Answer :
# Solving Linear Inequalities in Two Variables using the Graph Method
## Part A
### Inequalities:
1. [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex]
2. [tex]\(y \leq x + 4\)[/tex]
3. [tex]\(x < 4\)[/tex]
### Steps to Solve:
1. Graph the first inequality [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex]:
- Plot the line [tex]\(y = -\frac{1}{2}x - 2\)[/tex]. Since the inequality is [tex]\(\geq\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\(y = -\frac{1}{2}(4) - 2 = -4\)[/tex].
2. Graph the second inequality [tex]\(y \leq x + 4\)[/tex]:
- Plot the line [tex]\(y = x + 4\)[/tex]. Since the inequality is [tex]\(\leq\)[/tex], shade the region below this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex].
- When [tex]\(x = -4\)[/tex], [tex]\(y = 0\)[/tex].
3. Graph the third inequality [tex]\(x < 4\)[/tex]:
- Draw the vertical line [tex]\(x = 4\)[/tex]. Since the inequality is [tex]\(<\)[/tex], shade the region to the left of this line.
4. Find the intersection of the shaded regions:
- The solution to this system is the region where all three shaded areas overlap.
### Solution:
- The solution is the region that is:
- Above the line [tex]\(y = -\frac{1}{2}x - 2\)[/tex]
- Below the line [tex]\(y = x + 4\)[/tex]
- To the left of the line [tex]\(x = 4\)[/tex].
### Visual Representation:
- The region of overlap can be seen on the graph by shading the corresponding areas accordingly. The solution is the triangular region where the conditions [tex]\(x < 4\)[/tex], [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex], and [tex]\(y \leq x + 4\)[/tex] are satisfied.
## Part B
### Inequalities:
1. [tex]\(y > -\frac{3}{2}x - 6\)[/tex]
2. [tex]\(y \geq \frac{3}{2}x\)[/tex]
3. [tex]\(y < 5\)[/tex]
### Steps to Solve:
1. Graph the first inequality [tex]\(y > -\frac{3}{2}x - 6\)[/tex]:
- Plot the line [tex]\(y = -\frac{3}{2}x - 6\)[/tex]. Since the inequality is [tex]\(>\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -6\)[/tex].
- When [tex]\(x = -4\)[/tex], [tex]\(y = 0\)[/tex].
2. Graph the second inequality [tex]\(y \geq \frac{3}{2}x\)[/tex]:
- Plot the line [tex]\(y = \frac{3}{2}x\)[/tex]. Since the inequality is [tex]\(\geq\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 0\)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\(y = 6\)[/tex].
3. Graph the third inequality [tex]\(y < 5\)[/tex]:
- Draw the horizontal line [tex]\(y = 5\)[/tex]. Since the inequality is [tex]\(<\)[/tex], shade the region below this line.
4. Find the intersection of the shaded regions:
- The solution to this system is the region where all three shaded areas overlap.
### Solution:
- The solution is the region that is:
- Above the line [tex]\(y = -\frac{3}{2}x - 6\)[/tex]
- Above the line [tex]\(y = \frac{3}{2}x\)[/tex]
- Below the line [tex]\(y = 5\)[/tex].
### Visual Representation:
- The region of overlap can be seen on the graph by shading the corresponding areas accordingly. The solution is the area bounded by [tex]\(y > -\frac{3}{2}x - 6\)[/tex], [tex]\(y \geq \frac{3}{2}x\)[/tex], and [tex]\(y < 5\)[/tex], forming a polygonal region.
In both parts, by plotting the lines and shading the appropriate regions, the solution to the system of inequalities is the area where all conditions are satisfied simultaneously. A graphing calculator or software can assist in visually verifying the solution.
## Part A
### Inequalities:
1. [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex]
2. [tex]\(y \leq x + 4\)[/tex]
3. [tex]\(x < 4\)[/tex]
### Steps to Solve:
1. Graph the first inequality [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex]:
- Plot the line [tex]\(y = -\frac{1}{2}x - 2\)[/tex]. Since the inequality is [tex]\(\geq\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\(y = -\frac{1}{2}(4) - 2 = -4\)[/tex].
2. Graph the second inequality [tex]\(y \leq x + 4\)[/tex]:
- Plot the line [tex]\(y = x + 4\)[/tex]. Since the inequality is [tex]\(\leq\)[/tex], shade the region below this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex].
- When [tex]\(x = -4\)[/tex], [tex]\(y = 0\)[/tex].
3. Graph the third inequality [tex]\(x < 4\)[/tex]:
- Draw the vertical line [tex]\(x = 4\)[/tex]. Since the inequality is [tex]\(<\)[/tex], shade the region to the left of this line.
4. Find the intersection of the shaded regions:
- The solution to this system is the region where all three shaded areas overlap.
### Solution:
- The solution is the region that is:
- Above the line [tex]\(y = -\frac{1}{2}x - 2\)[/tex]
- Below the line [tex]\(y = x + 4\)[/tex]
- To the left of the line [tex]\(x = 4\)[/tex].
### Visual Representation:
- The region of overlap can be seen on the graph by shading the corresponding areas accordingly. The solution is the triangular region where the conditions [tex]\(x < 4\)[/tex], [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex], and [tex]\(y \leq x + 4\)[/tex] are satisfied.
## Part B
### Inequalities:
1. [tex]\(y > -\frac{3}{2}x - 6\)[/tex]
2. [tex]\(y \geq \frac{3}{2}x\)[/tex]
3. [tex]\(y < 5\)[/tex]
### Steps to Solve:
1. Graph the first inequality [tex]\(y > -\frac{3}{2}x - 6\)[/tex]:
- Plot the line [tex]\(y = -\frac{3}{2}x - 6\)[/tex]. Since the inequality is [tex]\(>\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -6\)[/tex].
- When [tex]\(x = -4\)[/tex], [tex]\(y = 0\)[/tex].
2. Graph the second inequality [tex]\(y \geq \frac{3}{2}x\)[/tex]:
- Plot the line [tex]\(y = \frac{3}{2}x\)[/tex]. Since the inequality is [tex]\(\geq\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 0\)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\(y = 6\)[/tex].
3. Graph the third inequality [tex]\(y < 5\)[/tex]:
- Draw the horizontal line [tex]\(y = 5\)[/tex]. Since the inequality is [tex]\(<\)[/tex], shade the region below this line.
4. Find the intersection of the shaded regions:
- The solution to this system is the region where all three shaded areas overlap.
### Solution:
- The solution is the region that is:
- Above the line [tex]\(y = -\frac{3}{2}x - 6\)[/tex]
- Above the line [tex]\(y = \frac{3}{2}x\)[/tex]
- Below the line [tex]\(y = 5\)[/tex].
### Visual Representation:
- The region of overlap can be seen on the graph by shading the corresponding areas accordingly. The solution is the area bounded by [tex]\(y > -\frac{3}{2}x - 6\)[/tex], [tex]\(y \geq \frac{3}{2}x\)[/tex], and [tex]\(y < 5\)[/tex], forming a polygonal region.
In both parts, by plotting the lines and shading the appropriate regions, the solution to the system of inequalities is the area where all conditions are satisfied simultaneously. A graphing calculator or software can assist in visually verifying the solution.
