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An object must travel at a speed of at least $11.2$ kilometers per second to escape Earth's gravitational field. At Gravitasi Z, engineers built a rocket and were tasked with the mission of exploring celestial objects faraway from Earth. However, there is a big problem: The rocket was made to travel at a speed of only $8$ kilometers per second!

Gravitasi Z's engineers plan to improve the rocket in order to accomplish its mission. Solve the following predicaments to help them succeed.

Solution

a It is known that the rocket can reach a speed of $8$ kilometers per second. Let $s$ represent the additional speed that the rocket will gain after improvements. Then, the sum of $s$ and $8$ will be the speed of the rocket after the improvements.

$$\begin{array}{c}\underline{\text{Final Speed of the Rocket}}\\ s+8\end{array}$$

The company needs the final speed to be at least $11.2$ kilometers per second. The phrase at least means greater than or equal to. Therefore, the inequality is non-strict and the symbol must be $\ge .$

$$\begin{array}{c}\underline{\text{Inequality}}\\ s+8\ge 11.2\end{array}$$

b It is asked to find the minimum speed that should be added to the rocket, the minimum value of $s.$ To do so, the inequality written in Part A should be solved. Using the Subtraction Property of Inequality, $s$ can be isolated.

$s+8\ge 11.2$

SubIneq

$\text{LHS}-8\ge \text{RHS}-8$

$s\ge 11.2-8$

SubTerm

Subtract term

$s\ge 3.2$

The boundary point is $3.2,$ representing the minimum speed that needs to be added to the rocket. This inequality also means that improvements that cause the rocket to speed up greater than or equal to $3.2$ kilometers per second will enable the rocket to start its mission.

c The graph of the inequality written in Part A should be determined.

$$\begin{array}{c}s+8\ge 11.2\end{array}$$

To graph this inequality on a number line, the first step is to determine its type. It is a non-strict inequality because it involves the symbol $\ge .$ Its solution set and boundary point were found in Part B.

$s+8\ge 11.2$
Solution Set	Boundary Point
$s\ge 3.2$	$3.2$

Since the inequality is non-strict, a closed circle will be drawn on the number line on its boundary point $3.2.$

Now, the rest of the solution set needs to be shaded. Because the speed added needs to be greater than or equal to $3.2,$ the region on the right of the boundary point will be shaded.

This corresponds to Graph II.

Ignacio is an employee at Gravitasi Z. He feels unsure about the success of their space exploration, so he decides to diversify his money making opportunities by investing in cryptocurrency. He has found a crypto exchange company that does all of the actual investing work for him.

The exchange company charges a subscription of $\$299$ plus a commission of $\$5$ for every cryptocurrency bought. Also, for an additional fee of $\$35,$ each cryptocurrency can be sold again later. Ignacio wants to determine the number of cryptocurrencies that will make this investment profitable. Fractions of cryptocurrencies can be bought.

Solution

a Let $c$ be the number of cryptocurrencies. Because the company charges a subscription of $\$299$ plus $\$5$ per cryptocurrency bought, an expression for the total investment can be written by adding the product of $5$ and $c$ to the subscription cost.

$$\begin{array}{c}\underline{\text{Total Investment}}\\ 299+5c\end{array}$$

Additionally, it is given that each cryptocurrency can be later sold for $\$35$ each. Therefore, multiplying $c$ and $\$35$ will give the total amount recovered after selling all the cryptocurrencies.

$$\begin{array}{c}\underline{\text{Total Sales}}\\ 35c\end{array}$$

Finally, to make a profit, the total sales need to be greater than the total investment. That means the inequality is strict and its symbol is $>.$ The inequality representing the number of cryptocurrencies needed to buy to make a profit can be written using this information.

$$\begin{array}{c}35c>299+5c\end{array}$$

b To help Ignacio find the minimum number required to expect make some profit, the inequality needs to be solved. To do so, the Properties of Inequalities can be used. In this case, the Subtraction Property of Inequality should be used first.

$35c>299+5c$

SubIneq

$\text{LHS}-5c>\text{RHS}-5c$

$35c-5c>299$

SubTerm

Subtract term

$30c>299$

DivIneq

$\text{LHS}/30>\text{RHS}/30$

$c>\frac{299}{30}$

UseCalc

Use a calculator

$c>9.9\overline{6}$

RoundDec

Round to $2$ decimal place(s)

$c>9.97$

The boundary point is about $9.97.$ This means that as long as Ignacio bought more than $9.97$ cryptocurrencies, he can expect to make a profit. Therefore, the minimum he needs to buy is $10.$

c Consider the inequality found in Part A.

$$\begin{array}{c}35c>299+5c\end{array}$$

To graph the solution set of this inequality, the first step is determining its type. Because the inequality symbol is $>,$ it is a strict inequality. The solution set and the boundary point of this inequality were found in the previous part.

$35c>299+5c$
Solution Set	Boundary Point
$c>9.97$	$9.97$

Since the inequality is strict, an open circle will be drawn on the number line on its boundary point $9.97$.

Now, the rest of the solution set needs to be shaded. Because the number of cryptocurrencies bought needs to be greater than $9.97,$ the region on the right side of the boundary point will be shaded.

This corresponds Graph I.

Gravitasi Z wants to sale coffee mugs to the families of their astronauts — at a low price, of course. A company that produces mugs wants to charge Gravitasi Z a fixed amount of $\$185$ for production plus a tax of $\$5$ per mug. Gravitasi Z plans to sell each mug for $\$18.$

The space exploration company, busy with making rockets, decides to hire a financial advisor to make these calculations. They hope to make at least a small profit. Find the answers to the following situations to become Gravitasi Z's trusted financial advisor.

Solution

a Let $m$ be the number of mugs that they will order from the production company. The cost of production is $\$185$ plus a tax of $\$5$ per mug. Then, the total expenses can be expressed as the sum of $185$ and the product of $5$ and $m.$

$$\begin{array}{c}\underline{\text{Total Expenses}}\\ 185+5m\end{array}$$

Since each coffee mug can be sold for $\$18,$ assuming that all mugs will be sold, the total sales can be expressed by the product of $18$ and the number of mugs $m.$

$$\begin{array}{c}\underline{\text{Total Sales}}\\ 18m\end{array}$$

In order for Gravitasi Z to profit from this project, the total expenses should be less than the total sales. Therefore, the relationship between these two quantities can be shown by the inequality symbol $<.$

$$\begin{array}{ccc}\text{Total Expenses}& & \text{Total Sales}\\ 185+5m& <& 18m\end{array}$$

b Gravitasi Z wants the minimum number of coffee mugs that guarantee that they will make a profit. To do so, the inequality written in the previous part will be solved using the Properties of Inequalities.

$185+5m<18m$

▼

Solve for $m$

SubIneq

$\text{LHS}-5m<\text{RHS}-5m$

$185<18m-5m$

SubTerm

Subtract term

$185<13m$

DivIneq

$\text{LHS}/13<\text{RHS}/13$

$\frac{185}{13}<m$

UseCalc

Use a calculator

$14.230769\dots <m$

RoundInt

Round to nearest integer

$14<m$

RearrangeIneq

Rearrange inequality

$m>14$

From this inequality, it can be seen that $14$ is the boundary point. Recall that $m$ represents the number of mugs. Therefore, $m$ must be a positive integer. Combining these findings, the solution set of the inequality can be written as follows.

$$\begin{array}{c}\underline{\text{Solution Set}}\\ \{x|x\text{ is an integer greater than }14\}\end{array}$$

The minimum number of mugs that Gravitasi Z should order from the production company is $15.$

c To find the answer, the solution set of the inequality will be graphed. The solution set and the boundary point of the inequality were found in the previous part.

$185+5m<18m$
Solution Set	Boundary Point
$\{x|x\text{ is an integer greater than }14\}$	$14$

Since the inequality is strict, an open circle will be drawn on the number line on its boundary point $14.$

Now the solution set needs to be shaded on the number line. Since integers represent the number of mugs, only the points representing integers should be plotted on the number line instead of shading the region greater than $14.$ The points will be closed as they are the part of the solution set.

This corresponds to Graph IV. Congratulations! Gravitasi Z was impressed by this data, and it looks like someone has a new job as financial advisor.