Answer
(a) $x=\left\{-2, 1\right\}$
(b) $[-2, 1]$
Work Step by Step
To solve the given equation graphically, perform the following steps:
(1) Treat each side of the equation as a function and graph them on the same coordinate plane.
Use a graphing calculator to graph:
$y=x^2$ (red graph)
$y=2-x$ (blue graph)
(refer to the image below for the graph)
(2) Identify the point/s where the graphs intersect
(a)
The points where the two graphs intersect represents the instance where the value of $x^2$ is equal to the value of $2-x$. The x-coordinate of this point is the solution to the given equation.
Notice that the graphs intersect at the points $(-2, 4)$ and $(1, 1)$.
The x-coordinates of these points are $-2$ and $1$.
Thus, the solution to the given equation is $x=\left\{-2, 1\right\}$.
(b)
The region where the graph of the function $y=x^2$ is lower than or equal to the graph of the function $y=2-x$ is the region where $x^2 \le 2-x$.
Notice that the red graph ($y=x^2$) is lower than or equal to the blue graph ($y=2-x$) in the region $(-2, 1)$. Since the inequality involves $\le$, then the endpoints $-2$ and $1$ are part of the solution set.
Thus, the solution to the given inequality is $[-2, 1]$.
