/tex]:
- Passes through [tex]\((0, 0)\)[/tex] and [tex]\((2, 1)\)[/tex].
- Use a dashed line and shade below it.
The solution region is where the two shaded areas intersect. This should look like a triangular region on the coordinate plane, below both lines and to the left of their intersection point.
To summarize:
1. Graph [tex]\( y = -x - 3 \)[/tex] as a solid line and shade below.
2. Graph [tex]\( y = \frac{x}{2} \)[/tex] as a dashed line and shade below.
3. The solution is the region where the shading overlaps—typically a portion below both lines.
This graphical approach helps to visualize the solution set for the system of inequalities.
The system of inequalities is:
1. [tex]\( x + y \leq -3 \)[/tex]
2. [tex]\( y < \frac{x}{2} \)[/tex]
### Graphing the First Inequality: [tex]\( x + y \leq -3 \)[/tex]
1. Rewrite the inequality: We can rewrite it in slope-intercept form, which is [tex]\( y \leq -x - 3 \)[/tex].
2. Graph the boundary line: The boundary line for this inequality is [tex]\( y = -x - 3 \)[/tex]. This is a straight line with a slope of [tex]\(-1\)[/tex] and a y-intercept of [tex]\(-3\)[/tex].
- When [tex]\( x = 0 \)[/tex], then [tex]\( y = -3 \)[/tex]. This gives the point [tex]\((0, -3)\)[/tex].
- When [tex]\( y = 0 \)[/tex], then [tex]\( x = -3 \)[/tex]. This gives the point [tex]\((-3, 0)\)[/tex].
Draw the line through these points.
3. Determine the shading: Since the inequality is [tex]\( y \leq -x - 3 \)[/tex], we shade below the line.
### Graphing the Second Inequality: [tex]\( y < \frac{x}{2} \)[/tex]
1. Rewrite the inequality: The inequality is already in slope-intercept form as [tex]\( y < \frac{x}{2} \)[/tex].
2. Graph the boundary line: The boundary line for this inequality is [tex]\( y = \frac{x}{2} \)[/tex]. This is a straight line with a slope of [tex]\(\frac{1}{2}\)[/tex] and a y-intercept of [tex]\(0\)[/tex].
- When [tex]\( x = 0 \)[/tex], then [tex]\( y = 0 \)[/tex]. This gives the point [tex]\((0, 0)\)[/tex].
- When [tex]\( x = 2 \)[/tex], then [tex]\( y = 1 \)[/tex]. This gives the point [tex]\((2, 1)\)[/tex].
Draw the line through these points but use a dashed line, indicating that the points on the line are not included (since it's a strict inequality).
3. Determine the shading: Since the inequality is [tex]\( y < \frac{x}{2} \)[/tex], we shade below the line.
### Finding the Intersection of the Two Regions
To find the solution to the system, we need to find where the shaded regions of both inequalities overlap.
1. Shade the region below the line [tex]\( y = -x - 3 \)[/tex].
2. Shade the region below the line [tex]\( y < \frac{x}{2} \)[/tex].
The overlapping region between these two shaded areas will be the solution to the system of inequalities.
### Sketch of the Graph
1. Graph the line [tex]\( y = -x - 3 \)[/tex]:
- Passes through [tex]\((0, -3)\)[/tex] and [tex]\((-3, 0)\)[/tex].
- Use a solid line and shade below it.
2. Graph the line [tex]\( y = \frac{x}{2} \)[/tex]:
- Passes through [tex]\((0, 0)\)[/tex] and [tex]\((2, 1)\)[/tex].
- Use a dashed line and shade below it.
The solution region is where the two shaded areas intersect. This should look like a triangular region on the coordinate plane, below both lines and to the left of their intersection point.
To summarize:
1. Graph [tex]\( y = -x - 3 \)[/tex] as a solid line and shade below.
2. Graph [tex]\( y = \frac{x}{2} \)[/tex] as a dashed line and shade below.
3. The solution is the region where the shading overlaps—typically a portion below both lines.
This graphical approach helps to visualize the solution set for the system of inequalities.
