Inequality problem, can someone help me understand?

Question

This is a 3 part problem..

a.)suppose A and B are positive numbers and a 1/a > 1/b or 1/a<1/b? Explain how you know based on explanation of the transformations that produce equivalent inequalities.

B.) Suppose B is a positive number and A is a negative number. Answer the same question in part (a) and give a similiar explanation.

C.)Suppose both A and B are negative. answer the same question and explain.

Answer 1

For all of these examples, a and b must be whole numbers. They cannot be fractions (or rational numbers).

A) for positive rational numbers, if the numerator stays the same, the number will become smaller and smaller as the denominator gets larger and larger. in other words:

1/1>1/2>1/3>1/4>1/5>1/6...see?

So for A) 1/a > 1/b since a
B) Negative numbers are always smaller than positive numbers.

1/a<1/b

C) For negative rational numbers the relationship in A) is reversed. Since negative numbers get smaller as the absolute value gets larger. In other words:

-1>-2>-3>-4... and -1/1<-1/2<-1/3<-1/4...

you can also realise that as negative ratios get smaller they get closer to zero, which is closer to being positive, therefore they must be getting larger.

So for C) 1/a<1/b

Answer 2

look at a number line when you think of this .....

when b is farther away from zero than a ( B > A)... the reciprocal (1/b,1/a) is "flipped" so the 1/a is farther away from zero than 1/b

ex .... if b=4 and a=2 ... then 1/4 < 1/2 (more denominator... more pieces ... less value

when a=(-) and b=(+) then the reciprocals come together next to zero, but the negative stays on the negative side and the same for positive....so the positive will always be larger then the negative

when a and b are both negative, its the same as condition 1... they come together between 0 and -1.... with the smaller number being the larger valve ...
-1/4 > - 1/2 ( closer to the zero is larger )...

*** if you take an inequality and divide/ multiply by a negative... you must reverse he sign of the inequality ***
(-1) -1/4 > -1/2 = 1/4 < 1/2 ( this is true)


hope this helps

Answer 3

Here is a general way of looking at this, you will have to see if you can answer yourself though...

If a>b
Then 1/a < 1/b (always)

for example: 4>2
and 1/4 < 1/2

If both A and B are negative, and
a>b

Then
1/a > 1/b (always)
example: -8 > -4
and -1/8 > -1/4 (this is the opposite of the above)


If a is a postive number and b is a negative number,
Then a>b

Therefore 1/a >1/b (always)

example: a=6 and b= -10
then 1/6 > -1/10

I hope this helps.
Note, I didnt answer your questions directly!

Answer 4

a) 1/a > 1/b. To see this, start with a
b) Start with a < b. Divide by a, and you get 1 > b/a, because a being negative changes the direction of the inequality. Then divide by b, and you get 1/b > 1/a, which you would expect since 1/b > 0 and 1/a < 0.

c) 1/a > 1/b. Divide a b/a, because a being negative changes the direction of the inequality. Then divide by b, and you get 1/b < 1/a, and the direction of the inequality changes again, since b is also negative.

Answer 5

a)
a then
1/a>1/b
the best way to understand this is with an example:
a=2 b=5,
then
1/2 > 1/5

b) b is positive and a is negative:
0 so a<0 a now
a/b<1,
but 1/b>1/a,
see the example:
a=-4, b=5,
then 1/5> - 1/4

c) a then a/b>1
1/b<1/a
again, an example is very illustrative:
a=-5, b=-3,
then -1/3 < -1/5