Answer
(a) $x=2$
(b) $[2, +\infty)$
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 $y=-2x+3$ (red graph) and $y=3x-7$ (blue graph)
(refer to the image below for the graph)
(2) Identify the point/s where the graphs intersect
(a) The point/s where the two graphs intersect represents the instance where the $-2x+3$ is equal to $3x-7$. The x-coordinate of this point is the solution to the given equation.
Notice that the graphs intersect at the point $(2, -1)$.
The x-coordinate of this point is $2$.
Thus, the solution to the given equation is $x=2$.
(b) The region where the graph of the function $y=-2x+3$ is lower than or equal to the graph of the function $y=3x-7$ is the region where $-2x+3 \le 3x-7$.
Notice that the red graph ($y=-2x+3$) is lower than or equal to the blue graph ($y=3x-7$) in the region $(2, +\infty)$. Since the inequality involves $\le$, then the endpoint $2$ is part of the solution set.
Thus, the solution to the given inequality is $[2, +\infty)$.
