Q41. - Algebra 2

Question

41. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.

How do inequalities apply to fantasy football?

Include the following in your answer:

• An inequality, and an explanation on how you obtained it, to represent a good game for Randy Moss in Dana’s fantasy football league,
• A graph of your inequality (remember that the number of touchdowns cannot be negative, but receiving yardage can be), and
• Which of the games in statistics in the table qualify as good games.

Step-by-Step Solution

Answer

The inequality which represents a good game for Randy Moss in Dana’s fantasy football league is $5x+100y\ge 1000$, where $x$ is the number of receiving yards that Moss gets in a game and $y$ is the number of touchdowns that Moss scores in a game.

The graph of this inequality is given as:

Out of the three games whose statistics are given in the table, only Game 1 is a good game.

Step 1 – State the concept

An inequality is an algebraic expression with one or more variables, containing one of the symbols; $<,>,\le ,\ge$

Step 2 &ndash; List the given data

A receiving yard is worth 5 points, a touchdown is worth 100 points and a game with 1000 points or more is considered to be a good game. 

The statistics of 3 games is given as:

Step 3 &ndash; Construct an inequality for a good game

Let $x$ be the number of receiving yards that Moss gets in a game and $y$ be the number of touchdowns that Moss scores in a game.

Since each receiving yard is worth 5 points and each touchdown is worth 100 points, the total points earned in a game is evaluated as $5x+100y$.

It is given that a game in which 1000 points or more are earned is a good game. Then, by the problem, a game is a good game if $5x+100y\ge 1000$.

This is the required inequality.

Step 4 &ndash; Graph the inequality

It is obvious that the number of touchdowns cannot be negative. So, $y\ge 0$. However, receiving yardage may be negative or positive. So, there is no restriction on $x$.

Graph the obtained inequality $5x+100y\ge 1000$ with the restriction $y\ge 0$ as follows:

Step 5 &ndash; Selecting good games from the table

It is clear, from the table that for Game 1, $x=168$ and $y=3$.

Put $x=168$ and $y=3$ in the obtained inequality $5x+100y\ge 1000$ to get,

$\begin{array}{l}5\left(168\right)+100\left(3\right)\ge 1000\\ 840+300\ge 1000\\ 1140\ge 1000\end{array}$

This implies that the given inequality is true for $x=168$ and $y=3$. Thus, Game 1 is a good game.

Similarly, from the table, for Game 2, $x=144$ and $y=2$.

Put $x=144$ and $y=2$ in the obtained inequality $5x+100y\ge 1000$ to get,

$\begin{array}{l}5\left(144\right)+100\left(2\right)\ge 1000\\ 720+200\ge 1000\\ 920\ge 1000\end{array}$

This is a contradiction and thus implies that the given inequality is false for $x=144$ and $y=2$. So, Game 2 is not a good game.

Finally, from the table, for Game 3, $x=136$ and $y=1$.

Put $x=136$ and $y=1$ in the obtained inequality $5x+100y\ge 1000$ to get,

$\begin{array}{l}5\left(136\right)+100\left(1\right)\ge 1000\\ 680+100\ge 1000\\ 780\ge 1000\end{array}$

This is a contradiction and thus implies that the given inequality is false for $x=136$ and $y=1$. So, Game 3 is also not a good game.

Thus, only Game 1 is a good game.