Question

I need help with One-Variable Compound Inequalities. Please show your steps:

-16 < less than or equal to 2+9x < greater than or equal to 11

Thank you!

Answer 1

It is written this way it is -16 less than or equal to 2+9x < 11.
The underline will not show up both parts is less than or equal to.
-16 <2+9x < 11.

Answer 2

So in the long run...we want to solve for 'x' in the inequality... so we have

\[-16 \le 2 + 9x \ge 11\]

We want to get 9x by itself in the middle...so we need to subtract 2 from everything! why? because

2 + 9x = 2 = 9x....just 9x so...

\[-16 -2 \le 2 + 9x - 2 \ge 11 - 2\]
and this becomes

\[-18 \le 9x \ge 9\]

Now...we want to solve for 'x' right? not 9x...so lets divide EVERYTHING by 9...because 9x/9 = 1x...or just x right? so we have

\[-18 /9 \le 9x / 9\ge 9/9\]

\[? \le x \ge ?\]

What are the 2 ?'s replaced with?

Answer 3

Wait...BOTH parts are less than or equal to???

Answer 4

I got the first part -18 but did not get the 9’s. I thought everything was subtracted by 2 and divided by 2.

Answer 5

4.5

Answer 6

Well then if THAT is correct....then in my entire explaination *it stays the same* just read it as

\[\le x \le\]

And no...everything IS subtracted by 2...*shown above* but then everything is divided by 9...after you subtract 2 from everything...it is gone! the only thing left is 9x

Answer 7

9 is the denominator leaving 9x. If I graph this it is less than 9x?

Answer 8

Let me rewrite the steps...maybe that will clear it up

Answer 9

Ok

Answer 10

\[-16 \le 2 + 9x \le 11\]
\[-16 - 2 \le 2 - 2 + 9x \le 11-2\]
\[-18 \le 9x \le 9\]

With me so far? All I did was subtract EVERYTHING by 2 okay?

Answer 11

Thanks John.

Can you help with the second part of my equation?

12-x>15 or 4x-13>7

Answer 12

\[\frac{ -18 }{ 9 } \le \frac{ 9x }{ 9 } \le \frac{ 9 }{ 9 }\]

This is me dividing EVERYTHING by 9...this will isolate 'x' after that..we'll have

\[-2 \le x \le 1\]

better?

Answer 13

Thank you so very much! You are great!

Answer 14

Okay now onto that 2nd part....what do you think you would do first?

Answer 15

I am thinking the opposition add instead of subtract?

Answer 16

12-x>15 or 4x-13>7
this means we have 2 equations to solve

\[12 - x > 15\]
\[4x-13 > 7\]

Lets solve that first one to begin with....
what would you get?

Answer 17

On the first answer when I draw the number line to reflect the answer what are my two points that I color in?

Answer 18

The first answer meaning the first problem we solved? *not the one we're working on now*?

Answer 19

Yes I have to draw the answer on the number line also on both equations.

Answer 20

Okay...that very first problem we solved...we came up with

\[-2 \le x \le 1\]

So you can split this into 2 equations as well

\[x \le 1\]
and
\[x \ge -2\]

where would your 2 points be?

Answer 21

Wow that is the answer I got but thought that it was incorrect! I am so happy you were still up to help me! Yeah!

Answer 22

On the number line I would show -2 through 1, color it in at those points.

Answer 23

That would be correct...because if you say it outloud...

'x' is greater than or equal to -2 AND LESS than or equal to 1...so it would be all points between the 2

Answer 24

Thank you that is what I wrote and I was just about to give up but thought I would ask for the correct steps and you provided that. Thank you so much! Can we solve the last two?

Answer 25

Absolutely....no like I wrote above...we have 2 equations

\[12 - x > 15\]

\[4x - 13 > 7\]

What do you get when you solve that first one?

Answer 26

Subtract 12 from all parts?

Answer 27

We can do that yes....

\[12 - 12 - x > 15 - 12\]
\[- x > 3\]

remember...we aren't solving for -x...we want 'x'...so what next?

Answer 28

We do the same as above, subtract and then divide.

Answer 29

Right we subtracted 12 from both sides...*done above* and now we divide by '-' to solve for 'x'

****what do we do when we divide or multiply by a negative number in inequalities??****

Answer 30

3

Answer 31

I think when we multiply or divide we change the inequity to a positive number so we divide by 3, I think.

Answer 32

Not quite....

when we multiply or divide by a negative number....we flip the direction of the inequality sign...here I'll do this one for you

\[-x > 3\]

Divide both sides by -1

\[\frac{ -1x }{ -1 } > \frac{ 3 }{ -1 }\]

this becomes

\[x < -3\]

Answer 33

This is a difficult concept but I am very happy to have your help.

Answer 34

No problem..I hope you understand it....we have 1 more equation to solve though remember?

\[4x - 13 > 7\]

what would you do first?

Answer 35

I know I was just trying to figure out the one you just listed and my head is spinning. I do not understand so I am writing all of the steps down as you write them.

Answer 36

Want me to run over the steps again? I can probably explain it a little better! :)

Answer 37

Yes, if you can walk me through the

Answer 38

last one I will call it a day. the 4x<13<7

Answer 39

I mean: 4x-13>7

Answer 40

Okay...and at the end I'll remind you about what I did for that 2nd one I just finished!

so

\[4x - 13 > 7\]

what do we do?

Answer 41

Subtract 4 from all parts and then divide.

Answer 42

Not quite...let's not touch the 'x' number....we want to solve for 'x' eventually so lets get rid of everything ELSE first

Answer 43

Oh yeah we are going to flip the inequality symbol over so it points to the other direction.

Answer 44

Hang on hang on....lets take it step by step

Answer 45

\[4x - 13 > 7\]

We don't touch the 'x' number here....we want to get rid of EVERYTHING else first...

next to the 4x...we have a -13...how do we get rid of that?

Answer 46

I cannot pretend to know the answer because I don't

Answer 47

For the last two the assignment is asking for the solution set of both of the last equations with the number line shaded in the answer.

Answer 48

Okay...well just remember...since we want 4x to be alone....we want to get rid of the stuff next to it...

\[4x - 13 > 7\]

If we add 13 to both sides...what do we get?

\[4x - 13 + 13 > 7 + 13\]

notice how -13 + 13 = 0 right? so we'll have

\[4x > 20\]

Answer 49

so the answer is 4x is greater than 20 showing 4-20 on the number line for this one?

Answer 50

After that...we want to solve for x....not 4x....so we divide everything by 4

\[\frac{ 4x }{ 4 } > \frac{ 20 }{ 4 }\]

this becomes

\[x > 5\]

Answer 51

x>5 is the final answer and the points to graph!

Answer 52

I really hope you're understanding this.....

so

we have *remember we started with 2 equations*

our 2 answers are

x < -3
and
x > 5

so what points do you plot?

Answer 53

The first set will show -3 on the graph the second will begin at 5 and go beyond that

Answer 54

Right....one part says "x is less than -3".....or "x is greater than 5"

so the left hand would be an open circle at -3 and shading to the left....and the right side will have an open circle at 5 and shading to the right

Answer 55

John, thank you so very much I have all ready tagged you to be a fan! I will write it all out for my assignment tomorrow. Thank you!

Answer 56

No problem! and if you need more or an explanation or anything...feel free to tag me in a post or message me!

Answer 57

I will definitely do that. I don't know why it says that you are a novice, you seem like an expert!

Answer 58

Haha don't worry about that....I'm just so used to doing these problems from middle / high school...it is like second nature now...and one day soon...you will be able to do them like nothing as well :)

Answer 59

LOL! I don't know about that nearly every assignment I turn in there is something wrong. To contact you in the future do I just click on your name above?

Answer 60

Yeah you can click on my name and click send message...or you could post a question and if you wanted just tag me by using the '@' button followed by my name

Answer 61

Ok! I am so happy to have found you! LOL! Talk with you soon! Tuiti

Answer 62

Haha have a good night!

Answer 63

You too!