Question

If $\left| 2x+3 \right|-\left| 2x-5 \right|=6$ , then the number of values of x are:
A. 1
B. 2
C. 3

Answer 1

Hint: We will break the above equation into 4 parts and with that we will solve each equation by following the given condition for each part and then we will only take that value of x which is different.

Complete step by step answer:
$\left| x \right|$ can be broken as,
x for x > 0.
-x for x < 0.
Now we are going to use the above definition of mod x to solve this question.
Let’s first write the four equations and their conditions,
2x + 3 – 2x + 5 = 6, for 2x + 3>0 and 2x – 5>0
8 = 6, which is not possible so in this case the value of x does not exist.
Now,
$2x+3-\left( -2x+5 \right)$= 6 , for 2x + 3>0 and 2x – 5<0
4x – 2 = 6, for $x>\dfrac{-3}{2}$ and $x<\dfrac{5}{2}$
$x=\dfrac{8}{4}=2$
From this we get x = 2 which satisfies both the above conditions.
Now,
$-\left( 2x+3 \right)-\left( -2x+5 \right)=6$ for 2x + 3<0 and 2x – 5<0
-8 = 6, which is not possible so in this case the value of x does not exist.
Now,
$-\left( 2x+3 \right)-\left( 2x-5 \right)=6$ , for 2x + 3<0 and 2x – 5>0
$-4x+2=6$ , for $x<\dfrac{-3}{2}$ and $x>\dfrac{5}{2}$
There is no value of x which satisfies the above condition.
Hence, from all the four cases only one case has given the value of x while satisfying the condition.
Hence, the correct answer is (a).

Note: Here we used the definition of mod x to divide our given question into four parts and then we have solved each part one by one. While opening the mod one should always be careful with the sign of mod and the conditions of x for which it is valid.