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(8x5)/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 
$\displaystyle x+2>\frac {5x}{2}\frac{8x5}{4}$ $\displaystyle x+2>\frac {10x}{4}\frac{8x5}{4}$ $\displaystyle x+2>\frac{10x8x+5}{4}$ $\displaystyle (x+2)×4>2x+5$ $\displaystyle 4x+8>2x+5$ $\displaystyle 4x2x>58$ $\displaystyle 2x > 3$ $\displaystyle x>\frac {3}{2}$ 
August 16th, 2017, 10:39 AM  #3  
Math Team Joined: Jul 2011 From: Texas Posts: 2,818 Thanks: 1463  Quote:
$0 > \dfrac{x}{2}  \dfrac{3}{4}$ $0 < \dfrac{2x+3}{4} \implies 2x+3 > 0 \implies x > \dfrac{3}{2}$  
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 
Math Team Joined: Jul 2011 From: Texas Posts: 2,818 Thanks: 1463  
August 16th, 2017, 11:52 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,302 Thanks: 1974 
Putting all the terms on the lefthand 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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