Step 1
Find the equation of the line of the inequality
Let
[tex]A(-3,-1)\ B(3,3)[/tex]
Find the slope of the line
The slope is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute the values
[tex]m=\frac{3+1}{3+3}[/tex]
[tex]m=\frac{4}{6}[/tex]
[tex]m=\frac{2}{3}[/tex]
Find the equation of the line into point-slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{2}{3}[/tex]
[tex](x1,y1)=B(3,3)[/tex]
substitute in the equation
[tex]y-3=\frac{2}{3}(x-3)[/tex]
[tex]y=\frac{2}{3}x-2+3[/tex]
[tex]y=\frac{2}{3}x+1[/tex]
Find the equation of the inequality
The solution is the shaded area above the dotted line
so the inequality is
[tex]y>\frac{2}{3}x+1[/tex]
If a point is the solution of the inequality, then it must satisfy the inequality. Let's check each of the points
Step 2
case A) [tex](0,0)[/tex]
Substitute the values of x and y in the inequality
[tex]x=0\ y=0[/tex]
[tex]0>\frac{2}{3}*0+1[/tex]
[tex]0>1[/tex]-----> is not true
therefore
the point [tex](0,0)[/tex] is not solution of the inequality
Step 3
case B) [tex](3,3)[/tex]
Substitute the values of x and y in the inequality
[tex]x=3\ y=3[/tex]
[tex]3>\frac{2}{3}*3+1[/tex]
[tex]3>3[/tex]-----> is not true
therefore
the point [tex](3,3)[/tex] is not solution of the inequality
Step 4
case C) [tex](-3,-1)[/tex]
Substitute the values of x and y in the inequality
[tex]x=-3\ y=-1[/tex]
[tex]-1>\frac{2}{3}*-3+1[/tex]
[tex]-1>-1[/tex]-----> is not true
therefore
the point [tex](-3,-1)[/tex] is not solution of the inequality
Step 5
case D) [tex](0,5)[/tex]
Substitute the values of x and y in the inequality
[tex]x=0\ y=5[/tex]
[tex]5>\frac{2}{3}*0+1[/tex]
[tex]5>1[/tex]-----> is true
therefore
the point [tex](0,5)[/tex] is a solution of the inequality
therefore
the answer is
[tex](0,5)[/tex]
