[tex]c)y\ge\frac{1}{4}x-4[/tex]
Explanation
Step 1
let's find the equation of the line
a)slope
the slope of a line is given by:
[tex]\begin{gathered} \text{slope}=\frac{chang\text{e in y}}{\text{change in x}}=\frac{y_2-y_{1|}}{x_2-x_1} \\ \text{where} \\ P1(x_1,y_1) \\ \text{and } \\ \text{P2(x}_2,y_2) \\ \text{are 2 points from the line} \end{gathered}[/tex]
so
pick up 2 points from the line
let
P1=(0,-4)
P2=(4,-3)
now, replace in the equation to find the slope
[tex]\begin{gathered} \text{slope}=\frac{y_2-y_{1|}}{x_2-x_1} \\ \text{slope}=\frac{-3-(-4)}{4-0}=\frac{-3+4}{4}=\frac{1}{4} \end{gathered}[/tex]
Step 2
b) the equation of the line, it can be found by using the poitn slope formula
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ \text{where m is the slope and } \\ P1(x_1,y_1) \end{gathered}[/tex]
then, let
P1=(0,-4)
slope=1/4
replace and isolate y
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-(-4)=\frac{1}{4}(x-0) \\ y+4=\frac{1}{4}x \\ \text{subtract 4 in both sides} \\ y+4-4=\frac{1}{4}x-4 \\ y=\frac{1}{4}x-4 \end{gathered}[/tex]
Step 3
finally, we need the shaded region, it means all the values over the line, in other words, the values greater than the function, so
[tex]\begin{gathered} y=\frac{1}{4}x-4\Rightarrow shaded\text{ region }\Rightarrow y\ge\frac{1}{4}x-4 \\ \text{the line is continous so}\Rightarrow\ge \end{gathered}[/tex]
therfore, the answer is
[tex]c)y\ge\frac{1}{4}x-4[/tex]
I hope this helps you
