Question

Please help with telling me how to write a system of inequalities to represent the shaded portion of the graph that I have sent as a picture.

Answer 1

To get the system of equation, we have to get the equation of both lines

For the first line, that is the thick line, let us get the slope of the line

Let's take two points on the line (0, 4) and (4, 0)

We have the slope as

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ m=(0-4)/(4-0) \\ m=(-4)/(4) \\ m=-1 \end{gathered}`

So, the slope of the first line is - 1,

Now equation of a line in slope-intercept form is given as

`\begin{gathered} y=mx+b \\ \text{where m is the slope and b is the intercept on the y axis} \end{gathered}`

So, since the line cuts the y-axis at 4, the y-intercept is 4

So, the equation of the line is

`\begin{gathered} y=mx+b \\ y=-1x+4 \\ y=-x+4 \end{gathered}`

But since, it is an inequality, it is a thick line and the shaded part downward to the left, the equation of this line becomes

`y\le-x+4`

Now, for the second line, this line passes through the origin, so the y-intercept is 0,

So, the equation of the line should be

`\begin{gathered} y=mx \\ \text{but this line intercept the other line at (3, 1) } \\ x=3,y=1 \\ \text{substituting we have } \\ y=mx \\ 1=3m \\ m=(1)/(3) \\ so\text{ the slope of this line is }(1)/(3) \\ \text{The equation becomes } \\ y=(1)/(3)x \\ y=(x)/(3) \end{gathered}`

But since this line is a broken line (less than/greater than) and the shaded part is above, to the right (greater than)

The equation of the line becomes

`y>(x)/(3)`

Hence, our answer is the system of equations for the shaded portion of the line is

`y\le-x+4\text{ and }y>(x)/(3)`