Answer :
To solve the system of inequalities graphically and find a point in the solution set, follow these detailed steps:
### Step 1: Plot the Boundary Lines
1. Inequality [tex]\( y \geq 2x - 4 \)[/tex]:
- The boundary line is [tex]\( y = 2x - 4 \)[/tex].
- To plot this line, find two points on the line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2(0) - 4 = -4 \)[/tex]. So, one point is [tex]\( (0, -4) \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 2(2) - 4 = 0 \)[/tex]. So, another point is [tex]\( (2, 0) \)[/tex].
- Plot these points and draw the line through them. Since the inequality is [tex]\( y \geq 2x - 4 \)[/tex], shade the region above the line.
2. Inequality [tex]\( y > -x + 8 \)[/tex]:
- The boundary line is [tex]\( y = -x + 8 \)[/tex].
- To plot this line, find two points on the line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -0 + 8 = 8 \)[/tex]. So, one point is [tex]\( (0, 8) \)[/tex].
- When [tex]\( x = 8 \)[/tex], [tex]\( y = -8 + 8 = 0 \)[/tex]. So, another point is [tex]\( (8, 0) \)[/tex].
- Plot these points and draw the line through them. Since the inequality is [tex]\( y > -x + 8 \)[/tex], draw the line as a dashed line (indicating that the line itself is not included in the solution) and shade the region above the line.
### Step 2: Identify the Solution Region
- The solution to the system of inequalities is the region where the shaded areas overlap.
- The region where [tex]\( y \geq 2x - 4 \)[/tex] (above the line [tex]\( y = 2x - 4 \)[/tex]) and [tex]\( y > -x + 8 \)[/tex] (strictly above the line [tex]\( y = -x + 8 \)[/tex]) overlap will be the solution region.
### Step 3: Find a Point in the Solution Set
- Choose a point within the overlapping region that satisfies both inequalities.
- A suitable point can be found by inspection. Let's determine if the point [tex]\( (2, 5) \)[/tex] lies in the solution set:
- Substitute [tex]\( x = 2 \)[/tex] into [tex]\( y \geq 2x - 4 \)[/tex]:
[tex]\[ y \geq 2(2) - 4 \][/tex]
[tex]\[ y \geq 4 - 4 \][/tex]
[tex]\[ y \geq 0 \][/tex]
- Since [tex]\( y = 5 \)[/tex] (for the point [tex]\( (2, 5) \)[/tex]), [tex]\( 5 \geq 0 \)[/tex] is true.
- Substitute [tex]\( x = 2 \)[/tex] into [tex]\( y > -x + 8 \)[/tex]:
[tex]\[ y > -(2) + 8 \][/tex]
[tex]\[ y > -2 + 8 \][/tex]
[tex]\[ y > 6 \][/tex]
- Since [tex]\( y = 5 \)[/tex] (for the point [tex]\( (2, 5) \)[/tex]), [tex]\( 5 > 6 \)[/tex] is not true.
- This choice of point doesn't satisfy the set, so let's choose another point: [tex]\( (3, 5) \)[/tex]
- Substitute [tex]\( x = 3 \)[/tex] into
[tex]\( y \geq 5 \)[/tex]
[tex]\( y > 5 \)[/tex]
These tests show that some of the inequalities could be mismatched, by curving the whole solution set [tex]\( (5,6) \)[/tex]
### Final Solution
- A point in the solution set is [tex]\( (4, 6) \)[/tex].
### Step 1: Plot the Boundary Lines
1. Inequality [tex]\( y \geq 2x - 4 \)[/tex]:
- The boundary line is [tex]\( y = 2x - 4 \)[/tex].
- To plot this line, find two points on the line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2(0) - 4 = -4 \)[/tex]. So, one point is [tex]\( (0, -4) \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 2(2) - 4 = 0 \)[/tex]. So, another point is [tex]\( (2, 0) \)[/tex].
- Plot these points and draw the line through them. Since the inequality is [tex]\( y \geq 2x - 4 \)[/tex], shade the region above the line.
2. Inequality [tex]\( y > -x + 8 \)[/tex]:
- The boundary line is [tex]\( y = -x + 8 \)[/tex].
- To plot this line, find two points on the line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -0 + 8 = 8 \)[/tex]. So, one point is [tex]\( (0, 8) \)[/tex].
- When [tex]\( x = 8 \)[/tex], [tex]\( y = -8 + 8 = 0 \)[/tex]. So, another point is [tex]\( (8, 0) \)[/tex].
- Plot these points and draw the line through them. Since the inequality is [tex]\( y > -x + 8 \)[/tex], draw the line as a dashed line (indicating that the line itself is not included in the solution) and shade the region above the line.
### Step 2: Identify the Solution Region
- The solution to the system of inequalities is the region where the shaded areas overlap.
- The region where [tex]\( y \geq 2x - 4 \)[/tex] (above the line [tex]\( y = 2x - 4 \)[/tex]) and [tex]\( y > -x + 8 \)[/tex] (strictly above the line [tex]\( y = -x + 8 \)[/tex]) overlap will be the solution region.
### Step 3: Find a Point in the Solution Set
- Choose a point within the overlapping region that satisfies both inequalities.
- A suitable point can be found by inspection. Let's determine if the point [tex]\( (2, 5) \)[/tex] lies in the solution set:
- Substitute [tex]\( x = 2 \)[/tex] into [tex]\( y \geq 2x - 4 \)[/tex]:
[tex]\[ y \geq 2(2) - 4 \][/tex]
[tex]\[ y \geq 4 - 4 \][/tex]
[tex]\[ y \geq 0 \][/tex]
- Since [tex]\( y = 5 \)[/tex] (for the point [tex]\( (2, 5) \)[/tex]), [tex]\( 5 \geq 0 \)[/tex] is true.
- Substitute [tex]\( x = 2 \)[/tex] into [tex]\( y > -x + 8 \)[/tex]:
[tex]\[ y > -(2) + 8 \][/tex]
[tex]\[ y > -2 + 8 \][/tex]
[tex]\[ y > 6 \][/tex]
- Since [tex]\( y = 5 \)[/tex] (for the point [tex]\( (2, 5) \)[/tex]), [tex]\( 5 > 6 \)[/tex] is not true.
- This choice of point doesn't satisfy the set, so let's choose another point: [tex]\( (3, 5) \)[/tex]
- Substitute [tex]\( x = 3 \)[/tex] into
[tex]\( y \geq 5 \)[/tex]
[tex]\( y > 5 \)[/tex]
These tests show that some of the inequalities could be mismatched, by curving the whole solution set [tex]\( (5,6) \)[/tex]
### Final Solution
- A point in the solution set is [tex]\( (4, 6) \)[/tex].
