Question

The sum of two positive integers, x and y, is not more than 40. The difference of the two integers is at least 20. Chancee chooses x as the larger number and uses the inequalities y ≤ 40 – x and y ≤ x – 20 to determine the possible solutions. She determines that x must be between 0 and 10 and y must be between 20 and 40. Determine if Chancee found the correct solution. If not, state the correct solution.

Answer 1

No, Chancee mixed up the variables, the correct solution is that x must be between 20 and 40 and y must be between 0 and 10.

Explanation:

Here, x and y are positive integers,

Such that, their sum is not more than 40,

i.e. x + y ≤ 40 where x, y ≥ 0,

⇒ y ≤ 40 - x ...............(1),

Their difference is at least 20,

i.e. x - y ≥ 20

-y ≥ 20 - x

⇒ y ≤ x - 20 ............(2),

Graphing y ≤ 40 - x :

Related equation of inequality (1),

y = 40 - x

if x = 0, y = 40,

if y = 0, 0 = 40 - x ⇒ x = 40,

'≤' represents a solid line,

Join the points (0, 40) and (40, 0) by solid line.

Graphing y ≤ x - 20:

Related equation of inequality (1),

y = x - 20,

if x = 0, y = -20,

if y = 0, 0 = x - 20 ⇒ x = 20,

'≤' represents a solid line,

Join the points (0, -20) and (20, 0) by solid line.

By graphing we obtained a feasible region that is the solution of the system of inequalities (1) and (2) ( shown below ),

In which,

20 ≤ x ≤ 40 and 0 ≤ y ≤ 30

Hence, Chancee did not find the correct solution.