Question

A clothing store has a going-out-of business sale. They are selling pants for $8.99 and shirts for $3.99. You can spend as much as $60 and want to buy at least two pairs of pants. Please write the system of inequalities equations and state at least two possible answers.

Answer 1

We can answer this question as follows:

We have that the clothing store is on sale and is selling pants, p, for $8.99, and shirts, s, for $3.99, and that we can spend no more than $60 (as much as $60).

We need to buy at least two pairs of pants.

Then, the system of inequalities is:

`8.99p+3.99s\leq60`

And at least two pairs of pants

`p\ge2`

We can solve this system of inequalities graphically as follows:

And we can see that the darkest region is the region of the possible answers to our question. If we see the region, we can have the following answers:

If we buy 5 pants, and 3 shirts, we have (x for pants and y is for shirts):

`f(x,y)=8.99x+3.99y\Rightarrow f(5,3)=8.99(5)+3.99(3)`

Then

`f(5,3)=44.95+11.97=56.92`

[Notice that the solutions must be for positive values for shirts (the possible negative solutions for y are nonsense in this situation: a negative value for a shirt?).]

Therefore, we can say that the latter is a possible solution since we spend less than $60 and we buy 5 pairs of pants and 3 shirts.

The other answer could be 3 pairs of pants and 8 shirts:

`f(3,8)=8.99\cdot3+3.99\cdot8=58.89`

And also other answers are possible as this one:

`f(4,5)=8.99\cdot(4)+3.99\cdot(5)=55.91`

In summary, we have that after solving the system of inequalities, we have that we can buy:

• Five, ,pants, and ,three, ,shirts, (,5 pants, 3 shirts,)

,

• Three pants, and ,eight shirts, (,3 pants, 8 shirts,)

And also

• Four pants, and ,five shirts, (,4 pants, 5 shirts,), and so on.

[The key in this exercise is to graph the system of inequalities and then find, in the shaded area in which both inequalities form a dark region, the possible values for the solution of the system.]