Answer
$\text{a)
NOT a solution
}\\\text{b)
NOT a solution
}\\\text{c)
solution
}\\\text{d)
solution
}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Substitute the given points in the given inequality, $
x+y\gt0
.$ If the inequality is satisfied, then the given point is a solution. Otherwise, the given point is not a solution.
$\bf{\text{Solution Details:}}$
a) Substituting the given point, $(
0,0
)$ in the given inequality results to
\begin{array}{l}\require{cancel}
x+y\gt0
\\\\
0+0\gt0
\\\\
0\gt0
\text{ (FALSE)}
.\end{array}
Hence, $(
0,0
)$ is NOT a solution.
b) Substituting the given point, $(
-2,1
)$ in the given inequality results to
\begin{array}{l}\require{cancel}
x+y\gt0
\\\\
-2+1\gt0
\\\\
-1\gt0
\text{ (FALSE)}
.\end{array}
Hence, $(
-2,1
)$ is NOT a solution.
c) Substituting the given point, $(
2,-1
)$ in the given inequality results to
\begin{array}{l}\require{cancel}
x+y\gt0
\\\\
2+(-1)\gt0
\\\\
2-1\gt0
\\\\
1\gt0
\text{ (TRUE)}
.\end{array}
Hence, $(
2,-1
)$ is a solution.
d) Substituting the given point, $(
-4,6
)$ in the given inequality results to
\begin{array}{l}\require{cancel}
x+y\gt0
\\\\
-4+6\gt0
\\\\
2\gt0
\text{ (TRUE)}
.\end{array}
Hence, $(
-4,6
)$ is a solution.
Hence,
\begin{array}{l}\require{cancel}
\text{a)
NOT a solution
}\\\text{b)
NOT a solution
}\\\text{c)
solution
}\\\text{d)
solution
}
\end{array}
