Answer
The graph is shown below:
Work Step by Step
Let us consider the inequalities $\begin{align}
& {{x}^{2}}+{{y}^{2}}\le 16 \\
& x+y<2 \\
\end{align}$
Put the equals symbol in place of the inequality and rewrite the equation as given below:
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=16 \\
& x+y=2 \\
\end{align}$
The equation shows a circle with origin at $\left( 0,0 \right)$ of radius $4$.
Second equation shows a line passing through the coordinates $\left( 0,2 \right)$.
Now, take origin $\left( 0,0 \right)$ as a test point for the equations and check the region in the graph to shade:
$\begin{align}
& {{x}^{2}}+{{y}^{2}}\le 16\text{ and }x\text{+}y<2 \\
& {{0}^{2}}+{{0}^{2}}\le 16\text{ and 0}<2 \\
& 0\le 16\text{ and 0}<2 \\
\end{align}$
As we see that the test point satisfies both inequalities, therefore the shaded region will contain the test point; that is, the origin in both cases.
Thus, the graph of the provided inequality is plotted.
