Question

yasmani uses a gift card to buy video games. the initial value of the gift card is $250. the function v(x)=250 - 15x represents the amount of money, v, in dollars, that is still left on the gift card after purchasing x video games that cost $15 each. describe the constraints. write the constraints in set - builder notation. graph the function in terms of the context. use the graph to find the solution to v(x)=70. label the point. what does this point mean in the context of the problem?

Explanation:

Step1: Determine non - negativity of x

The number of video games $x$ cannot be negative, so $x\geq0$.

Step2: Determine upper - bound of x

The amount of money left on the gift card $V(x)\geq0$. So, $250 - 15x\geq0$. Solving for $x$:
\[

$$\begin{align*} 250-15x&\geq0\\ - 15x&\geq - 250\\ x&\leq\frac{250}{15}=\frac{50}{3}\approx16.67 \end{align*}$$

\]
Since $x$ represents the number of video games, $x$ is a non - negative integer. So the constraints are $0\leq x\leq16$ and $x\in\mathbb{Z}$.

Step3: Write in set - builder notation

The set - builder notation for the constraints is $\{x\in\mathbb{Z}:0\leq x\leq16\}$.

Step4: Graph the function

The function $V(x)=250 - 15x$ is a linear function. When $x = 0$, $V(0)=250$. When $V(x)=0$, $0=250 - 15x$, so $x=\frac{250}{15}=\frac{50}{3}\approx16.67$. Plot the points $(0,250)$ and $(16,10)$ and draw a line segment between them (since $x$ is an integer between 0 and 16 inclusive).

Step5: Solve $V(x)=70$

Set $V(x)=70$, so $250 - 15x=70$.
\[

$$\begin{align*} 250-15x&=70\\ -15x&=70 - 250\\ -15x&=-180\\ x& = 12 \end{align*}$$

\]
On the graph, find the point where the line $V(x)=250 - 15x$ intersects the horizontal line $V(x)=70$. The point is $(12,70)$.

Step6: Interpret the point

The point $(12,70)$ means that after purchasing 12 video games, there is $70$ dollars left on the gift card.

Answer:

• Constraints description: The number of video games $x$ is a non - negative integer and $x\leq16$.
• Set - builder notation: $\{x\in\mathbb{Z}:0\leq x\leq16\}$
• Graph: A line segment with endpoints $(0,250)$ and $(16,10)$ (where $x$ is an integer between 0 and 16 inclusive).
• Solution to $V(x)=70$: The point is $(12,70)$
• Interpretation: After buying 12 video games, there is $70$ dollars left on the gift card.