We are asked to determine an inequality that does not contain the following points as solutions:
[tex](0,0);(0,-1);(0,1)[/tex]
And the following are solutions:
[tex](1,1);(3,-1);(-1,3)[/tex]
First, we will plot the points:
Therefore, we can use a line that has y-intercept 2 and x-intercept 2, like this:
To determine the equation of the line we use the intercept form of a line equation:
[tex]\frac{x}{a}+\frac{y}{b}=1[/tex]
Where:
[tex]\begin{gathered} a=\text{ x-intercept} \\ b=\text{ y-intercept} \end{gathered}[/tex]
Now, we substitute the intercepts:
[tex]\frac{x}{2}+\frac{y}{2}=1[/tex]
Now, we multiply both sides by 2:
[tex]x+y=2[/tex]
Now, we subtract "x" from both sides:
[tex]y=2-x[/tex]
Now, since the points are included in the line and we need the solutions to be above the line we use the inequality sign "greater or equal to":
[tex]y\ge2-x[/tex]
The graph is the following:
Thus we get the required inequality.
