Answer:
The system of linear inequalities to each graph is the following
1) (Largest graph)
(1) y≤-3x
(2) y≥2x-4
2) (Smallest graph)
(1) y≤x+2
(2) y≤-x-3
Explanation:
1) Largest graph
We can see two right lines
a) One of the lines (Line 1) goes through:
Origin: O=P1=(0,0)=(x1, y1)→x1=0, y1=0
P2=(2, -6)→x2=2, y2=-6
Slope line 1: m1=(y2-y1)/(x2-x1)
Replacing the known values:
m1=(-6-0)/(2-0)
m1=(-6)/(2)
m1=-3
Equation line 1:
y-y1=m1 (x-x1)
y-0=-3(x-0)
y=-3x
The region is below this right line, then the first inequality is:
(1) y≤-3x
b) The other line (Line 2) goes through:
P3=(2,0)=(x3, y3)→x3=2, y3=0
P4=(5, 6)→x4=5, y4=6
Slope line 2: m2=(y4-y3)/(x4-x3)
Replacing the known values:
m2=(6-0)/(5-2)
m2=(6)/(3)
m2=2
Equation line 2:
y-y3=m2 (x-x1)
y-0=2(x-2)
y=2x-4
The region is above this right line, then the second inequality is:
(2) y≥2x-4
Then, the system of linear inequalities for the largest graph is:
(1) y≤-3x
(2) y≥2x-4
2) Smallest graph
We can see two right lines
a) One of the lines (Line 1) goes through:
P1=(0, 2)=(x1, y1)→x1=0, y1=2
P2=(5, 7)→x2=5, y2=7
Slope line 1: m1=(y2-y1)/(x2-x1)
Replacing the known values:
m1=(7-2)/(5-0)
m1=(5)/(5)
m1=1
Equation line 1:
y-y1=m1 (x-x1)
y-2=1(x-0)
y-2=1(x)
y-2=x
y-2+2=x+2
y=x+2
The region is below this right line, then the first inequality is:
(1) y≤x+2
b) The other line (Line 2) goes through:
P3=(0,-3)=(x3, y3)→x3=0, y3=-3
P4=(5, -8)→x4=5, y4=-8
Slope line 2: m2=(y4-y3)/(x4-x3)
Replacing the known values:
m2=(-8-(-3))/(5-0)
m2=(-8+3)/(5)
m2=(-5)/5
m2=-1
Equation line 2:
y-y3=m2 (x-x1)
y-(-3)=-1(x-0)
y+3=-1(x)
y+3=-x
y+3-3=-x-3
y=-x-3
The region is below this right line, then the second inequality is:
(2) y≤-x-3
Then, the system of linear inequalities for the smallest graph is:
(1) y≤x+2
(2) y≤-x-3