Question

(absolute value of x minus 5) is less than or equal to 2
How many integers satisfy the inequality above?


(A) None
(B) One
(C) Two
(D) Three
(E) Five

Answer 1

sorry I x-5 I less than or equal to 2

Answer 2

\(|x-5| \le 2 \)

for |a| <=b, we have a<=b or a>=-b

so what about \(|x-5| \le 2 \) ??

Answer 3

how many integers satisfy the equation?

Answer 4

lets find out!
\(|x-5|\le 2 \implies (x-5)\le2 \quad or \quad (x-5)\ge -2 \)
can you soilve them individually ?

Answer 5

**solve

Answer 6

i think so...
not so sure...

Answer 7

x-5 <= 2
add 5 to both sides, what u get ?

Answer 8

you get x = 7

Answer 9

not x=7

\(x-5 \le 2\)
adding 5 to both sides,
\(x-5+5 \le 2+5 \implies x \le7 \)
you don't loose <= sign

Answer 10

oh ya i forgot. *facepalm*

Answer 11

similarly what u get after adding 5 to both sides of
x-5 >= -2 ??

Answer 12

you would get +3

Answer 13

x>=3, right ?
so, x>=3 and x<=7 gives you,
x=3,4,5,6,7 ----> 5 values

Answer 14

A great way of solving this problem would be to visually interpret it. When we say:
\[\bf |x-5| \le 2\]We're really finding values of 'x' which are within a distance of 2 unit from 5. The only values within 2 units from 5 are 7 and 3. So really, we are looking at all integers within 2 units to the left of 5 on the number line and within 2 units on the right of the number line.Which integers are in that range once you look at the number line? @betterwiththelightsoff99

Answer 15

oh i understand now.... thank you hartnn

Answer 16

@genius12 wouldnt that be a little more time-consuming?

Answer 17

Nope. Once you understand how that works, you can work it out in your head without having to write or type a word. Think about it. We are looking at integers within 2 units of 5. How many are there? 3, 4, 6, 7. 2 units to the left and 2 units to the right. Easy. You can always doing it algebraically like @hartnn but understanding what the |x - 5| really means can be really helpful as you can solve the inequality in your head without much work.

@betterwiththelightsoff99

Answer 18

oh! ok thank you @genius12

Answer 19

But remember, because its less than or equal to 2, we are finding values that are 2 units or less away from 5. Because its 2 units or less, we include the 3 and 7. If it was just less than 2 units, then the only integers satisfying the inequality would 4 and 6.

Answer 20

alright got it thanks for the help @genius12

Answer 21

np