5

I have been trying to reconcile this solution with my work for hours now. Could someone please help me confirm whether the solution provided below is incorrect?

I get a solution of X > -3/2, the provided solution is X < 3/2

I am too new to embed images

Parcly Taxel
  • 105,904
Heidi Good
  • 85
  • That was my question initially. The solution shown is my professor's. My calculations do not do this. I've been trying to figure out where she got the 3x for hours now. – Heidi Good Oct 22 '16 at 17:15
  • She made a typo. $\frac 58 - \frac x3 = \frac {35 - 8x}{3*5}$. Once that mistake happened all things that followed will not be correct. – fleablood Oct 22 '16 at 17:20
  • 1
    @fleablood, denominator should be 38, not 35. – Ameet Sharma Oct 22 '16 at 17:23
  • 1
    Like I said. Once a mistake happens... things. follow. the professor made an error. I made an error. Neither of us got the right answer because of it. – fleablood Oct 22 '16 at 19:34

4 Answers4

6

The solution is incorrect. In the first passage both sides of $$\frac 58-\frac x3>\frac18-\frac23x$$ are multiplied by $24$. But in the proposed solution "$-\frac x3$" becomes "$-8\cdot 3x$", instead of "$-8x$".

  • 1
    I am most grateful for a confirmation that the solution provided is in error and my own calculations are correct. It is a relief to my aching brain. – Heidi Good Oct 22 '16 at 17:19
  • Could you provide for me the rule which causes 12>-8x to become x>-3/2. My remaining confusion with this problem is that although I can see that this is true I cannot extrapolate exactly the procedure where the x changes sides and the signs remain the same. – Heidi Good Oct 22 '16 at 17:23
  • 1
    @HeidiGood, the rule is... whenever you multiply or divide by a negative number, the less than sign turns into greater than, and vice versa... 12>-8x. divide both sides by -8, so we change signs 12/-8 < x. -3/2<x. or x>-3/2. – Ameet Sharma Oct 22 '16 at 17:28
  • Ah.. and then the sign changes back to the original only because we flipped it to read the X value the other way around. Thank you. That will also help me a lot. – Heidi Good Oct 22 '16 at 17:29
  • 1
    If you want to get down to brass tacks. $a < b$ means $a + x < b + x$ no matter what $x$ is (positive, negative or zero). So $a < b$ means $a - a < b - a$ means $0 < b-a$ means $0 -b < b - a -b $ means $-b < -a$ means $-a > -b$. So "taking negatives flip the sign". We also know if $n > 0$ and $a < b$ then $na < nb$. This is because... well, it's an axiom. But if you think of $n$ as an positive integer it is $na = a+a+a+..< b+b+b+.... =nb$. For other reals... well, "it follows" so $a < b => an < bn => -na > -nb$. That's hand-wavy but I hope it helps. "negative flip" it's a rule. – fleablood Oct 22 '16 at 19:32
  • More explanations on why an inequality symbol flips from division by a negative: http://math.stackexchange.com/questions/1543722/why-does-the-sign-have-to-be-flipped-in-this-inequality – Daniel R. Collins Oct 22 '16 at 22:24
4

\begin{align} \frac{5}{8} - \frac{x}{3} &> \frac{1}{8} -\frac{2x}{3} && \text{Multiply by } 24 \\ 15 -8x &> 3 -16x \\ 12 &> -8x && \text{Divide by $-8$ which changes the sign}\\ -\frac{3}{2} &< x \end{align}

null
  • 1,540
Zubzub
  • 4,213
2

Hint:

You have a mistake at the first step:

$$ \frac{5}{8}-\frac{x}{3}>\frac{1}{8}-\frac{2x}{3} \iff 5\cdot 3 \color{red}{-8\cdot x} >3 -8\cdot(2x) $$

so the final step becomes $$ 12 >-8x $$

Emilio Novati
  • 64,377
1

The solution is very simple let me show you

$5/8-x/3>1/8-2x/3$

now we interchange

$-x/3+2x/3>1/8-5/8$

$x/3>-4/8$

$x>-12/8$

$x>-3/2$

get it

just see this once
https://www.wolframalpha.com/input/?i=5%2F8%E2%88%92x%2F3%3E1%2F8%E2%88%922x%2F3

Marble
  • 948