Answer :
To solve the system of inequalities [tex]\( y > -5x + 2 \)[/tex] and [tex]\( y < 2x - 5 \)[/tex], we need to find the region in the coordinate plane where both inequalities are true at the same time.
Let's solve each inequality step by step and analyze their graphical representation.
### Step 1: Graph the first inequality [tex]\( y > -5x + 2 \)[/tex]
1. Graph the boundary line [tex]\( y = -5x + 2 \)[/tex]:
- This is a straight line with slope [tex]\(-5\)[/tex] and y-intercept [tex]\(2\)[/tex].
- To graph it, find two points on the line, for example:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. So, one point is [tex]\((0, 2)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -5(1) + 2 = -3\)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
2. Draw the boundary line, typically with a dashed line because the inequality is strict ([tex]\(>\)[/tex], not [tex]\(\geq\)[/tex]).
3. Shade the region above the line (since [tex]\( y > -5x + 2 \)[/tex]), which is the region where the inequality holds.
### Step 2: Graph the second inequality [tex]\( y < 2x - 5 \)[/tex]
1. Graph the boundary line [tex]\( y = 2x - 5 \)[/tex]:
- This is a straight line with slope [tex]\(2\)[/tex] and y-intercept [tex]\(-5\)[/tex].
- To graph it, find two points on the line, for example:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -5 \)[/tex]. So, one point is [tex]\((0, -5)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 2(1) - 5 = -3\)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
2. Draw the boundary line, again with a dashed line because the inequality is strict ([tex]\(<\)[/tex], not [tex]\(\leq\)[/tex]).
3. Shade the region below the line (since [tex]\( y < 2x - 5 \)[/tex]), which is the region where the inequality holds.
### Step 3: Find the intersection of the shaded regions
The solution to the system of inequalities will be the intersection of the regions from both inequalities.
- From [tex]\( y > -5x + 2 \)[/tex], we have the region above the line [tex]\( y = -5x + 2 \)[/tex].
- From [tex]\( y < 2x - 5 \)[/tex], we have the region below the line [tex]\( y = 2x - 5 \)[/tex].
### Step 4: Find the points of intersection, if necessary
To determine the points of intersection between the lines [tex]\( y = -5x + 2 \)[/tex] and [tex]\( y = 2x - 5 \)[/tex]:
1. Set the equations equal to each other to find the point of intersection:
[tex]\[ -5x + 2 = 2x - 5 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ -5x + 2 = 2x - 5 \][/tex]
[tex]\[ 2 + 5 = 2x + 5x \][/tex]
[tex]\[ 7 = 7x \][/tex]
[tex]\[ x = 1 \][/tex]
3. Substitute [tex]\( x = 1 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) - 5 = -3 \][/tex]
So, the lines intersect at the point [tex]\((1, -3)\)[/tex].
### Step 5: Conclusion
The intersection of the regions from both inequalities (the solution set) will be the area above the line [tex]\( y = -5x + 2 \)[/tex] and below the line [tex]\( y = 2x - 5 \)[/tex].
In summary:
The solution to the system of inequalities [tex]\( y > -5x + 2 \)[/tex] and [tex]\( y < 2x - 5 \)[/tex] is the region in the coordinate plane that lies above the line [tex]\( y = -5x + 2 \)[/tex] and below the line [tex]\( y = 2x - 5 \)[/tex].
Graphically, the solution is the region that intersects both shaded areas and does not cross the boundary lines (since they are strict inequalities). This region is visually a narrow band that widens as you move away from the origin along the intersection guidelines formed by the given inequalities.
Let's solve each inequality step by step and analyze their graphical representation.
### Step 1: Graph the first inequality [tex]\( y > -5x + 2 \)[/tex]
1. Graph the boundary line [tex]\( y = -5x + 2 \)[/tex]:
- This is a straight line with slope [tex]\(-5\)[/tex] and y-intercept [tex]\(2\)[/tex].
- To graph it, find two points on the line, for example:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. So, one point is [tex]\((0, 2)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -5(1) + 2 = -3\)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
2. Draw the boundary line, typically with a dashed line because the inequality is strict ([tex]\(>\)[/tex], not [tex]\(\geq\)[/tex]).
3. Shade the region above the line (since [tex]\( y > -5x + 2 \)[/tex]), which is the region where the inequality holds.
### Step 2: Graph the second inequality [tex]\( y < 2x - 5 \)[/tex]
1. Graph the boundary line [tex]\( y = 2x - 5 \)[/tex]:
- This is a straight line with slope [tex]\(2\)[/tex] and y-intercept [tex]\(-5\)[/tex].
- To graph it, find two points on the line, for example:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -5 \)[/tex]. So, one point is [tex]\((0, -5)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 2(1) - 5 = -3\)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
2. Draw the boundary line, again with a dashed line because the inequality is strict ([tex]\(<\)[/tex], not [tex]\(\leq\)[/tex]).
3. Shade the region below the line (since [tex]\( y < 2x - 5 \)[/tex]), which is the region where the inequality holds.
### Step 3: Find the intersection of the shaded regions
The solution to the system of inequalities will be the intersection of the regions from both inequalities.
- From [tex]\( y > -5x + 2 \)[/tex], we have the region above the line [tex]\( y = -5x + 2 \)[/tex].
- From [tex]\( y < 2x - 5 \)[/tex], we have the region below the line [tex]\( y = 2x - 5 \)[/tex].
### Step 4: Find the points of intersection, if necessary
To determine the points of intersection between the lines [tex]\( y = -5x + 2 \)[/tex] and [tex]\( y = 2x - 5 \)[/tex]:
1. Set the equations equal to each other to find the point of intersection:
[tex]\[ -5x + 2 = 2x - 5 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ -5x + 2 = 2x - 5 \][/tex]
[tex]\[ 2 + 5 = 2x + 5x \][/tex]
[tex]\[ 7 = 7x \][/tex]
[tex]\[ x = 1 \][/tex]
3. Substitute [tex]\( x = 1 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) - 5 = -3 \][/tex]
So, the lines intersect at the point [tex]\((1, -3)\)[/tex].
### Step 5: Conclusion
The intersection of the regions from both inequalities (the solution set) will be the area above the line [tex]\( y = -5x + 2 \)[/tex] and below the line [tex]\( y = 2x - 5 \)[/tex].
In summary:
The solution to the system of inequalities [tex]\( y > -5x + 2 \)[/tex] and [tex]\( y < 2x - 5 \)[/tex] is the region in the coordinate plane that lies above the line [tex]\( y = -5x + 2 \)[/tex] and below the line [tex]\( y = 2x - 5 \)[/tex].
Graphically, the solution is the region that intersects both shaded areas and does not cross the boundary lines (since they are strict inequalities). This region is visually a narrow band that widens as you move away from the origin along the intersection guidelines formed by the given inequalities.
