Exercise 53 Page 302

To solve the given inequality using a graphing calculator, we need to treat each side of the inequality symbol as its own individual equation. Then we can look for the values of $x$ where the value of $\frac{4}{3}x+1$ is $\text{greater}$ $\text{than}$ $\text{or}$ $\text{equal}$ $\text{to}$ the value of $\frac{2}{3}x+2.$ In other words, we want to find the values of $x$ where the graph of $y=\frac{4}{3}x+1$ lies $\text{above}$ the graph of $y=\frac{2}{3}x+2.$

Graphing the Functions

Press $\boxed{\text{Y=}}$ on the calculator to access the screen to input our functions.

To draw the graph, press the $\boxed{\text{GRAPH}}$ button.

Finding the Solution Set

We see that the graphs intersect at a point. The $x\text{-}$coordinate of this point can be found by pressing $\boxed{2\text{ND}}$ and then $\boxed{\text{CALC}}.$

After selecting the intersect option, we choose the first and second curves. Then, the calculator asks for a guess where the intersection point might be. After that, it will calculate the exact point for us.

The lines intersect at $x=1.5,$ or $x=\frac{3}{2}.$ Tracing the lines on the calculator screen using the $\boxed{\text{TRACE}}$ button, we can see that $y=\frac{4}{3}x+1$ is the upper line after the intersection. Since the inequality is non-strict, it is also true at the point of intersection. This means that the given inequality holds true when $x\ge \frac{3}{2}.$ We can now write the solution set for our inequality.