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# Help

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Find all $$x$$ such that $$-4<{1\over{x}}<3$$.

Feb 14, 2022

#1
+363
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ok..

i don't know if this is possible..

x can equal 1 to infinity and it works

Feb 15, 2022
#2
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Note that $$1 \over {1 \over3}$$ is 3. When $$x$$ gets bigger, the value of the expression gets smaller.

Likewise, $$1 \over -{1 \over 4}$$ is -4. As $$x$$ gets smaller than $$-{1 \over 4}$$, the value of the equation gets bigger, but never is greater than 0.

So, the 2 ranges that work are $$\color{brown}\boxed{{x<-{1\over4} \space \text {and} \space{x>{1\over 3}}}}$$

BuilderBoi  Feb 15, 2022
edited by BuilderBoi  Feb 15, 2022
#3
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Rewrite  -4  <  1/x  < 3  into its two parts:  -4  <  1/x   and   1/x  <  3

Divide the problem into two parts:

Part 1:  x < 0:

(Remember that when you multiply both sides of an inequality by a negative number,

you must change the sense of the inequality.

Also remember that, for this part, x is negative.)

-4 < 1/x   --->   -4x > 1   --->   x < -1/4

1/x < 3   --->   1 > 3x   --->   3x < 1   --->   x < 1/3

Since these two inequalities are combined by the word and, the solution to this

part is the more restrictive of the two answers:  x < -1/4

Part 2:  x > 0

-4 < 1/x   --->   -4x < 1   --->   x > -1/4

1/x < 3   --->   1 < 3x   --->   3x > 1   --->   x > 1/3

The more restrictive of these two is:  x > 1/3

Since the answer consists of two parts, either Part 1 or Part 2, the final answer is the

region:  x < -1/4  or  x > 1/3

which is:  (-inf, -1/4)  union  (1/3, inf)

Feb 15, 2022