Question

For
\[\left| {\dfrac{x}{{x + 1}}} \right| < {\left( {10} \right)^{ - 5}}\] hold if
\[\left( 1 \right)\] \[ - {(10)^{ - 5}} < x + 1 < {(10)^{ - 4}}\]
\[\left( 2 \right)\] \[ - {\left( {100001} \right)^{ - 1}} < x < {\left( {99999} \right)^{ - 1}}\]
\[\left( 3 \right)\] \[\dfrac{1}{{10000}} < x < 1\]
\[\left( 4 \right)\] \[{(99999)^{ - 1}} < x < {(100001)^{ - 1}}\]

Answer 1

Hint: We have to find the value of \[x\] form the given expression \[\left| {\dfrac{x}{{x + 1}}} \right| < {\left( {10} \right)^{ - 5}}\] . We solve this question using the concept of solving the linear equations , the concept of linear inequality and the concept of splitting of the modulus function . First we will simplify the left hand side of the given expression such that we will make the expression in terms of \[x\] only by simplifying the terms . For that we will add \[ \pm 1\] in the numerator of the given expression and then we will solve the expression by taking the denominator separately under the terms of the numerator . Then splitting the modulus and changing the sign of the inequality , and hence we would compute the range of \[x\] for the given expression .

Complete answer: Given :
\[\left| {\dfrac{x}{{x + 1}}} \right| < {\left( {10} \right)^{ - 5}}\]

Now , on adding \[ \pm 1\] in the numerator , we get the expression as :
\[\left| {\dfrac{{x + 1 - 1}}{{x + 1}}} \right| < {\left( {10} \right)^{ - 5}}\]

Now , we know that on splitting the modulus function , we add \[ \pm \] to the given expression . Also , we know that as the given expression is of linear inequality , we would also obtain a linear inequality after splitting the modulus function .

So , on splitting the modulus function , we get the expression as :
\[ - {\left( {10} \right)^{ - 5}} < \dfrac{{x + 1 - 1}}{{x + 1}} < {\left( {10} \right)^{ - 5}}\]

Now , taking the denominator separately under the numerator , we get the expression as :
\[ - {\left( {10} \right)^{ - 5}} < \dfrac{{x + 1}}{{x + 1}} - \dfrac{1}{{x + 1}} < {\left( {10} \right)^{ - 5}}\]

On further simplifying , we get the above expression as :
\[ - {\left( {10} \right)^{ - 5}} < 1 - \dfrac{1}{{x + 1}} < {\left( {10} \right)^{ - 5}}\]

Subtracting \[1\] from the inequality , we get the expression as :
\[ - 1 - {\left( {10} \right)^{ - 5}} < \dfrac{{ - 1}}{{x + 1}} < {\left( {10} \right)^{ - 5}} - 1\]

As , we know that when we change the sign of the initial condition of the inequality the signs of the inequality also changes .

Using this property of inequality .
Now multiplying the inequality by \[\left( { - 1} \right)\] , we get the expression as :
\[1 - {\left( {10} \right)^{ - 5}} < \dfrac{1}{{x + 1}} < {\left( {10} \right)^{ - 5}} + 1\]
\[1 - \dfrac{1}{{{{\left( {10} \right)}^5}}} < \dfrac{1}{{x + 1}} < \dfrac{1}{{{{\left( {10} \right)}^5}}} + 1\]

On further simplifying the expression , we get
\[\dfrac{{99999}}{{{{\left( {10} \right)}^5}}} < \dfrac{1}{{x + 1}} < \dfrac{{100001}}{{{{\left( {10} \right)}^5}}}\]

Again using the property of the inequality of change in sign .

Taking the reciprocal of the inequality , we get the expression as :
\[\dfrac{{{{\left( {10} \right)}^5}}}{{100001}} < x + 1 < \dfrac{{{{\left( {10} \right)}^5}}}{{99999}}\]

Subtracting \[1\] from the inequality , we get the expression as :
\[\dfrac{{ - 1}}{{100001}} < x < \dfrac{1}{{99999}}\]

We also , know that the inverse of a function can be represented as :
\[\dfrac{1}{a} = {a^{ - 1}}\]

Using this formula , we get the value of x as :
\[ - {\left( {100001} \right)^{ - 1}} < x < {\left( {99999} \right)^{ - 1}}\]

Hence , the value of x for the given expression \[\left| {\dfrac{x}{{x + 1}}} \right| < {\left( {10} \right)^{ - 5}}\] is \[ - {\left( {100001} \right)^{ - 1}} < x < {\left( {99999} \right)^{ - 1}}\] .
Thus , the correct option is \[\left( 2 \right)\] .

Note:
Modulus function : It is a function which always gives a positive value when applied to a function irrespective of the values of the function . The graph of a modulus function is a V shaped graph where the tip is the point of contact on the graph . We add \[ \pm \] for removing the modulus function as we don’t know the value was taken as negative or positive , so to remove errors while solving we add \[ \pm \] sign and solve it for two cases separately .
Example : The value of a mod function is as given below
\[\left| { - 1} \right| = 1\]
\[\left| 1 \right| = 1\]
We get the value as \[1\] for both \[ + 1\] or \[ - 1\] .