Kiani is shopping at a bookstore with a budget of $60. Hardbacks cost $12 and paperbacks cost $5. Write a two-variable inequality and determine the constraints of the problem. Which of the following points represents a viable solution?(1 point)

Question

Responses

(1,4.5)
left parenthesis 1 comma 4.5 right parenthesis

(6,6)
left parenthesis 6 comma 6 right parenthesis

(2,4)
left parenthesis 2 comma 4 right parenthesis

(−2,6)

Answer 1

Let's define the variables for the problem:

• Let $x$ be the number of hardbacks purchased.
• Let $y$ be the number of paperbacks purchased.

The cost of hardbacks is $12 each, and the cost of paperbacks is $5 each. Kiani has a budget of $60. This can be expressed as the following inequality:

$$12x+5y\le 60$$

Next, we need to consider the constraints of the problem:

1. Non-negativity constraints: Since Kiani cannot purchase a negative number of books, we have:
   • $x\ge 0$
   • $y\ge 0$

Now we have the following system of inequalities:

$$\begin{array}{rlrl}12x+5y& \le 60x& \ge 0y& \ge 0\end{array}$$

Next, we can evaluate the proposed points to see which one satisfies all the inequalities.

1. Point (1, 4.5): $$12(1)+5(4.5)=12+22.5=34.5(\text{True: }34.5\le 60)$$ Non-negativity: $x=1\ge 0,,y=4.5\ge 0\text{(True)}$

2. Point (6, 6): $$12(6)+5(6)=72+30=102(\text{False: }102\le 60)$$ Non-negativity: $x=6\ge 0,,y=6\ge 0\text{(True)}$

3. Point (2, 4): $$12(2)+5(4)=24+20=44(\text{True: }44\le 60)$$ Non-negativity: $x=2\ge 0,,y=4\ge 0\text{(True)}$

4. Point (−2, 6): $$12(-2)+5(6)=-24+30=6(\text{True: }6\le 60)$$ Non-negativity: $x=-2<0\text{(False)}$

After evaluating all the points, the viable solutions are:

• (1, 4.5)
• (2, 4)

The only point listed as an option that represents a viable solution is:

(2, 4).