Answer
Verified
431.7k+ views
Hint: The above inequality can be solved by solving all the values of x. Hence calculate their magnitudes according to the inequality given and sum up their magnitudes.
Complete step-by-step answer:
Here, it is given \[{\log _{{x^2}}}\left| {x - 1} \right| > 0\]
From the given inequality equation, it is observed that \[{x^2} \ne 1\]that means x is neither equal to 1 nor equal to -1.
\[ \Rightarrow x \ne \pm 1\] Which means \[{x^2} > 1\]
As, \[\left| {x - 1} \right| > 0\] where modulus will open two times: one for positive and other for negative.
So, for positive it will open as \[x - 1 > 0\] which implies \[x > 1\]
And other for negative \[ - \left( {x - 1} \right) > 0\]
On further solving inequality we get, \[ - x + 1 > 0\] which implies \[x < - 1\]
By concluding from all the positive and negative modulus we can say that
\[\left| {x - 1} \right| > 1\]
And hence, by opening this we get, \[ - 1 > x - 1 > 1\]
On further simplifying this equality, \[0 > x > 2\] which means x is neither equal to 1 nor equal to 2.
\[ \Rightarrow x \ne 1,2\]
Hence, the magnitude of x comes out to be -1, 1, and 2.
So, the sum of their magnitudes = 1-1+2 = 2.
Additional information:
The inequality is used for comparing numbers with help of different signs. Example: \[ > , < , \le {\rm{and}} \ge \].These signs help us in finding the numbers which are less than or greater than the variables given in the question. The simple inequalities can be solved by just adding and subtracting the values by taking variables at one side and finding the result on the other side.
Note: In these types of questions we need to solve the inequality where x is present. We can calculate the magnitudes by solving the inequality. At last, after calculating the magnitudes sum up their magnitudes to get the desired result.
Complete step-by-step answer:
Here, it is given \[{\log _{{x^2}}}\left| {x - 1} \right| > 0\]
From the given inequality equation, it is observed that \[{x^2} \ne 1\]that means x is neither equal to 1 nor equal to -1.
\[ \Rightarrow x \ne \pm 1\] Which means \[{x^2} > 1\]
As, \[\left| {x - 1} \right| > 0\] where modulus will open two times: one for positive and other for negative.
So, for positive it will open as \[x - 1 > 0\] which implies \[x > 1\]
And other for negative \[ - \left( {x - 1} \right) > 0\]
On further solving inequality we get, \[ - x + 1 > 0\] which implies \[x < - 1\]
By concluding from all the positive and negative modulus we can say that
\[\left| {x - 1} \right| > 1\]
And hence, by opening this we get, \[ - 1 > x - 1 > 1\]
On further simplifying this equality, \[0 > x > 2\] which means x is neither equal to 1 nor equal to 2.
\[ \Rightarrow x \ne 1,2\]
Hence, the magnitude of x comes out to be -1, 1, and 2.
So, the sum of their magnitudes = 1-1+2 = 2.
Additional information:
The inequality is used for comparing numbers with help of different signs. Example: \[ > , < , \le {\rm{and}} \ge \].These signs help us in finding the numbers which are less than or greater than the variables given in the question. The simple inequalities can be solved by just adding and subtracting the values by taking variables at one side and finding the result on the other side.
Note: In these types of questions we need to solve the inequality where x is present. We can calculate the magnitudes by solving the inequality. At last, after calculating the magnitudes sum up their magnitudes to get the desired result.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE