Question

Three consecutive integers are such that the first plus one-half the second plus seven less than twice the third is 2101. What are the integers?

Answer 1

How would you represent the first of the 3 integers? Then, how do you represent consecutive integers in terms of that first integer?

Answer 2

would this help x+(x-1)+(x-2)=2101?

Answer 3

cos im not so sure on it

Answer 4

You are close. You have correctly identified the 3 consecutive integers:

x, x+1, and x+2

That first integer will be "x". The second one will be "x+1".

Ok, with that identification what the second consecutive integer is, "x + 1", how do you represent half of that?

Answer 5

sorry im a little confused

Answer 6

Hint: you can take the expression for a number and put that over "2". How do you write that fraction?

Answer 7

"expression" / 2

Now, fill in what the expression for the second integer is.

Answer 8

Did we say it was "x + 1" ?

Answer 9

sorry im trying to figure it out

Answer 10

yes

Answer 11

so would we add add all the X's?

Answer 12

The first integer is "x", the second is "x + 1". So, one more than that, for the 3rd in the series, is "x + 2".

So, we're still on the second integer:

(x + 1) / 2

is the expression for the second of the consecutive integers, but now divided by 2.

Do you want to take a stab at the 3rd expression?

Answer 13

so we can start with the 3rd expression first?

Answer 14

(x) + (x + 1)/2 2(x + 2) - 7 = 2101

Answer 15

is it \(x+ \frac{1}{2}(x+1)+(7-2(x+2))\)

Answer 16

are both methods the same?

Answer 17

oh nvm i think @tcarroll010 is right in where the -7 goes

Answer 18

"seven less than twice the third" is

2(x + 2) - 7

So, now just take that equation I wrote out and solve for "x" :

(x) + (x + 1)/2 2(x + 2) - 7 = 2101

Answer 19

oky one sec sorry bout tht

Answer 20

Are you able to simplify that equation, or would you like a little more help on that? @bbb911

Answer 21

im almost getting my answer just need a few seconds

Answer 22

np, just take your time, that's perfectly fine.

Answer 23

thnks :)

Answer 24

okay i think im wrong i got a decimal number

Answer 25

That's ok, you are trying. I'll write out how I got it and you can either go with that method, or just use it to check your own work.

Answer 26

i got 525.75

Answer 27

(x) + (x + 1)/2 2(x + 2) - 7 = 2101 -> multiply both sides by 2

2x + (x + 1) 4(x + 2) - 14 = 2(2101) -> combine terms

(2x + x + 4x) + (1 + 8 - 14) = 4202 -> simplify

7x - 5 = 4202 -> isolate the "x" term

7x = 4207 -> divide

x = 4207 / 7 -> last simplification

x = 601 -> identification of that first integer

Answer 28

So, since your first integer is 601 and since each of the next is consecutive:

601, 602, and 603

Answer 29

i see what i did wrong i didn't know you had to multiply the 2 to the 2101

Answer 30

Checking your work:

601 + 602/2 + 2(603) - 7 = 2101

Answer 31

in 602 would you have to divide it by 2 right?

Answer 32

You would divide 602 by 2 when calculating the right side, 2101.

But in the concept of consecutive integers, you are eventually looking for:

x, x+1, and x+2

Answer 33

oh okay i see thank you so much :) ill put this on note . YOU sir are the best :) Thanks again :)

Answer 34

And in my above post "checking your work" you will see that I did divide 602 by 2.

Uw! It was very nice working with you! I wish you well in your studies! @bbb911

Answer 35

Thanks Tcarroll :) great help

Answer 36

:-)

Answer 37

I don't know why one of my "plus signs" didn't print, so I'll write that out again:

(x) + (x + 1)/2 + 2(x + 2) - 7 = 2101 -> multiply both sides by 2

2x + (x + 1) + 4(x + 2) - 14 = 2(2101) -> combine terms

(2x + x + 4x) + (1 + 8 - 14) = 4202 -> simplify

7x - 5 = 4202 -> isolate the "x" term

7x = 4207 -> divide

x = 4207 / 7 -> last simplification

x = 601 -> identification of that first integer

Answer 38

oh ok Thanks :)