What system of linear inequalities is shown in the graph?   Enter your answers in the boxes. there should be two, I have an idea but still a little confused XP

In order to determine the system of inequalities that have been graphed, we need to find the equations of the two lines. We can then use the equations to write the corresponding inequalities.

[tex]<b><u>Equation of the dashed line</u></b>[/tex]

The equation of this line is given by the formula,

[tex]y = mx + c[/tex]
where m is the slope and c is the y -intercept.

From the graph,

[tex]m = \frac{rise}{run} = \frac{ + 4}{ - 1} = - 4[/tex]
and
[tex]c = 1[/tex]
Hence the equation of this line is
[tex]y = - 4x + 1[/tex]
Since the line is not solid it means the possible inequalities are

[tex]y < - 4x + 1 \: or \: y > - 4x + 1[/tex]

From the graph one of the solution sets is the origin, which has coordinates,
[tex](0,0) [/tex]
We can test the two inequalities with this point to determine the right one. That is,

[tex]0< - 4(0)+ 1 \: or \: 0> - 4(0)+ 1[/tex]

[tex]0< 0+ 1 \: or \: 0 > 0+ 1[/tex]

[tex]0< 1 \: or \: 0 > 1[/tex]

We can see that the first statement is true but the second is false.

This implies that the right inequality is;

[tex]y < - 4x + 1[/tex]

[tex]<b><u>Equation of the solid line</u></b>[/tex]

With the solid line,the slope is
[tex]m=\frac{1}{2}[/tex] and

[tex]c=-1[/tex]

Hence the equation is,
[tex]y=\frac{1}{2}x-1[/tex]

Since this line is solid, the possible inequalities are;

[tex]y\ge \frac{1}{2}x-1 \:or\:y\le \frac{1}{2}x-1[/tex]

Testing with the origin again, we have;
[tex]0\ge \frac{1}{2}(0)-1 \:or\:y\le \frac{1}{2}(0)-1[/tex]

[tex]0\ge -1 \:or\:y\le -1[/tex]

The second statement is true.

Hence the inequaliy of the solid line is,

[tex]y \geqslant \frac{1}{2} x - 1[/tex]

Therefore the system of inequalities are;

[tex]y < - 4x + 1[/tex]

and

[tex]y \geqslant \frac{1}{2} x - 1[/tex]