Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that $f\left(x\right)=3x+5$and  $g\left(x\right)=-2x+15$

(a)  Solve $f\left(x\right)=0$                   (b) Solve $f\left(x\right)<0$ 

(c) Solve $f\left(x\right)=g\left(x\right)$                (d) Solve $f\left(x\right)\ge g\left(x\right)$

(e) Graph $y=f\left(x\right)$ and $y=g\left(x\right)$ and label the point that represents the solution to the equation $f\left(x\right)=g\left(x\right)$.

Step-by-Step Solution

Answer

Part  (a) $x=-\frac{5}{3}$

Part  (b) $x<-\frac{5}{3}$

Part  (c) $x=2$

Part  (d) $x\ge 2$

Part  (e) Point of intersection is $\left(2,11\right)$

Part (a). Step 1. Given Information .

There are two functions $f\left(x\right)=3x+5$ and $g\left(x\right)=-2x+15$.

Part (a). Step 2. Explanation.

Solve $f\left(x\right)=0$

we have $f\left(x\right)=3x+5$

$$\begin{gathered} 3x+5=0 \\ 3x=-5 \\ x=-\frac{5}{3} \end{gathered}$$

Part (a). Step 3. Conclusion

At $x=-\frac{5}{3}$, $f\left(x\right)=0$

Part (b). Step 1. Explanation

To solve $f\left(x\right)<0$

We have

 $$\begin{gathered} 3x+5<0 \\ 3x<-5 \\ x<-\frac{5}{3} \end{gathered}$$

Part (b). Step 2. Conclusion.

For $x<-\frac{5}{3}$, $f\left(x\right)<0$

Part (c). Step 1. Explanation

To solve $f\left(x\right)=g\left(x\right)$

We have $f\left(x\right)=3x+5$ and $g\left(x\right)=-2x+15$

$$\begin{gathered} 3x+5=-2x+15 \\ 3x+2x=15-5 \\ 5x=10 \\ x=2 \end{gathered}$$

Part (c). Step 2. Conclusion.

For $x=2$,  $f\left(x\right)=g\left(x\right)$

Part (d). Step 1. Explanation

To solve $f\left(x\right)\ge g\left(x\right)$

$$\begin{gathered} 3x+5\ge -2x+15 \\ 5x\ge 10 \\ x\ge 2 \end{gathered}$$

Part (d). Step 2. Conclusion.

As $x\ge 2$, $f\left(x\right)\ge g\left(x\right)$

Part (e). Step 1. Explanation.

Make the table for $y=f\left(x\right)$

x	$y=3x+5$
-2	-1
0	5
2	11

Make the table for $y=g\left(x\right)$

x	$y=-2x+15$
-2	19
0	15
2	11

the point that represents the solution to the equation $f\left(x\right)=g\left(x\right)$ is intersection of graphs $f\left(x\right), g\left(x\right)$.

Part (e). Step 2. Conclusion.

The graph is as below and the point that represents the solution to the equation $f\left(x\right)=g\left(x\right)$ is $\left(2,11\right)$