Question

Can someone explain inequalities to me..
y² - 8y - 9 ≤ 0
(y + 1)(y - 9) ≤ 0
but why is the answer -1 ≤ y ≤ 9
and not y ≤ 9, y ≤ -1 ?

Answer 1

(y + 1)(y - 9) ≤ 0

First, we notice that product of two number is negative ONLY when both the numbers are of opposite signs.

Answer 2

for example :

(2)(-3) = -6

the product is negative cuz `2` and `-3` are of opposite signs.

Answer 3

uhm.. i think that doesnt answer my question but thanks anyway! :D

Answer 4

thats just a starter :)

(y + 1)(y - 9) ≤ 0

can be broken into two cases :

1) y+1 <= 0 and y-9 >= 0
2) y+1 >=0 and y-9 <= 0

Answer 5

solve each case and see what you get

Answer 6

uhm.. i got
1. y <= -1 and y >= 9
2. y >= -1 and y <= 9

Answer 7

Excellent ! take a look at 1st case, the solution we have gotten is just nonsense (why ?) so discard it

only the 2nd case is valid.

Answer 8

i noticed that in each case there are ≤ and ≥.
can i ask how does the ≥ come about? Since the original equation only gave ≤, where
(y + 1)(y - 9) ≤ 0

Answer 9

thats a good question :)

Answer 10

actually there is another method which won't make u ask these questions, but i don't like that method... lets look at your question :)

Answer 11

we have below inequality to solve :
(y + 1)(y - 9) ≤ 0

Answer 12

It is a product of two individual factors: (y+1) and (y-9)

what do we know about the `signs` of these individual factors ?

Answer 13

For the product to be negative, these factors must have opposite `signs` right ?

(if one factor is <=0, then other factor has to be >=0)

Answer 14

yes

Answer 15

(y+1) meaning that the answer will be -1 and 9 for (y-9)

Answer 16

yes so did u get how we got these two cases ?

(y + 1)(y - 9) ≤ 0

1) y+1 <= 0 and y-9 >= 0
2) y+1 >=0 and y-9 <= 0

Answer 17

if it helps, we're using this fact :

\(\large - \times + = -\)

\(\large + \times - = -\)

Answer 18

waaat :O

Answer 19

nvm im too dumb.. cuz all i see is to shift from left side of the equation to right side of the equation ._.

Answer 20

thats why i cant see how the 2 cases and the ≥ come about @@

Answer 21

can you try telling me the other method maybe i wont see how does the ≥ come about ? :x

Answer 22

Ahh I can see how that could be confusing...

Answer 23

but i don't really like the other method as it involves more new terms and more explanation - IMHO this method is simple but it has to makes sense first -.-

Answer 24

oh okie >.<

Answer 25

@HatcrewS
Let me try to explain to you what you don't understand.
@ganeshie8 did a great job with this problem, and you followed it well.

Your question above is:
"i noticed that in each case there are ≤ and ≥. can i ask how does the ≥ come about? Since the original equation only gave ≤, where (y + 1)(y - 9) ≤ 0"

Answer 26

When you have the product (y + 1)(y - 9) ≤ 0, think of the "≤ 0" part as meaning it's 0 or a negative number. A number less than zero is negative.

Answer 27

The question is what two numbers can you multiply together and get a negative result?
The answer is one must be positive and one must be negative.

When you get an answer in the form y >= -1 and y <= 9, instead of thinking of the inequality sign having flipped, think of it as meaning the answer is any number between -1 and 9. It's just that to write that down we need to flip the inequality sign.

Answer 28

omg #mind blown. thank you so much @mathstudent55! i have a much better understanding now :)

Answer 29

You're welcome.