Answer:
See explanation
Step-by-step explanation:
1. To determine whether the ordered pair is the solution to the inequality, just substitute the coordinates of the ordered pair into the inequality
[tex]y>-2x+y\g\le 4[/tex]
A. For the ordered pair (-2,-3),
[tex]x=-2, y=-3[/tex]
Thus,
[tex]-3>-2\cdot (-2)+(-3)\le 4\\ \\-3>-4-3\le 4\\ \\-3>-7\le 4[/tex]
This option is true, because -3>-7 and -7≤4
B. For the ordered pair (0,-4),
[tex]x=0, y=-4[/tex]
Thus,
[tex]-4>-2\cdot 0+(-4)\le 4\\ \\-4>-4\le 4[/tex]
This option is false, because -4=-4 (not -4>-4)
C. For the ordered pair (1,5),
[tex]x=1, y=5[/tex]
Thus,
[tex]5>-2\cdot 1+5\le 4\\ \\5>-2+5\le 4\\ \\5>3\le 4[/tex]
This option is true, because 5>3 and 3≤4
D. For the ordered pair (1,3),
[tex]x=1, y=3[/tex]
Thus,
[tex]3>-2\cdot 1+3\le 4\\ \\3>-2+3\le 4\\ \\3>1\le 4[/tex]
This option is true, because 3>1 and 1≤4
2. The inequality [tex]y<-12x+2y\ge -32x+2[/tex] is equivalent to the system of two inequalities
[tex]\left\{\begin{array}{l}-12x+2y>y\\ \\-12x+2y\ge -32x+2\end{array}\right.\Rightarrow \left\{\begin{array}{l}-12x+y>0\\ \\20x+2y\ge 2\end{array}\right.[/tex]
Plot the dotted line [tex]-12x+y=0[/tex] and shade the upper region. Plot the solid line [tex]20x+2y=2[/tex] and shede the right part. The intersection of these two regions is the solution set to the inequality [tex]y<-12x+2y\ge -32x+2[/tex] (see attached diagram)
