Answer :
Inequalities
We are given a graph where a shaded area is shown. It's bounded by a solid line.
We first need to find the equation of that line by selecting any pair of clear points on the line.
For example, I can see the points (-4,1) and (0,3).
To find the equation of the line passing through them we use the following equation.
The equation of a line passing through points (x1,y1) and (x2,y2) can be found as follows:
[tex]\displaystyle y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]Substituting the given points:
[tex]\displaystyle y-1=\frac{3-1}{0+4}(x+4)[/tex]Simplifying:
[tex]\begin{gathered} \displaystyle y-1=\frac{2}{4}(x+4) \\ \displaystyle y-1=\frac{1}{2}(x+4) \\ 2y-2=x+4 \\ 2y=x+6 \\ y=\frac{1}{2}x+3 \end{gathered}[/tex]Since the shaded area is below the line, the inequality is:
[tex]y\leq\frac{1}{2}x+3[/tex]Note the symbol is 'less or equal' and not 'less' because the line is solid, which includes equality, or the points in the line.