Answer:
The system of inequalities:
[tex]3x+5y\leq 10[/tex] .......[1]
[tex]x-y<-1[/tex] .......[2]
First graph each inequality:
The graph of x-y<-1 is dashed and is not included in the graph of the solution as shown below in the Figure 1.
The intercept for the graph is the point where the graph crosses the axis
x-intercept of a graph is the point where the graph crosses the x-axis i.e, plug in 0 for y and solve for x.
y-intercept of a graph is the point where the graph crosses the y -axis i.e,
plug in 0 for x and solve for y.
The related equation for the inequalities [2] is;
x-y =-1
for x =0,
0-y=-1
⇒ y =1
Then, the y-intercept is (0, 1).
To find the x-intercept;
Put y =0 in above related equation;
x-0 =-1
x=-1
Therefore, the x-intercept is (-1, 0)
Also,
The graph [tex]3x+5y\leq 10[/tex] is solid and is included in the graph of the solution as shown below in Figure 2.
The related equation for the inequalities [1] is;
3x+5y =10
for x =0,
0+5y=10 or
5y=10 [divide by 5 both sides]
⇒ y =2
Then, the y-intercept is (0, 2).
To find the x-intercept;
Put y =0 in above related equation;
3x+ 0=10
3x= 10 [divide by 3 both sides ]
⇒ x =3.33..
Therefore, the x-intercept is (3.333, 0).
The solution of the system is the set of ordered pairs in the intersection of the graphs of [tex]3x+5y\leq 10[/tex] and [tex]x-y<-1[/tex].
So, the solution region is the darkest of the shaded area in the graph below in Figure-3.
Therefore, the only option D is correct.
