Answer
(a) $x=\left\{-4.32, -1.12, 1.44\right\}$
(b) $[-4.32, -1.12] \cup [1.44, +\infty)$
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:
$y=x^3+3x^2$ (red graph)
$y=-x^2+3x+7$ (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 $x^3+3x^2$ is equal to the value of $-x^2+3x+7$. The x-coordinate of this point is the solution to the given equation.
Notice that the graphs intersect at the points $(-4.319, -24.615)$, $(-1.123, 2.368)$, and $(1.443, 9.247)$.
The x-coordinates of these points are $-4.319, -1.123$, and $1.443$.
Thus, the approximate solutions to the given equation are $x=\left\{-4.32, -1.12, 1.44\right\}$.
(b)
The region/s where the graph of the function $y=x^3+3x^2$ is higher than or equal to the graph of the function $y=-x^2+3x+7$ is the region where $x^3+3x^2 \ge -x^2+3x+7$.
Notice that the red graph ($y=x^3+3x^2$) is higher than or equal to the blue graph ($y=-x^2+3x+7$) in the regions $(-4.319, -1.123)$ and $(1.443, +\infty)$. Since the inequality involves $\ge$, then the endpoints $-4.319, -1.123,$ and $1.443$ are part of the solution set.
Thus, the solution (rounded off to 2 decimal places) to the given inequality is:
$[-4.32, -1.12] \cup [1.44, +\infty]$
