Question

Mrs. Gibbs goes to the store to buy AA and 9V batteries. She needs at least 6 AA batteries. In all, she wants to buy at least 20 batteries. The store has 12 AA batteries and 15 9V batteries in stock.
The graph shows the feasible region, where x represents the number of AA batteries and y represents the number of 9V batteries.

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

Select each correct answer.



(7, 13)

(8,11)

(10,16)

(12,12)

Answer 1

The answer is (7,13) and (12,12). 


Take a look at the snip for proof ;)

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Answer 2

Let

x------> represents the number of AA batteries

y------> represents the number of 9V batteries

we know that

The constraints are

$x\geq 6$

$x\leq 12$

so

$6\leq x\leq 12$ --------> inequality A    

$y\leq 15$ --------> inequality B

$(x+y)\geq 20$        

$(x+y)\leq 27$

so

$20\leq (x+y) \leq 27$-----> inequality C

The solution of the system of inequalities is the shaded area in the attached figure

If a ordered pair meet all the constraints and make sense in the context of the situation, then the ordered pair is a solution of the system of inequalities

let's check each of the ordered pairs

case A) point $(7,13)$      

$x=7\ y=13$

Verify inequality A

$6\leq 7\leq 12$--------> is true

Verify inequality B

$13\leq 15$ -------> is true

Verify inequality C

$20\leq (7+13) \leq 27$

$20\leq (20) \leq 27$------> is true

The ordered pair  $(7,13)$ meets all restrictions, therefore the ordered pair is a solution of the system of inequalities

case B) point $(8,11)$      

$x=8\ y=11$

Verify inequality A

$6\leq 8\leq 12$--------> is true

Verify inequality B

$11\leq 15$ -------> is true

Verify inequality C

$20\leq (8+11) \leq 27$

$20\leq (19) \leq 27$------> is not true

The ordered pair  $(8,11)$   does not meet all the restrictions, therefore the ordered pair is not a solution of the system of inequalities.

case C) point $(10,16)$      

$x=10\ y=16$

Verify inequality A

$6\leq 10\leq 12$--------> is true

Verify inequality B

$16\leq 15$ -------> is not true

Verify inequality C

$20\leq (10+16) \leq 27$

$20\leq (26) \leq 27$------> is not true

The ordered pair  $(10,16)$   does not meet all the restrictions, therefore the ordered pair is not a solution of the system of inequalities.

case D) point $(12,12)$      

$x=12\ y=12$

Verify inequality A

$6\leq 12\leq 12$--------> is true

Verify inequality B

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

Verify inequality C

$20\leq (12+12) \leq 27$

$20\leq (24) \leq 27$------> is true

The ordered pair   $(12,12)$  meets all restrictions, therefore the ordered pair is a solution of the system of inequalities

the answer is

$(7,13)$

$(12,12)$