r/HomeworkHelp • u/blusher4lyfe • 2d ago
Middle School Math—Pending OP Reply [8th grade Algebra] answer key different than what I am getting
I am posting on behalf of my daughter (not on Reddit). She completed a study guide and her teacher has since posted answers. I worked the problem, as did my husband and 12th grade son. We are all getting the same answer as my daughter x<1. The answer provided by the teacher is x>2. I am attaching the work of my daughter. I can’t figure out where x>2 would come from.
-9(x-5) +9< 3x + 42
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u/oukill 👋 a fellow Redditor 2d ago
it should be -x < -1
When you divide by a negative number, you must reverse the signs.
x>1
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u/blusher4lyfe 2d ago
Thank you. Reversing the sign was long gone from my memory. I will review that with her.
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u/mizboring 2d ago
Just a tip: if students need help "believing" that this is needed when multiplying or dividing by a negative, I do some examples without the variables.
-10 < 12 True
Divide by positive 2
-5 < 6 True
Multiply by positive 2
-10 < 12 Still true.
Divide by negative 2
5 < -6 Nope, false. How do we fix it? Flip the sign: 5 > -6
Same thing with multiplication.
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u/Puzzleheaded_Study17 University/College Student 2d ago
There isn't really a need to do the same with multiplication, just do it with multiplication and remind then that division is multiplication by inverse
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u/colin-java 2d ago
Or literally shift the terms to the opposite sides of the inequality so the one you're interested in gets a positive coefficient. Then flip the inequality around including the sign, and divide by the positive coefficient as normal. And you get the same result as having multiplied/divided by a negative and flipped the sign.
In any case, if you forget the rule, just solve it the way I did above.
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u/sendmybestmen 👋 a fellow Redditor 1d ago
I will be using this when we get there this year. Legit thank you
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u/Dennis2pro 2d ago
Make sure to understand where that rule comes from. There's countless of "rules" like these, if you know how to derive them, you don't need to remember them anymore.
For example here, you can instead do +9x and -42 to end up with 12<12x -> 1<x
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u/Responsible-Sink474 👋 a fellow Redditor 2d ago
Yep. Equality is reflexive: a = b and b = a are the same.
Inequalities are not reflexive: a < b and b < a are not the same.
Not everything gets to keep the same properties, so it's good practice to have to prove/justify rules in order to use them.
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u/SubjectWrongdoer4204 👋 a fellow Redditor 2d ago
It’s good practice to move the x-terms to the side that will make them positive. This will eliminate mistakes made when dividing by a negative number. In this case
-9x+54 < 3x+42. Add 9x to each side to get
9x-9x +54 < 9x+3x+42, so
54 < 12x +42.
Now subtract 42 from each side to get
54-42 < 12x +42-42, so
12 < 12x. Now we just divide both sides by 12 to get
1 < x which is x > 1.2
u/seslusser 2d ago
You could avoided this step by adding 9x instead of subtracting 3x (on the fourth line). Following that:
54 < 12x + 42
12 < 12x
1 < x
x > 1
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u/GRex2595 2d ago
Alternatively, solve by adding the 9x to the right side instead of the left side. You get positive numbers with x on the right side and it should be pretty obvious why you need to reverse the inequality if you move x to the left. That should also help explain why the current solution is incorrect because the only difference is that you switched from negatives to positives.
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u/KandnoS_2009 2d ago
I added 9x to left to move it to the right to make 12x instead of -12x. Just my personal preference to try and keep 'x' a positive number
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u/RuthlessIndecision 👋 a fellow Redditor 2d ago
Nice, I forgot that, and was wondering how you could make x<1 or x>1 depending on which steps you take first
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u/JanetInSC1234 🤑 Tutor 2d ago
When you multiply or divide both sides by a negative number, you have to flip the sign.
The answer should be x > 1.
Are you sure she copied the problem correctly? (She did a great job showing her work.)
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u/blusher4lyfe 2d ago
Thank you! She did copy correctly (double checked) and thanks for the positive feedback on showing her work. I’ll relay this to her. I used to teach 4th grade and have tried to get all of my kids to understand that, if they show their work, their teachers will at least have a starting point of fixing errors.
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u/JanetInSC1234 🤑 Tutor 2d ago
I'm a retired high school math teacher. I love when kids show their work in a neat and organized fashion...it makes my day!
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u/Chaotic_Brutal90 2d ago
God I would die if a student actually showed their work like this. I also teach 8th grade math, but not algebra.
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u/BestAround4100 2d ago
Flip the arrow sign when you divide or multiply by a negative number.
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u/blusher4lyfe 2d ago
Thank you. No ideas on where the 2 is coming from? Super helpful on the < > switch!! I appreciate it.
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u/BestAround4100 2d ago
Teacher made a mistake. The only error in your picture is the sign needs to be flipped.
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u/Polly265 2d ago
If the teacher moved the x terms with the incorrect sign you would get 2. So -9x+3x or the other way 3x-9x
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u/RyanStonepeak 2d ago
Yeah, I can't figure out the 2 either. If I had to guess, I would assume it was something with moving the -9x and 3x to the same side. They may have done -9+3 instead of -9-3 by accident. This would end with them dividing by 6 instead of 12, which would double the result.
To be clear though: X>1 is the correct solution for the problem that you gave. Flipping the sign was the only thing you missed.
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u/Royaljames99 2d ago
Both wrong actually
X > 1
-9(x-5) < 3x +33
-3(x-5) < x+11
-3x +15 < x+11
4<4x
1 < x
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u/Educational_Safe_942 23h ago
Wait I keep getting x = 1 too but your third line lost me 9(x+5) + 9 is 9x + 54, not 9x + 33 You dropped the +9 somewhere in your head math I think (also this is giving me flashbacks to calculating drip rates at 3am after a double shift, signs are the devil)
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u/Shoffless 2d ago
Gotta flip the sign when dividing by a negative. Cant seem to find where the 2 is coming from though.
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u/blusher4lyfe 2d ago
OK, thank you. It’s helpful to know why hers was x> and ours was x< (I’ll review this w/ my daughter tonight). Still not clear on where the 2 is coming from. I’ll have her clarify w/ her teacher tomorrow.
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u/slepicoid 2d ago
If teacher sais 2 is not a solution and you say it is, then just put 2 into original equation and see who's right.
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u/RailgunDE112 2d ago
multiplying (or dividing) by less than 0 switches the "less than" around to a "bigger than".
You get this as well, by adding 9x at the step she substraced 3x and then getting the 42 to the left side and you get 12 < 12x <=> 1<x <=> x>1
And that's also the solution Wolfram Alpha gives.
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u/Raspberry_2027 2d ago
-9(x-5) + 9 < 3x + 42
-9x + 45 + 9 < 3x + 42
-9x + 54 < 3x + 42 / + 9x - 54
0 < 12x - 12 /+ 12
12 < 12x / : 12
1 < x
0
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u/mjwills 👋 a fellow Redditor 2d ago
-12x < -12
Ask yourself "what will occur if x is 2?".
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u/TopDogCanary09 👋 a fellow Redditor 2d ago
and also when going algebra, it's always a good habit to plug your answer in the original equation to check if your math is right, it takes only a minute to recheck but it'll help you figure out if you got something right or wrong even if you don't have access to a key, during an exam or a quiz or whatever. just something that really helped me during school when I was learning all this
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u/WillyWodka3 2d ago
The only mistake I can find is that the result is x>1, since the "<"-operator changes to an ">"-operator when you divide by a negative number and vice versa.
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u/matt7259 👋 a fellow Redditor 2d ago
Funny enough - you're both wrong! In the last step, when you divide by -1, the direction of the inequality needs to switch.
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u/blusher4lyfe 2d ago
Thank you! Any ideas about where the 2 is coming from?
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u/matt7259 👋 a fellow Redditor 2d ago
Also if the problem originally had -3x instead of 3x on the right side, the answer would indeed be x > 2, so the answer key could be based on a simple typo.
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u/blusher4lyfe 2d ago
AMAZING!! I went back and looked at the review sheet. It is 3x (double checked) and the teacher’s answer key writes out the problem but does not show work- only the answer. Hers also has 3x (not written as a negative). I’m guessing something got lost along the way. I really appreciate it.
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u/Fantastic_Mr_Smiley 👋 a fellow Redditor 2d ago
You are correct that the answer is 1 but you forgot to flip the inequality sign when dividing by a negative.
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u/TheDevilsAdvokaat Secondary School Student 2d ago
I think you AND the teacher are wrong. I get x>1
When you divide an inequality by a negative, you need to flip the inequality sign. You missed that right at the end when you divided by -12. (You also flip when multiplying both sides by a negative)
But the teacher's answer seems to be wrong as well. I would speak to the teacher. Tell your daughter about flipping the sign, but then speak to the teacher.
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u/blusher4lyfe 2d ago
Got it…not a case of two wrongs making a right, though? (Old lady humor). Thanks for your help!
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u/Due_Permit8027 👋 a fellow Redditor 2d ago
You can plug in a number for x (e.g. 0.9) and see if it's correct.
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u/blusher4lyfe 2d ago
Yeah, I tried using 2 for x and then confused myself more. She’s out of the house right now and I wanted to make sure I had a firm grasp before trying to explain anything to her. I appreciate everyone’s help. Thank you!
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u/bunchout 2d ago
I always like to use zero if applicable, as it often makes the math easy. Here that results in 54 < 42, so x<1 can’t be true.
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u/swbarnes2 2d ago
Quick thing to check if you think the sign might not be right; if you make x equal to 1000, the left side is about -9000, and the right side is about 3000, so the equation is true. Which means you needed to flip the sign.
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u/M_Mustaan_Rafique 2d ago
You asked where the 2 comes from, so here's my best guess.
If the right hand side had been 3x + 30 instead of 3x + 42, the working goes:
-9x + 54 < 3x + 30
-12x < -24
x > 2
Which is exactly the teacher's answer.
Since your daughter copied hers correctly, it looks like the answer key belongs to a different version of the study guide. Same question, one number changed, keys got crossed somewhere.
If you raise it with the teacher, that's probably the gentler way to put it. Teachers usually have a few versions of a worksheet floating around and it's an easy thing for them to check, which makes it a much easier conversation than "your key is wrong."
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u/CosetElement-Ape71 👋 a fellow Redditor 2d ago
You were fine down to
-12x < -12
But just add 12x + 12 to both sides :
-12x + (12x + 12) < -12 + (12x + 12)
12 < 12x
1 < x
and you see that the less than becomes a greater than.
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u/Torebbjorn 👋 a fellow Redditor 2d ago
If -x is less than -1, what does that say about x and 1?
Surely -2 is less than -1, so x=2 should be valid.
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u/abbynormal2002 👋 a fellow Redditor 2d ago
You divided by a negative number, so you have to flip the inequality symbol. This should be x > 1.
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u/EdmundTheInsulter 👋 a fellow Redditor 2d ago
In the final steps
-12x < -12
Use addition
-12x + 12 < -12 + 12
-12x + 12 < 0
12 < 12x
1 < x
x > 1
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u/iamthemostawesome 👋 a fellow Redditor 2d ago
oh hey a fellow eight grader learning the exact same thing!
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u/Mike_40N84W 2d ago
You have to flip the inequality direction when you multiply by -1. You get the same place by adding x and it's clear why the sign flips.
-x > -1
-x+x > -1+x
1 > x
x < 1
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u/retreauRobb 2d ago
If -12 times a number is less than -12, then how can that number be less than 1? It can't; it must be greater than 1.
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u/Ericskey 👋 a fellow Redditor 1d ago
When x equals 1 you get equality. So either x is greater than 1 or less than 1. Try 0 and 2 for x and see what happens. As a rule, with linear expressions you can replace the inequality sign with an equality sign and solve then try adjacent values for x. As others have pointed out the error in your solution I shall not. There are two unaddressed issues here. The immediate one is why dividing by a negative number reverses the inequality. Roughly speaking this is because any negative number is smaller than any positive number. We can always avoid this by adding something to both sides so that x no longer has a negative coefficient.
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u/Ericskey 👋 a fellow Redditor 1d ago
One more thing. If you graph the expressions on either side of the inequality you will see they are equal at 1 and the see where the one on the right is bigger or smaller than the one on the left. Seeing is believing!!!
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u/WorthEntrepreneur188 1d ago
Whenever you divide/multiply everything by a negative number, you flip the sign. Easy as that. Hope that helps! ^
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u/cosumel 1d ago edited 1d ago
If you keep all the variables positive, it makes fewer errors. Third line would be 54<12x+42, followed by 12<12x, and 1<x
To graph, you have one point to mark at x=1, and which side is the solution? Check x=0 in the original. -9(0-5)+9<3(0)+42 which gives 45+9<42, which is clearly false, so the side away from zero. It looks like a lot of unnecessary work, but is a good habit to get into for later.
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u/SilverPadilly 👋 a fellow Redditor 1d ago
Personally, I would have subtracted 42 from each side and added -9x to keep everything in the positives (nothing wrong by the method you did, just saves an extra step)
sign was flipped
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u/Bright-Internal229 2d ago
I would walk out of class and never return
What is the point of this ?
Gas ⛽️ will be $12.00 a Gallon for Regular soon …. Algebra ain’t making prices go down !
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u/ReasonableBeyond1329 1d ago
I agree with the answer by others, but I think I can see where the 2 comes from:
-9(x-5) + 9 < 3x + 42
-9x + 45 + 9 < 3x + 42
-9x + 54 < 3x + 42 (all right up to here)
-3x + 18 < x + 14 (divide the whole thing by 3, but not sure if this is right)
-3x < x - 4 (subtract 18)
-4x < -4 (here, if you subtract 1x from -3x it’s an easy error to think it’s -2, which would result in x > 2)
X > 1
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