The difference between two numbers is 15 if eight is added to twice the greater number the result is less than three times the lesser number find the two numbers

Answer 1

It is given:

x-y = 15

2x+8 < 3y <-- adding 8 to twice the larger is less

than 3 times the smaller

the first equation says: x-y-15=0

solving for y : y = x-15 <--- please label this equation ALPHA

solving for x : x = y+15 <--- please label this equaiton BETA

plugs ALPHA into inequality

2x+8 < 3(x-15)

2x+8 < 3x-45

8 < x-45

53 < x

so x>53 <-- please label this inequality GAMMA

solves the original inequality for y:

2x+8 < 3y

2x < 3y -8

x < (3y-8)/2

plugs in eqaution BETA:

y+15 < (3y-8)/2

2y +30 < 3y-8

30 < y-8

38<y

So y>38 <--- please label this inequality DELTA

THerefore:

x>53, y>38, and x-y=15 <-- combines inequalities GAMMA and DELTA

check:

let x=55 and y=40 so that x-y=15

then 2(55)+8 = 118 while 3y = 120

and the inequality does hold