Answer
(a) $x=\left\{1, 3\right\}$
(b) $[1, 3]$
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)
$y=3-4x$ (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 $3-4x$. The x-coordinate of this point is the solution to the given equation.
Notice that the graphs intersect at the points $(1, -1)$ and $(3, -9)$.
The x-coordinates of these points are $1$ and $3$.
Thus, the solution to the given equation is $x=\left\{1, 3\right\}$.
(b)
The region where the graph of the function $y=-x^2$ is higher than or equal to the graph of the function $y=3-4x$ is the region where $-x^2 \ge 3-4x$.
Notice that the red graph ($y=-x^2$) is lower than or equal to the blue graph ($y=3-4x$) in the region $(1, 3)$. Since the inequality involves $\ge$, then the endpoints $1$ and $3$ are part of the solution set.
Thus, the solution to the given inequality is $[1, 3]$.
