Answer :
To determine the solution to the system of linear inequalities graphically, let's go step by step and first understand the given inequalities and their corresponding lines.
### Step-by-Step Solution:
1. Identify the boundary lines:
- For the inequality \( y \leq 2x - 5 \):
The boundary line is \( y = 2x - 5 \).
- For the inequality \( y > -3x + 1 \):
The boundary line is \( y = -3x + 1 \).
2. Find the intersection point of the lines:
To find where the lines intersect, we can set the equations equal to each other:
[tex]\[ 2x - 5 = -3x + 1 \][/tex]
Solving for \( x \):
[tex]\[ 2x + 3x = 1 + 5 \][/tex]
[tex]\[ 5x = 6 \][/tex]
[tex]\[ x = \frac{6}{5} = 1.2 \][/tex]
Substitute \( x = 1.2 \) back into either of the original equations to solve for \( y \):
Using \( y = 2x - 5 \):
[tex]\[ y = 2(1.2) - 5 \][/tex]
[tex]\[ y = 2.4 - 5 \][/tex]
[tex]\[ y = -2.6 \][/tex]
Therefore, the intersection point of the lines is \( (1.2, -2.6) \).
3. Graph the boundary lines:
- Plot the line \( y = 2x - 5 \). This line will be solid because the inequality includes \( y \leq \).
- Plot the line \( y = -3x + 1 \). This line will be dashed because the inequality is strict \( y > \).
4. Shade the appropriate regions:
- For \( y \leq 2x - 5 \):
Shade the region below the solid line \( y = 2x - 5 \).
- For \( y > -3x + 1 \):
Shade the region above the dashed line \( y = -3x + 1 \).
5. Determine the solution region:
The solution to the system of inequalities is the region where the shaded areas overlap. This is the area that lies below the line \( y = 2x - 5 \) and above the line \( y > -3x + 1 \).
### Conclusion:
The solution to the system of linear inequalities [tex]\( \begin{array}{l} y \leq 2x - 5 \\ y > -3x + 1 \end{array} \)[/tex] is the region that lies below the line [tex]\( y = 2x - 5 \)[/tex] and above the line [tex]\( y = -3x + 1 \)[/tex], with the boundary at the intersection point [tex]\((1.2, -2.6)\)[/tex].
### Step-by-Step Solution:
1. Identify the boundary lines:
- For the inequality \( y \leq 2x - 5 \):
The boundary line is \( y = 2x - 5 \).
- For the inequality \( y > -3x + 1 \):
The boundary line is \( y = -3x + 1 \).
2. Find the intersection point of the lines:
To find where the lines intersect, we can set the equations equal to each other:
[tex]\[ 2x - 5 = -3x + 1 \][/tex]
Solving for \( x \):
[tex]\[ 2x + 3x = 1 + 5 \][/tex]
[tex]\[ 5x = 6 \][/tex]
[tex]\[ x = \frac{6}{5} = 1.2 \][/tex]
Substitute \( x = 1.2 \) back into either of the original equations to solve for \( y \):
Using \( y = 2x - 5 \):
[tex]\[ y = 2(1.2) - 5 \][/tex]
[tex]\[ y = 2.4 - 5 \][/tex]
[tex]\[ y = -2.6 \][/tex]
Therefore, the intersection point of the lines is \( (1.2, -2.6) \).
3. Graph the boundary lines:
- Plot the line \( y = 2x - 5 \). This line will be solid because the inequality includes \( y \leq \).
- Plot the line \( y = -3x + 1 \). This line will be dashed because the inequality is strict \( y > \).
4. Shade the appropriate regions:
- For \( y \leq 2x - 5 \):
Shade the region below the solid line \( y = 2x - 5 \).
- For \( y > -3x + 1 \):
Shade the region above the dashed line \( y = -3x + 1 \).
5. Determine the solution region:
The solution to the system of inequalities is the region where the shaded areas overlap. This is the area that lies below the line \( y = 2x - 5 \) and above the line \( y > -3x + 1 \).
### Conclusion:
The solution to the system of linear inequalities [tex]\( \begin{array}{l} y \leq 2x - 5 \\ y > -3x + 1 \end{array} \)[/tex] is the region that lies below the line [tex]\( y = 2x - 5 \)[/tex] and above the line [tex]\( y = -3x + 1 \)[/tex], with the boundary at the intersection point [tex]\((1.2, -2.6)\)[/tex].
