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December 18th, 2017, 04:08 AM  #1 
Newbie Joined: Dec 2017 From: Trinidad Posts: 2 Thanks: 0  Determine the interval of x
Hello, I have a question here that I have no previous knowledge of, I turn to the internet for help because I am currently studying on my own for a test. I have a question here that to be honest I have no idea about. It states: Determine the interval of x for which (i) x(x+2)> 0 (ii) x(x+4)â‰¤ 5 Now all I know is the basic outline of what an interval is, on a graph the regions where say a number x, is either increasing or decreasing, hence positive or negatives intervals of the number x. But that's about it, I really do not know where to go from this point. Any help would on where to start and what this question is about would be appreciated, the answer is not important as much as understanding what this topic is. 
December 18th, 2017, 04:52 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,767 Thanks: 1018 Math Focus: Elementary mathematics and beyond  
December 18th, 2017, 05:26 AM  #3 
Newbie Joined: Dec 2017 From: Trinidad Posts: 2 Thanks: 0 
Hi, I have an update here, so apparently my friend explained to me here that, I should separate the equation and say Let x = 0 Let x = 2 Then draw a number line from 3 to 3, and mark the two places that x is equal to, then test the range which is less than 2, by picking a value and substituting it into the equation, to see whether it satisfies the inequality, and do so for the range that is greater than zero, and finally for the range in between 2 and 0. After doing this, I got the answer which is x < 2 and x > 0. And I was thinking that it was a quadratic equation before my friend explained it to me, I was looking at it and was wondering if I could bring across the 0 and use the quadratic formula, but even that I'm unsure of. Now those two answers I got there, are correct according to the book, but they go on to say that the union of those two inequalities such that X>0 âˆª x < 2 = {x: 2 â‰¤ x â‰¤ 0}^1. Can you possible explain how they got that final answer in the brackets, and also if I could have used the quadratic formula by chance? Last edited by skipjack; December 18th, 2017 at 06:18 PM. 
December 18th, 2017, 05:34 AM  #4  
Senior Member Joined: May 2016 From: USA Posts: 924 Thanks: 369  Quote:
$x \in (a,\ b) \iff a < x < b;$ $x \in [a,\ b] \iff a \le x \le b;$ $x \in [a,\ b) \iff a \le x < b; \text { and}$ $x \in (a,\ b] \iff a < x \le b.$ So in this problem you are being asked to find what intervals make the given inequalities true. Example. Find the interval or intervals such that $x^2  4 \ge 0.$ $x \ge 2 \implies x^2 \ge 4 \implies x^2  4 \ge 0 \implies x \in [2,\ \infty).$ $x \le \ 2 \implies x^2 \ge 4 \implies x^2  4 \ge 0 \implies x \in (\ \infty ,\ \ 2].$ Notice this problem has two intervals. Frequently the answer is given just by the intervals themselves and set notation $(\ \infty ,\ \ 2] \ \bigcup \ [2,\ \infty ).$  

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