# How do you solve an inequality that has a fraction 1/x+3< 0

### How do you solve an inequality that has a fraction 1/x+3< 0

Hi there,
i was hoping someone would be able to help me with an inequality that i have come across.
1/x+3< 0
Any help would be appreciated as I cannot find out how to solve this anywhere on the internet.

thank you.
Guest

### Re: how do you solve an inequality that has a fraction < 0?

Is this the equation?

$$\frac1x + 3< 0$$
Guest

yes
Guest

### Re: How do you solve an inequality that has a fraction 1/x+3

$$\frac{1 + 3x}{x} < 0$$

First we solve $$1+3x = 0$$
$$3x = -1$$
$$x = \frac{-1}{3}$$

The intervals are $$(-\infty; -\frac{1}{3}), (-\frac{1}{3}, 0)$$ and $$(0, +\infty)$$

Now we choose a random number from the last interval let it be for example 1
the equation
becomes $$\frac{1}{1}+3 > 0$$

So numbers in the intervals $$(0, +\infty)$$ and $$(-\infty; -\frac{1}{3})$$ are positive and
solutions are numbers within the interval

$$(-\frac{1}{3}, 0)$$
Guest

Is it clear?
Guest

### Re: How do you solve an inequality that has a fraction 1/x+3

Another way: The inequality is $\frac{1}{x}+ 3< 0$. Subtracting 3 from both sides, $\frac{1}{x}< -3$.

Now, an important way solving inequalities differs from solving equalities: Multiplying both sides of an inequality by a negative number reverses the direction of the inequality. Multiplying both sides of an inequality by a positive number does not change the direction and, of course, x cannot be 0 here.
So we consider two possibilities. If x> 0, then -3x< 0 so multiplying both sides of $\frac{1}{x}< -3$ by -3x gives $-3> x$ which is the same as $x< -3$. But then x is not positive so there is no solution here. If x< 0 then -3x> 0 so multiplying both sides of $\frac{1}{x}< -3$ bu -3x gives $-3< x$. Since we must have x< 0 for this solution so we must have $-3< x< 0$.

The solution to the inequality is "all x such that -3< x< 0".
HallsofIvy

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