Is 1/a-b < b-a?
Alright First off : Here is an important rule with absolute values
if |x| < a then it is always true that -a<x<a
if |x| > a then it is always true that x>a & x <-a
Memorize the two important statments above. It will apply to all problems with absolute values.
If you are a curious cat and want to know how we derived it check the link belowhttp://www.purplemath.com/modules/absineq.htm
Now getting to the problem
Is 1/a-b < b-a?
The question asks us if we can answer the above inequality with a resounding Yes or a resounding No.
Lets simplify the question stem first
Rewrite the question stem as (1/a-b) < (b-a/1)
Therefore cross multiplying we get
1/(a-b)(b-a) < 1
Now the new question stem -> Is 1/(a-b)(b-a) <1
Good Now lets look at Statement 1
if a< b --> a-b is -ve and b-a is +ve
If you are not sure plug in
a=2, b=3 implies 2-3(i.e a-b) is -ve and (3-2)(i.e b-a) is +ve
a=-4, b=-3 implies -4-(-3)(i.e a-b) is -ve and -3-(-4) which is b-a is +ve
What do we have in the Denominator of the new question stem we have one a-b term and one b-a term. So when a<b a-b is -ve and b-a is +ve so the product of the two is always -ve.
So in our new question stem the Left hand side is always negative which answers the question that it is < 1.
So statement 1 is good.
Ok now White wash your memory so that you forget statement 1
Lets looks at what we have in statement 2.
Absolute values and inequalities
Remember what we said about absolute values above
So from statement 2 --> a-b > 1 and a-b < -1 . In otherwords we have two ranges of solutions. Lets take them one at a time
a-b>1 implies a>b. So in In our new question stem a-b is +ve and b-a is -ve . Therefore the product of a-b x b-a is -ve . hence the answer is less than 1.
Take a-b< -1 . Lets multiply both sides by -1> remember to flip the sign of the inequality when multiplying by -ve so we have
-(a-b) > 1 which is b-a > 1 which is b> a.
So with this in our new question stem b-a is +ve and a-b is -ve. So the product of a-b x b-a is -ve. Hence the answer is -ve which is less than 1.
Therefore statement 2 is sufficient as well.
So answer choice is
Heman & the master of the universe