# Multiplying inequalities by x or ^2

If you have an ineqaulity like $\tfrac{2}{x} < 1$ I know you can multiply both sides by x^2 to solve for x but why can't I just multiply by x and then make seperate inequalities for when x is positive and negative for instance I multiply by x and assume it is positive and I get $2 < x$ and then I multiply by x and assume that x is negative and get $2 > x$. I know this doesn't work but why since am I not considering both values of x.

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Erdos
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• Erdos
+1

Let me know if you need any clarifications.

• Thanks for the speedy reply. I just have one question with case II we get 2 > x where does the x < 0 come from?

• Thanks for the speedy reply. I just have one question with case II we get 2 > x where does the x < 0 come from?

• Erdos
+1

Since multiplying by positive and negative numbers makes a difference in changing the direction of the inequality, the two cases are indeed created for x>0 and x<0. That's where this comes from.