GATE2013 CE: GA-7

Question

If  $\mid 4X-7 \mid =5$ then the values of $2\mid X\mid -\mid -X\mid$ is:

• A. $\quad 2,\left(\dfrac{1}{3}\right)$
• B. $\left(\dfrac{1}{2}\right),3$
• C. $\left(\dfrac{3}{2}\right),9$
• D. $\left(\dfrac{2}{3}\right),9$

Comments

We are getting x=1/2,3.both are positive

|X|= X, when X>0

         =- X when X<0

Similarly ,|-X| =-X when X <0

                     =-(-X) when X>0

2|X|-|X|=2X-(-(-X))=2X-X=X

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Answer here!   

Answer 1

$\mid 4X-7\mid = 5$  ,since it is in absolute form the regular form of this equation is given as

$(4X-7)= 5$

$(4X-7)=-5$

By solving the above equations we get the following answer

$X=\frac{1}{2}, X = 3$

Now $2\mid X\mid -\mid -X\mid = 2\mid X \mid  - \mid X| = \mid X \mid$.

So, our answer will be $B.$

Comments

4x-7= 5     if 4x-7 >0   here we get x=3

4x-7= -5     if 4x-7 <=0 here we get x =0.5

then they ask about  2|X| - |-X| 

        = 2 |X| - |X|

        = |X|  then           3          if x>0

                                    0.5      if x<=0      so B is anwser   @arjun , @arjun plz verify it

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Here why are giving "values" shouldn't it be ((3/2) when x=1/2   & (9) when x=3

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$|x| = \begin{cases}
    x, & \text{if } x \geq 0 \\
    -x, & \text{if } x < 0
\end{cases}$

$|-x| = \begin{cases}
    -(-x) = x, & \text{if } x \geq 0 \\
    -x, & \text{if } x < 0
\end{cases}$

$2|X| - |-X|$

Case 1: $X \geq 0$

$2|X| - |-X| = 2|X| - X = 2|X| - |X| = |X|$

Case 2: $X < 0$

$2|X| - |-X| = 2|X| - (-X) = 2|X| - |X| = |X|$

Hence,  $2|X| - |-X| = |X|$  in all cases.

Answer 2

4x-7=5  gives x=3

-4x=-2 gives x=1/2

then on substitution,we'll get 3,1/2

Answer 3

Here we are utilizing the defintion of |X| function, and then cross verifying if the solutions are within the assumed domains

If they are then the solutions are accepted else they are rejected.