Question 30212: Questions:
(x-5) (x+6)/ x-4 <0
Possible Answers:
(a) x<-6 or 4
(b) 4
(c) x>5 or -6
(d) x<-6 or x>-5
Please be specific as to which letter answer is correct. Thanks a bunch!
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! (x-5)(x+6)/(x-4) < 0 ----(1)
Since division by zero is not defined and we have (x-4) in the dr
this means (x-4) is not zero
That is x NOT EQUAL TO 4
Therefore either x > 4 or x < 4
Case 1:Let x >4 (which means (x-4) > 0 )
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Multiplying (1) by (x-4) we get
(x-5)(x+6) < 0 (on the LHS (x-4) cancels and on the right since zero mulitplied by anythings is zero, we get zero there and since multiplication by a positive quantity does not alter the inequality, the less than remains less than)
When two quantities are multiplied and the product is negative that means the quantities are of opposite signs.
That is (x-5) > 0 means (x+6) <0
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which further implies that x > 5 together with x < -6
which is an impossibility
(that which is to the right of 5 cannot be at the same time to the left of 6
(A person after finishing off loans has a deposit 5 million dollars plus can not owe any body 6 millions or more)
So this consideration possibility of the product < 0 fails
Let now (x-5) < 0 which means (x+6) > 0
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which further implies that x < 5 together with x > -6
which makes sense.
So we have -6 < x < 5 ----(*)
But all these results we got as a consequence of assuming x > 4 (which is the most vital information on x based on which we should decide the validity of (*)
So the verdict is 4 < x < 5
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Case 2:Let x < 4 (which means (x-4) < 0 )
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Multiplying (1) by (x-4) we get
(x-5)(x+6) > 0 (on the LHS (x-4) cancels and on the right since zero mulitplied by anythings is zero, we get zero there and since multiplication by a negative quantity alters the inequality, the less than becomes greater than)
When two quantities are multiplied and the product is positive that means the quantities are of the same sign.
That is (x-5) > 0 means (x+6) also > 0
-------------------------------------------
which further implies that x > 5 together with x > -6
And since anything to the right of 5 is automatically to the right of (-6)
the verdict is
x > 5 ----(**)
But this fails as all this result is got as a consequence of assuming x < 4 (which is the most vital information on x based on which we should decide the validity of (**)
When x is to the left of 4 it can not simultaneously be to the right of 5
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Let now (x-5) < 0 which means (x+6) also < 0
-----------------------------------------------
which further implies that x < 5 together with x < -6
And since anything to the left of -6 is automatically to the left of 5 and so
the verdict is
x < -6 ----(***)
which is very much true
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As this result is got as a consequence of assuming x < 4 (which is the most vital information on x based on which we should decide the validity of (***)
When x is to the left of -6 it is automatically to the left of 4
In conclusion: we get 4 < x < 5 and x < -6
Since we always go from the left to right on the number line the conclusion should be presented as
Answer: x < -6 and 4 < x < 5
Note: Here the word 'and' and 'or' are loosely used to convey the same meaning.
DO IT YOURSELF ACTIVITY: Please draw a horizontal line and mark a point O to represent the number zero and mark 1,2,3,... to the right of O and mark
-1,-2,-3,...to the left of O at every case consideration, at every verdict stage and actually visualise. Then the whole thing will be very clear and to keep intact what you have reasoned out and understood, in order for the concept to stay green in your memory try to tackle a problem on inequality every once in a while if the chapter on inequality accounts for a good grade!
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