Question

What is the intersection of the solution sets of the inequalities (2x-1)/(x+3)>_4 and (x+1)/(x+3)<_5

(>_is greater than or equal to)
the answer choices all have different types of parentheses...

Answer 1

Hmm...After reading this question again I am unsure if you have written it correctly. Please check it again :)

Answer 2

Solve both inequalities.
[(2x-1)-4(x+3)]/(x+3)>=0 and [(x+1)-5(x+3)]/(x+3)<=0

after getting both solutions sets to both problems
find the intersection of those sets

Answer 3

\[\frac{-(2x+13)}{x+3} \ge 0 \text{ and } -2 \cdot \frac{2x+7}{x+3} \le 0\]

Answer 4

by the way x cannot be -3

Answer 5

It's correct...but I got that there is no intersection.
the answer choices are:
(-6.5, -3)
[-6.5,-3.5]
(-6.5,-3.5)
[-6.5,-3.5)
[-6.5,-3)

Answer 6

so doing the first inequality we need to proceed as follows:
\[\frac{-(2x+13)}{x+3} \ge 0\]
so undefined at x=-3 and zero at x=-6.5

---|---|---
-6.5 -3

now we choose a number in each of the intervals above

Answer 7

if you want we can call \[f(x)=\frac{-(2x+13)}{x+3}\]
so we can choose numbers to test like x=-8,-4,0 (you didn't have to choose these numbers-just find numbers in those three intervals)

Answer 8

\[f(-8)=- ; f(-4)=+ ; f(0)=-\]

Answer 9

we wanted to find when f>=0

Answer 10

remember f(-4) was the only one to give us positive output

Answer 11

so the interval [-6.5,-3) is the solution to first inequality
now what did you get for the second inequality?

Answer 12

or how about you try the second inequality?

Answer 13

I got -6.5 and -3 as the interval for the two...but how do you read the parentheses?

Answer 14

[ ]
these brackets mean include the endpoint
( )
these paraethesis mean exclude the endpoint

so remember we didn't want to include -3 but we do want to include -6.5

so the answer is [-6.5,-3)

Answer 15

remember we wanted to find when f>=0

f=0 when x=-6.5

f is undefined at x=-3

Answer 16

ah...
thank you so much! :D

Answer 17

ok but we still need to solve the second inequality

Answer 18

...
doesn't it have the same solution?
i am confuzzled. :\

Answer 19

\[\text{ Let } g(x)=-2 \cdot \frac{2x+7}{x+3}\]
we wanted to find g(x)<=0

so we see that g is undefined at x=-3

and we see that g is zero at x=-3.5

Answer 20

no it doesn't have the same solution set

Answer 21

It has a different zero is one reason we have different solution sets.

Answer 22

it also a totally different function.

Answer 23

So anyways we still have 3 intervals to test since we have 1 undefined place in the function and 1 zero in the function
---|----|----
-3.5 -3

so I would choose numbers and you don't have to choose the same numbers as me just numbers in the intervals above
like x=-5,-3.2,0

Answer 24

... i graphed the two on a cartesian plane... the solution I got was [-6.5.-3)
??
very confuzzled.

Answer 25

so remember
\[\text{ Let } g(x)=-2 \cdot \frac{2x+7}{x+3} \]
\[g(-5)=- ;g(-3.2)=+ ; g(0)=-\]
so we are looking for when g<=0 and we see that we have - solutions for the following intervals (which makes the following intervals the solution set to the inequality)
\[(-\infty,-3.5] \cup (-3,\infty)\]

Answer 26

So solution from function f>=0 was
\[[-6.5,-3)\]
And solution from g<=0 was
\[(-\infty,-3.5] \cup (-3,\infty) \]


I think I will draw a rough graph so you can clearly see the intersection

-----|------|-----|------
-6.5 -3.5 -3
*~~~~~~~~~() <----f>=0
~~~~~~~~~~* ()~~~~~ <----g<=0

so intersection is
\[[-6.5,-3.5]\]

Answer 27

thank you! :D

Answer 28

hey one more thing...

Answer 29

so in both of those inequalities there is equals part right?

Answer 30

it's blank than or equal to for both signs.
sorry for not answering earlier.

Answer 31

what?

Answer 32

does _ this mean =?

Answer 33

I'm going to assume
<_ meant <=
and
>_ means >=