Question

Help for a medal, anyone?

I'm rusty on inequalities and need some help on a few problems to boost my memory.

Here's the first one...
-9 < 4x + 8 < 10

Answer 1

what are your thoughts ...

Answer 2

If I remember correctly, you're supposed to simplify the equation by adding, divining..etc terms on either side, but I'm not sure how to do that with an inequality.

Answer 3

dividing*

Answer 4

the math doesnt change ... what you do to one part, you do to all the parts to keep them "equal" right?

Answer 5

in this case, the equation is in the middle instead of to the left or right ... what would you think we should do first?

Answer 6

just make a guess at what you think we should do to start with ... what comes to mind?

4x+8 is what we want to manipulate, how do we work about getting 'x' all by itself?

Answer 7

Would you solve one equality at a time like
-9 < 4x + 8 or 4x + 8 < 10 ?

Answer 8

you can if you want, but i just work it all at once to avoid the extra work of writing it all out twice

Answer 9

-9 < 4x + 8 < 10
-8 -8 -8
----------------
-17< 4x < 2

Answer 10

Ah, I see. Then you would divide all 3 parts by 4 to get x by itself.

Answer 11

correct, the only thing to watch out for is when we have to mulitply or divide by a negative number ... which they really should include the rule when doing equalities

the sign reverses ... but we do not have to worry about that in this particular case do we

Answer 12

if -a < b, then a > -b

if -a > b, then a < -b

if -a = b, then a = -b

... the reversing of an equals sign just produces an equals sign, but it helps to make the rule consistent for all 3 equality signs.

Answer 13

So would
-17/4 < x < 1/2
be the final answer or is there another step?

Answer 14

thats the answer for me, but im not grading it

Answer 15

17/4 cant be reduced, but it can be written as 4 and 1/4 if need be

Answer 16

Okay, thank you. :)

Could you help with one other? It looks a bit different from the first.

Answer 17

i spose

Answer 18

\[\left|\frac{ x+4 }{ 5 } \right| < 6\]

Answer 19

absolute value, what does that mean to you?

Answer 20

I know that nothing inside of it can be negative (or at least that I recall)

Answer 21

first off |5| = 5 so we can go ahead and multiply thru by 5

|x+4| < 30

Answer 22

it can be negative inside of it, but the bars cancel out the sign

Answer 23

|-30| = 30

and

|30| = 30

when does x+4 = 30? or -30?

Answer 24

as a side note:

|a| < k can be written as:

-k < a < k

Answer 25

|x+4| < 30 can be written as

-30 < x+4 < 30

Answer 26

Then you just subtract 4 from both sides...
-34 < x < 26

Also does the |a| < k or -k < a < k
rule always apply?

Answer 27

it is a rule, so it always applies yes. its just my brain that doesnt always remember it in time lol

Answer 28

\[|\frac{x+4}{5}|<6\]
\[-6<\frac{x+4}{5}<6\]
\[-306<x+4<30\]

Answer 29

not -306, but yeah

Answer 30

Oh, okay.
So -34 < x < 26 would be the answer correct?

Answer 31

looks good to me

Answer 32

Thank you so much for your help. :)

Answer 33

youre welcome, and good luck :)