Question

Luis can spend up to $240 buying men’s and women’s shirts for his club. Men’s shirts cost $8 each and women’s shirts cost $12 each. He needs to buy at least 3 men’s shirts and at least 8 women’s shirts. Also, the women’s shirts must be at least twice the number of men’s shirts.

The graph shows the feasible region, where x represents the number of men’s shirts and y represents the number of women’s shirts.

Which ordered pairs meet all the constraints and make sense in context of the situation?

Select each correct answer.



(4, 9)

(7, 13)

(2, 15)

(5,12)

(6, 15)

Answer 1

Let

x------> the number of men’s shirts

y----->  the number of women’s shirts

we know that

$8x+12y \leq 240$ ----> constraint A    

$x\geq 3$ ----> constraint B

$y\geq 8$ ----> constraint C  

$y\geq 2x$ ----> constraint D  

using a graphing tool

see the attached figure N$1$

The solution is the shaded area

Remember that

if an ordered pair meets all restrictions and makes sense in the context of the situation, then the ordered pair is a system solution

we're going to proceed to verify each pair ordered

case A) $(4,9)$

Substitute the value of x and y in each  constraint

$x=4\\y=9$

Verify constraint A

$8*4+12*9 \leq 240$

$140 \leq 240$ -------> is true

Verify constraint B

$4\geq 3$ -------> is true

Verify constraint C

$9\geq 8$ ------> is true

Verify constraint D

$9\geq 2*4$

$9\geq 8$ -----> is true

therefore

the ordered pair $(4,9)$  is a system solution

case B) $(7,13)$

Substitute the value of x and y in each  constraint

$x=7\\y=13$

Verify constraint A      

$8*7+12*13 \leq 240$

$212 \leq 240$ -------> is true

Verify constraint B

$7\geq 3$ -------> is true

Verify constraint C

$13\geq 8$ ------> is true

Verify constraint D

$13\geq 2*7$

$13\geq 14$ -----> is not true

therefore

the ordered pair $(7,13)$ is not a system solution

case C) $(2,15)$

Substitute the value of x and y in each  constraint

$x=2\\y=15$

Verify constraint A

$8*2+12*15 \leq 240$

$196 \leq 240$ -------> is true

Verify constraint B

$2\geq 3$ -------> is not true

Verify constraint C

$15\geq 8$ ------> is true

Verify constraint D

$15\geq 2*2$

$15\geq 4$ -----> is true

therefore

the ordered pair $(2,15)$ is not a system solution  

case D) $(5,12)$

Substitute the value of x and y in each  constraint

$x=5\\y=12$

Verify constraint A

$8*5+12*12 \leq 240$

$184 \leq 240$ -------> is true

Verify constraint B

$5\geq 3$ -------> is true

Verify constraint C

$12\geq 8$ ------> is true

Verify constraint D

$12\geq 2*5$

$12\geq 10$ -----> is true

therefore

the ordered pair $(5,12)$ is  a system solution  

case E) $(6,15)$

Substitute the value of x and y in each  constraint

$x=6\\y=15$

Verify constraint A

$8*6+12*15 \leq 240$

$228 \leq 240$ -------> is true

Verify constraint B

$6\geq 3$ -------> is true

Verify constraint C

$15\geq 8$ ------> is true

Verify constraint D

$15\geq 2*6$

$15\geq 12$ -----> is true

therefore

the ordered pair $(6,15)$ is  a system solution  

therefore

the answer is

$(4,9)$

$(5,12)$

$(6,15)$

see the attached figure N$2$ to better understand the solution