Question

I need to answer as an interval, help please, will medal

Find all real numbers t such that (2/3 t )- 1 < t + 7 <= -2t + 15.

I think that the answer is 8 > t < = 24/9, but i dont know how to submit it in the right format.

Answer 1

Hint:
Break the compound inequality
\[\Large \frac{2}{3}t - 1 < t+7 \le -2t+15\]
into these two inequalities
\[\Large \frac{2}{3}t - 1 < t+7\]
and
\[\Large t+7 \le -2t+15\]

Answer 2

I just realized it mean less than or equal to, my bad.

Answer 3

How so you answer something in the form of an interval?

Answer 4

do you believe you have soled both inequalities correctly?

Answer 5

solved*

Answer 6

if so what did you get for both inequalities mentioned by jim?

Answer 7

assuming a is less than b:

interval notation for a<x<b is (a,b)
interval notation for a<=x<=b is [a,b]
interval notation for a<=x<b is [a,b)
interval notation for a<x<=b is (a,b]
interval notation for x<a or x>b is (-inf,a) union (b,inf)
interval notation for x<=a or x>b is (-inf,a] union (b,inf)
interval notation for x<=a or x>=b is (-inf,a] union [b,inf)
interval notation for x<a or x>=b is (-inf, a) union [b,inf)

i hope this helps
I can do some examples if you want actual numbers instead of a and b that represent some numbers where a is less than b

Answer 8

im not entirely sure, im quite confused. After I separate the compound inequality what do I
do?

Answer 9

solve both for t

Answer 10

let's play with the first mentioned one:
\[\frac{2}{3}t-1<t+7\]
if you don't like the fraction right now
multiply on through by 3 giving
\[2t-3<3t+21\]

Answer 11

your job is to isolate t

Answer 12

get your t-terms on one side
and your non t-terms on the opposing side by addition or subtraction

Answer 13

so 2/3t<t+8

Answer 14

ok that looks fine so far

Answer 15

and for inequality 2,
t<=-2t+8?

Answer 16

ok but you didn't finishing solving the first for t yet
but yes so far RIGHT(as in correct) on inequality 2 which is still is unfinished also

Answer 17

okay so to continue with 1, is it legal to say t<t+1/3t+8?

Answer 18

2/3 t < t+ 8
you want the t's on one side together
subtract t on both sides

Answer 19

\[\frac{2}{3} t < t+8 \\ \text{ subtract} t \text{ on both sides } \\ \frac{2}{3}t\color{red}{-t}<t\color{red}{-t}+8 \\ \\ \frac{2}{3}t-t<0+8 \\ \frac{2}{3}t-t<8\]

Answer 20

\[\frac{2}{3}t-t<8 \\ \frac{2}{3}t-1 t<8 \\ (\frac{2}{3}-1)t<8 \]

Answer 21

ohhhh okay...is that all for 1

Answer 22

no
you have to solve for t

Answer 23

this means the only next to t can be 1

you must solve for t aka 1t

Answer 24

oh duh whoops

Answer 25

you have (2/3-1) next to t
which is not 1

Answer 26

so 1>t<8

Answer 27

how did you that?

Answer 28

\[\frac{2}{3} t < t+8 \\ \text{ subtract} t \text{ on both sides } \\ \frac{2}{3}t\color{red}{-t}<t\color{red}{-t}+8 \\ \\ \frac{2}{3}t-t<0+8 \\ \]
\[\frac{2}{3}t-t<8 \\ \frac{2}{3}t-1 t<8 \\ (\frac{2}{3}-1)t<8 \]

do you know how to do 2/3-1 ?

Answer 29

-1/3?

Answer 30

right
\[-\frac{1}{3} t<8\]

Answer 31

lastly for the first inequality you need to multiply both sides by -3

Answer 32

what inequality do you have after doing that?

Answer 33

so t=24 for number 1

Answer 34

-24

Answer 35

wait

Answer 36

not exactly
you dropped your inequality part and replaced it with an equality sign
ok the -24 part is right

what sign should you have between t and -24 though ?

Answer 37

t<-24

Answer 38

there

Answer 39

so if 2<3 then you are saying -2<-3 (I multiplied both sides by a negative)?

Answer 40

what I'm saying is if you multiply or divided both sides by a negative number on an inequality then the sign flips

Answer 41

\[\frac{-1}{3} t<8 \\ t>8(-3) \\ t>-24\]

Answer 42

oh right, because we are multiplying by neg t would actually be bigger just like -2 is bigger than -3, so what i actually mean is t>-24

Answer 43

yah!

Answer 44

now that we have this t we solve inequality 2 the same right?

Answer 45

\[\Large t+7 \le -2t+15 \\ \text{ here you already subtracted 7 on both sides } \\ t \le -2t +8\]

Answer 46

isolate t again

Answer 47

you figure out a way to some how get all your t-terms on the same side

Answer 48

so 3t<=8?

Answer 49

beautiful so far
you added 2t on both sides
great deal

Answer 50

so t<=8/3

Answer 51

right

Answer 52

so we want all numbers t such that
\[t>-24 \text{ and } t \le \frac{8}{3}\]

Answer 53

can you write this as a compound inequality?

Answer 54

[-25,8/3)

Answer 55

oh

Answer 56

I bet you mean (-24,8/3]
we want to include 8/3 but not include -24
the brackets means to include (think equal here)
the parenthesis means to exclude ( think no equal there)

Answer 57

-24<t<=8/3

Answer 58

oh yeah:) thanks

Answer 59

so is that all?

Answer 60

yep the answer in inequality notation is -24<t<=8/3
while in interval notation (-24,8/3]

Answer 61

Thank you sososososososososo much!!!!!!!!!!!!!

Answer 62

np