How to solve the given inequality?

Question

The problem says:

1/x < 2 (Hint: Consider two cases.)

[Answers: x<0 or x>1/2]

I thought that one approach would be to have these equations:

1/x < 2 or 1/x > -2

1 < 2x or 1/x < -2x

1/2 < x or -1/2 > x

Which is not the answer. Can someone please explain?

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I also have a second question...

|x| = x if x ≥ 0

-x if x ≤ 0

Question:: |x| + |x-2| = 2

I decided to do:

x = x-2 = 2 -x+x-2 = 2

2x = 4 0=4

x=2

So the answer is 2..? But that isn't correct either. Can someone explain this one to me as well? Thank you so much!

Answer 1

1)  Well, normally you&#039;d multiply both sides by x to get it out of the denominator, but with an inequality there&#039;s a complication.  If you multiply by a negative, you have to reverse the direction of the inequality, and you don&#039;t know if x is positive or negative.  (At least you know it can&#039;t be 0.)  So we have to work it two ways:

first:  x &gt; 0, then 1 &lt; 2x, 1/2 &lt; x

If x is greater than 0 AND x is greater than 1/2, then x &gt; 1/2

Second:  x &lt; 0, then 1 &gt; 2x, 1/2 &gt; x

If x is less than 0 and x is less than 1/2, then x &lt; 0

So your solution would be x &lt; 0 and x &gt; 1/2.  Basically, all real numbers except those between 0 and 1/2, inclusive.

2)  The absolute value makes things tough, as you can see.  If what&#039;s in the absolute value sign is positive, then you can just drop it because it does nothing.  If what&#039;s in there is negative, then you have to reverse the sign.  Because you have two absolute values, you&#039;re going to have to work the problem three ways:

a)  x &gt;= 0 and x-2 &gt;= 0, in which case your problem becomes just

x + x - 2 = 2

2x = 4

x = 2.  This fits both of our criteria, so it&#039;s a valid solution.

b)  x &gt;= 0 and x - 2 &lt; 0, in which case your problem becomes

x + -(x - 2) = 2

x - x + 2 = 2

2 = 2

This is true regardless of the value of x, so that means that every number that fits our two criteria is a solution.  Our two criteria are x &gt;= 0 and x &lt; 2, so 0 &lt;= x &lt; 2 are all solutions.

c)  x &lt; 0 and x - 2 &lt; 0, in which case your problem becomes

-x - (x - 2) = 2

-x - x + 2 = 2

-2x = 0

x = 0

This does not fit either of our conditions, but it&#039;s already included in the solution to (b), so we&#039;ll ignore it.

Putting the three together, then, the solution set will be 0 &lt;= x &lt;= 2

Any number in that range will work.  Anything outside of it will not.