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 August 16th, 2017, 10:16 AM #1 Newbie   Joined: Aug 2017 From: Brazil Posts: 2 Thanks: 0 Inequations Hi, I'm having a really bad time with inequations; can someone help me with this one step by step? Thanks. x+2>5x/2-(8x-5)/4 Last edited by skipjack; August 16th, 2017 at 11:53 AM.
 August 16th, 2017, 10:34 AM #2 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 879 Thanks: 60 Math Focus: सामान्य गणित $\displaystyle x+2>\frac {5x}{2}-\frac{8x-5}{4}$ $\displaystyle x+2>\frac {10x}{4}-\frac{8x-5}{4}$ $\displaystyle x+2>\frac{10x-8x+5}{4}$ $\displaystyle (x+2)×4>2x+5$ $\displaystyle 4x+8>2x+5$ $\displaystyle 4x-2x>5-8$ $\displaystyle 2x > -3$ $\displaystyle x>-\frac {3}{2}$ Thanks from netomario
August 16th, 2017, 10:39 AM   #3
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Quote:
 x+2>5x/2-(8x-5)/4
$x+2 > \dfrac{5x}{2} - \dfrac{8x-5}{4}$

$0 > -\dfrac{x}{2} - \dfrac{3}{4}$

$0 < \dfrac{2x+3}{4} \implies 2x+3 > 0 \implies x > -\dfrac{3}{2}$

 August 16th, 2017, 10:58 AM #4 Newbie   Joined: Aug 2017 From: Brazil Posts: 2 Thanks: 0 Thanks so much, this really help me understand, just one thing, why does the -5 turns into +5?
August 16th, 2017, 11:28 AM   #5
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Quote:
 Originally Posted by netomario Thanks so much, this really help me understand, just one thing, why does the -5 turns into +5?
$- \dfrac{8x-5}{4} = \dfrac{-(8x-5)}{4} = \dfrac{-8x+5}{4} = -\dfrac{8x}{4} + \dfrac{5}{4}$

 August 16th, 2017, 11:52 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,302 Thanks: 1974 Putting all the terms on the left-hand side gives x + 2 - 5x/2 + (8x - 5)/4 > 0, i.e. x + 2 - 5x/2 + 2x - 5/4 > 0. Hence x/2 + 3/4 > 0, which implies x > -3/2.
 August 17th, 2017, 03:20 AM #7 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,139 Thanks: 721 Math Focus: Physics, mathematical modelling, numerical and computational solutions Remember that if you multiply by a negative number, you need to swap the sign of the inequality to keep the inequality consistent. For example: $\displaystyle 5 > 3$ Multiply both sides by -1: $\displaystyle -5 < -3$ you need to swap the > to a < because -5 is actually less than -3.
 August 17th, 2017, 06:02 AM #8 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 Another way to do it: first solve the related equation: I would start by multiplying both sides by 4: $\displaystyle 4x+ 8= 10x- 8x+ 5= 2x+ 5$ Subtract 8 and 2x from both sides: $\displaystyle 2x= -3$ So $\displaystyle x= -3/2$ is where the two sides are equal. The point is that this x separates "<" from ">". Now we just need to check the value for one x on each side. x= -2< -3/2. With x= -2, x+ 2= -2+ 2= 0 and $\displaystyle \frac{5x}{2}- \frac{8x- 5}{4}= -5+ \frac{21}{4}= \frac{-20+ 21}{4}= \frac{1}{4} which is greater than 0. x= 0> -3/2. With x= 0, x+ 2= 2 and$\displaystyle \frac{5x}{2}- \frac{8x- 5}{4}= 0- \frac{5}{4}= -\frac{5}{4}$which is less than 2$ $\displaystyle x+ 2> \frac{5x}{2}- \frac{8x- 5}{4}$ for all x> -3/2.

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