(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 $[2, +\infty)$.