List the negative integral values of x that render the algebraic expre

Solution

Hey Everyone,

Please find below the solution of the given problem.

Rewriting the inequality to easily identify the zero points

We want the given algebraic expression to be positive.

So, we need to find the range of values of x for the following inequality:

\((x – 5)^8 (x – 2)^3>0\)


After we find the range, we need to choose the negative integers in this range since the question asks for negative integral values that satisfy the above inequality.


Plotting the zero points and drawing the wavy line:



Required Range:

Per the wavy line drawn above, it is clear that 2 < x < 5 and x > 5 render the given expression positive.

However, question asks us to choose only negative integral values for x.
Note that these ranges do NOT contain any negative integers.

Therefore, no negative integral value exists for x such that the given expression can be rendered positive.

Correct Answer: Option E
_________________

Hi Abhimahna,

For x=1, the expression is negative whereas we want the resultant expression to be positive.

In my opinion, there are no negative integral solutions for this expression to result in a positive number.

Please cross-check by plugging in negative values (x=-1,-2 and so on)

Question States we need to find out the value of negative integral values of x such that \((x – 5)^8 (x – 2)^3\) >0

Since the power of (x-5) is EVEN, we don't need to consider the negative values between 2 and 5.

Substituting the values of x as 2 and 5 on number line, we will get the range of x as (-infinity, 2).

Please correct me if I am missing anything.
_________________

Yes, you are correct. I missed such a simple thing. Thanks Jurassic Park

Your 1st Kudo will be given by me.
_________________

Zero points are 2 and 5. Plot them on the number line.

Start from x=5 (top right). Power of (x-5) is even. So, the wavy line should bounce back at x=5.

From the graph, for any number > 2, the algebraic expression remains positive. So, solution would be all integers >2 (not = 2 since at 2, the expression becomes 0)

Hi Rajaram,

The question asks for negative integral values of x. Hence, the solution of all integers > 2 will be incorrect in this case.

Payal : can u plz elaborate on the negative integral values for an expression in general

Those negative integers (like -2, -3, -1..etc) for which the expression holds true. In this case, for all negative integers the expression is less than zero. Hence, there is no -ve integral / integer value for which expression is > 0 i.e positive.

Product of 2 terms is > 0 when both are +ve or both are -ve. In the given expression, first term is always +ve because of even power. That means second term also needs to be +ve. Therefore, x needs to be +ve since power of the 2nd term is odd. So, we can directly conclude, that no negative value of x will give is positive value for the given inequality.

List the negative integral values of x that render the algebraic expression \((x – 5)^8 (x – 2)^3\) positive.

(x-5)^8(x-2)^3 > 0
Since (x-5)^8(x-2)^2>=0
x-2>0
x>2

IMO E

I did not solve it graphically. Given, ((x–5)^8)*((x–2)^3) is positive. For every x (not equal to 0), (x–5)^8 is positive. That means, (x–2)^3 also has to be positive so that the product is more than 0. It is possible when x>2. Therefore, no negative value of x exists. Option E is correct.