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I was wondering if anyone knows how to solve $|ax+b|<cx+d$ type questions by using cases and the number line to finish. I am personally struggling with the number line, I have half-finished a sample question by I am unsure of how to use the number to finish it.

$|2x-4|<6x$

Case 1: $x\le 2$

$-2x+4<6x$

$x>{1\over2}$

Case 2:x>2

$2x-4\lt6x$

$x>-1$

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You have followed the right path.

Case 1: here, you have strictly restricted $x$ to less-than-or-equal-to $2$. The solution you got was $x > 1/2$. Keeping the initial restriction in mind, you have $1/2<x \le 2$. Remember that you have ignored the numbers that follow $2$ because they don't follow the original restriction $x \le 2$.

Case 2: here, you have strictly restricted $x$ to greater than $2$. The solution you got was $x > -1$. Keeping the initial restrictions in mind, you get $x > 2$ as you will ignore all the numbers between $-1$ and $2$ (since they don't follow your original restriction).

So, finally, you get the solution $2 \ge x> 1/2$ and $x >2$. Here is where the number line comes into play. Represent your solutions on a number line, and you'll notice that it's the same as $x > 1/2$.

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  • $\begingroup$ Hi thanks for your reply, but do you have any idea how i would use a number to work this out? $\endgroup$ – andrew12678 May 17 '14 at 23:41
  • $\begingroup$ A number? I'm not sure if I perfectly understand your query. Do you mean a number line? $\endgroup$ – Parth Kohli May 17 '14 at 23:42
  • $\begingroup$ yea number line sorry for the typo $\endgroup$ – andrew12678 May 17 '14 at 23:44
  • $\begingroup$ A number line wouldn't be much of a solution. It can only be used to represent numbers. Depending on different contexts, it's used in different ways. Here, the question might be asking you to use a number line for representing the solution of this inequality. $\endgroup$ – Parth Kohli May 17 '14 at 23:48
  • $\begingroup$ So why is it that when I enter the question in Wolfram Alpha I get x>1/2 $\endgroup$ – andrew12678 May 17 '14 at 23:52

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