Question

Find the solution sets for Ix^2-x+9I<3

Answer 1

\[|x^2-x+9|<3\]
\[-3<x^2-x+9<3\]
so you have two inequalities to solve:
\[x^2+x-9<3\] and \[-3<x^2-x+9\] and you have to take the intersection

Answer 2

oh so the 3 changes sign not the other side

Answer 3

first one:
\[x^2-x+9<3\]
\[x^2-x+6<0\]
\[(x-3)(x+2)<0\]

Answer 4

yes if
\[|blah|<p\]
then
\[-p<blah<p\]

Answer 5

oh ok that's where I got stuck I was doing IbahI<3 then -blah>3 or blah<3

Answer 6

the first one is zero if x =-2 or x = 3 and it is negative between them so the solution is
\[(-2,3)\]

Answer 7

we still have more work to do. (-2,3) is just the solution to the first inequality. you have to solve them separately

Answer 8

right the other is (3,4)

Answer 9

I just did it now.

Answer 10

the second inequality is
really?

Answer 11

How do I write that as an answer?

Answer 12

that sucks because that means there is no solution

Answer 13

let me try it hold on

Answer 14

wait what?

Answer 15

sorry 3, -4

Answer 16

oooh. let me try it anyway

Answer 17

so its (-4,-4)u(2,3)?

Answer 18

\[-3<x^2-x+9\]
\[0<x^2-x+12\]
\[0<(x-4)(x+3)\]

Answer 19

aaah i see your confusion

Answer 20

oh woops.

Answer 21

this inequality says that that product is positive

Answer 22

right.

Answer 23

so answer is not (-3,4) but rather
\[(-\infty,-3) \cup (4,\infty)\]

Answer 24

oh ok.

Answer 25

is that the whole answer?

Answer 26

so now we have to take the intersection of those two sets.

Answer 27

which looks empty to me. hold on let me try it again real quick. it is possible there is no solution

Answer 28

ok.

Answer 29

k we started with \[x^2-x+9\] yes?

Answer 30

yes

Answer 31

the minimum value of this thing is where
\[x=-\frac{b}{2a}\] which in this case is \[\frac{1}{2}\]

Answer 32

woah woah woah.

Answer 33

what's that?

Answer 34

plug in x = 1/2 get y = 35/4=8.75

Answer 35

which is bigger than 3!

Answer 36

did i lose you there?

Answer 37

yep.

Answer 38

ok let me ask you this. do you know what the graph of
\[y=x^2-x+9\] look like?

Answer 39

let me look up the answer in the answer key real quick

Answer 40

yes, a parabola

Answer 41

the answer is no solution. yes a parabola that faces upward yes?

Answer 42

because it faces upward, it has a minimum value, where the parabola "sits" that is called the vertex

Answer 43

yes.

Answer 44

the vertex of this parabola is
\[(\frac{1}{2},\frac{35}{4})\]

Answer 45

35/4=8.5

Answer 46

and of course 8.5 > 3

Answer 47

(a) (−3,−2) ∪ (3,4)

Answer 48

that's the answer...

Answer 49

so there is no way that this thing can be less than 3

Answer 50

no chance

Answer 51

here is a picture of it. you will see that it is never less than 3
http://www.wolframalpha.com/input/?i=y%3Dx^2-x%2B9

Answer 52

lol ok

Answer 53

the answer is "no solution"

Answer 54

ok.

Answer 55

we found that in any case. the intersection of the 3 intervals we found is empty

Answer 56

ok.

Answer 57

one was (-2,3) and the other two were \[(-\infty,-3)\]
and
\[(4,\infty)\]

Answer 58

that intersection is empty

Answer 59

ok

Answer 60

Thanks.