PAT inequality question

https://oxfordpat.wordpress.com/oxford-pat-2016-question-9/

on the answer, why is it 0 < x <2/3 rather than just x < 2/3

You can't allow x = 0, also looking at where the function is > 0 only applies to x > 0, once you look at x < 0 you have to switch to the 2nd inequality, which has no solutions. So the only good bits are x > 1 and that small region between 0 and 2/3.

You can't allow x = 0, also looking at where the function is > 0 only applies to x > 0, once you look at x < 0 you have to switch to the 2nd inequality, which has no solutions. So the only good bits are x > 1 and 0 < x < 2/3.

by 2nd inequality do you mean when the inequality sign is flipped?

Yes, when x flips to < 0 you have to look at where the function is < 0, which it never is. You were still looking at the 1st inequality for x > 0 (and forgot x =/= 0).

ah i see, so i have to be careful when i turn the inequality equation in to a quadratic equation by times x

That care is needed to form 2 inequalities instead of 1 (for x > 0 and x < 0). The next bit of care is in using these together with the graph, to find your solutions.

just want to say thank you again for helping