Answer :
To graph the system of linear inequalities and find a point in the solution region, follow these steps:
### Step 1: Graph the Boundary Lines
1. Graph the line for [tex]\( y < 4 - x \)[/tex]:
- Rewrite the boundary line equation as [tex]\( y = 4 - x \)[/tex].
- This is a linear equation with a y-intercept of 4 (the point (0, 4)) and a slope of -1.
- Plot the y-intercept (0, 4).
- Use the slope to find another point: starting from (0, 4), move down 1 unit and to the right 1 unit, reaching (1, 3). Plot this point as well.
- Draw a dashed line through these points to indicate that the inequality is [tex]\( y < 4 - x \)[/tex], not [tex]\( y \leq 4 - x \)[/tex].
- Shade the region below the line, as this represents [tex]\( y < 4 - x \)[/tex].
2. Graph the line for [tex]\( y > 2x - 4 \)[/tex]:
- Rewrite the boundary line equation as [tex]\( y = 2x - 4 \)[/tex].
- This is a linear equation with a y-intercept of -4 (the point (0, -4)) and a slope of 2.
- Plot the y-intercept (0, -4).
- Use the slope to find another point: starting from (0, -4), move up 2 units and to the right 1 unit, reaching (1, -2). Plot this point as well.
- Draw a dashed line through these points to indicate that the inequality is [tex]\( y > 2x - 4 \)[/tex], not [tex]\( y \geq 2x - 4 \)[/tex].
- Shade the region above the line, as this represents [tex]\( y > 2x - 4 \)[/tex].
### Step 2: Identify the Solution Region
- The solution region is where the shaded areas from both inequalities overlap. This is the area where [tex]\( y < 4 - x \)[/tex] and [tex]\( y > 2x - 4 \)[/tex] are both true.
### Step 3: Choose a Point in the Solution Region
- Select an example point from within the overlapping shaded region. A convenient point would be (2, 1).
### Step 4: Verify the Point Satisfies Both Inequalities
- Checking the first inequality [tex]\( y < 4 - x \)[/tex]:
- Substitute the point (2, 1) into the inequality.
- This gives [tex]\( 1 < 4 - 2 \)[/tex], which simplifies to [tex]\( 1 < 2 \)[/tex]. This is true.
- Checking the second inequality [tex]\( y > 2x - 4 \)[/tex]:
- Substitute the point (2, 1) into the inequality.
- This gives [tex]\( 1 > 2(2) - 4 \)[/tex], which simplifies to [tex]\( 1 > 4 - 4 \)[/tex], then to [tex]\( 1 > 0 \)[/tex]. This is also true.
Hence, point (2, 1) satisfies both inequalities and lies in the solution region.
### Conclusion
The point (2, 1) is in the solution region of the system of inequalities:
[tex]\[ \begin{array}{l} y < 4 - x \\ y > 2x - 4 \end{array} \][/tex]
We know this point satisfies the system because substituting it into both inequalities yields true statements for each.
### Step 1: Graph the Boundary Lines
1. Graph the line for [tex]\( y < 4 - x \)[/tex]:
- Rewrite the boundary line equation as [tex]\( y = 4 - x \)[/tex].
- This is a linear equation with a y-intercept of 4 (the point (0, 4)) and a slope of -1.
- Plot the y-intercept (0, 4).
- Use the slope to find another point: starting from (0, 4), move down 1 unit and to the right 1 unit, reaching (1, 3). Plot this point as well.
- Draw a dashed line through these points to indicate that the inequality is [tex]\( y < 4 - x \)[/tex], not [tex]\( y \leq 4 - x \)[/tex].
- Shade the region below the line, as this represents [tex]\( y < 4 - x \)[/tex].
2. Graph the line for [tex]\( y > 2x - 4 \)[/tex]:
- Rewrite the boundary line equation as [tex]\( y = 2x - 4 \)[/tex].
- This is a linear equation with a y-intercept of -4 (the point (0, -4)) and a slope of 2.
- Plot the y-intercept (0, -4).
- Use the slope to find another point: starting from (0, -4), move up 2 units and to the right 1 unit, reaching (1, -2). Plot this point as well.
- Draw a dashed line through these points to indicate that the inequality is [tex]\( y > 2x - 4 \)[/tex], not [tex]\( y \geq 2x - 4 \)[/tex].
- Shade the region above the line, as this represents [tex]\( y > 2x - 4 \)[/tex].
### Step 2: Identify the Solution Region
- The solution region is where the shaded areas from both inequalities overlap. This is the area where [tex]\( y < 4 - x \)[/tex] and [tex]\( y > 2x - 4 \)[/tex] are both true.
### Step 3: Choose a Point in the Solution Region
- Select an example point from within the overlapping shaded region. A convenient point would be (2, 1).
### Step 4: Verify the Point Satisfies Both Inequalities
- Checking the first inequality [tex]\( y < 4 - x \)[/tex]:
- Substitute the point (2, 1) into the inequality.
- This gives [tex]\( 1 < 4 - 2 \)[/tex], which simplifies to [tex]\( 1 < 2 \)[/tex]. This is true.
- Checking the second inequality [tex]\( y > 2x - 4 \)[/tex]:
- Substitute the point (2, 1) into the inequality.
- This gives [tex]\( 1 > 2(2) - 4 \)[/tex], which simplifies to [tex]\( 1 > 4 - 4 \)[/tex], then to [tex]\( 1 > 0 \)[/tex]. This is also true.
Hence, point (2, 1) satisfies both inequalities and lies in the solution region.
### Conclusion
The point (2, 1) is in the solution region of the system of inequalities:
[tex]\[ \begin{array}{l} y < 4 - x \\ y > 2x - 4 \end{array} \][/tex]
We know this point satisfies the system because substituting it into both inequalities yields true statements for each.
