Solving an Inequality: -9 < 1/x

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To solve the inequality -9 < 1/x, it is essential to consider the sign of x, as multiplying both sides by a negative value reverses the inequality. The correct approach involves splitting the problem into two cases: x > 0 and x < 0. For x < 0, the inequality becomes -9x < 1, leading to x > -1/9. This confirms that the solution is x < -1/9 when considering the conditions for x. Understanding these rules is crucial for correctly solving inequalities.
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Homework Statement



Solve the inequality -9 < 1/x

A simple inequality, I can see the solution is just x < -1/9 but I can't prove it at all.

The Attempt at a Solution



-9 < 1/x

-9x < 1

x > -1/9

Any helpful rules I am forgetting about inequalities? This was a problem in a review from high school set provided by my instructor for my introductory math class. Just curious about a solution, it's a calculus course that doesn't really test on this sort of thing but I figure I should actually try and figure out these basic things.
 
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brownman said:

Homework Statement



Solve the inequality -9 < 1/x

A simple inequality, I can see the solution is just x < -1/9 but I can't prove it at all.

The Attempt at a Solution



-9 < 1/x

-9x < 1

x > -1/9

Any helpful rules I am forgetting about inequalities? This was a problem in a review from high school set provided by my instructor for my introductory math class. Just curious about a solution, I have no

You are forgetting if you multiply both sides by x and x is negative you have to reverse the inequality. Split into two cases x>0 and x<0.
 


Oh okay, that makes sense, thanks for the help :).
 

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