Question

Use the drawing tools to form the correct answer on the graph. Graph the System of Inequalities and then draw a point on it.

Answer 1

Given the System of Inequalities:

`\begin{cases}y>1.5^x+4{} \\ y<{(2)/(3)x+6}\end{cases}`

• You can identify that the boundary curve of the first inequality is:

`y=1.5^x+4`

Notice that it is an Exponential Function with this base:

`b=1.5`

Since it is greater than 1, it shows an Exponential Growth.

Find three points on the curve in order to graph it by substituting these values of "x" into the equation and evaluating:

`\begin{gathered} x=-10 \\ x=0 \\ x=6 \end{gathered}`

You get:

`y=1.5^((-10))+4\approx4.017`
`y=1.5^((0))+4=1+4=5`
`y=1.5^((6))+4\approx15.391`

Since the symbol of the inequality is:

`>`

The shaded region is above the curve and the curve is dotted.

Notice that the boundary line of the second inequality is:

`y=(2)/(3)x+6`

It is written in Slope-Intercept Form:

`y=mx+b`

Where "m" is the slope and "b" is the y-intercept.

In this case:

`b=6`

By definition, the value of "y" is zero when the line intersects the x-axis. Then, in order to find the x-intercept, substitute this value into the equation and solve for "x":

`y=0`

You get:

`0=(2)/(3)x+6`
`\begin{gathered} (-6)((3)/(2))=x \\ \\ x=-(18)/(2) \\ \\ x=-9 \end{gathered}`

Since the inequality symbol of the second inequality is:

`<`

The shaded region is above the line and this is dotted.

Knowing the above, you get this graph:

• The intersection region is the solution to the System of Inequalities. Therefore, you can conclude any point inside that region is a solution. For example:

`(0,5.5)`

Then, you can plot it on the Coordinate Plane.

Hence, the answer is: