JEE Novice - (4)
∣ x − 1 ∣ = ∣ x − 2 ∣ + ∣ x − 3 ∣
How many solutions for x does the above equation has if x is a real number?
This question is a part of JEE Novices .
The answer is 2.
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4 solutions
Moderator note:
Simple approach to solve absolute value equations.
Graphical Approach :
Let ∣ x − 1 ∣ = f ( x ) and ∣ x − 2 ∣ + ∣ x − 3 ∣ = g ( x )
No. of real solutions of the equation f ( x ) = g ( x ) will be equal to the no. of points at which the graphs of y = f ( x ) and y = g ( x ) will intersect .
So lets, draw the graphs :
As clear from the graph above, the total number of solutions of the equation
:
(
∣
x
−
1
∣
=
∣
x
−
2
∣
+
∣
x
−
3
∣
)
is
2
enjoy !
Moderator note:
Can you solve this without graphing?
Yeah , drawing graphs is the simplest and time saving way of solving this problem , instead of check various cases by algebra.
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awesome problem
The problems were really amazing, I truly enjoyed them, and for this you recieve C H E E R S ! ! ! from me!
I'm sorry that I'm not able to post solutions as I am confined to a small screen, Keep up the good work and post harder problems! ⌣ ¨
P.S: Honestly I think I talked more than my age requirements!
how did you make the g(x) graph
There's a very simple approach by dividing the real number line into regions, namely,
x ≤ 1 , 1 ≤ x ≤ 2 , 2 ≤ x ≤ 3 , x ≥ 3
For each region, you can transform the given equation into a linear equation in one variable, then solve them and accept the solution if they fall within the region being examined.
The four cases give solutions x = 4 , x = 2 , x = 2 , x = 4 respectively. Matching these solutions with their respective examined regions, we conclude that the only real solutions are x = 2 , 4 .
can you tell me how did you draw its graph??????????
Same approach by me
Moderator note:
Why are you checking for only range of x in 2 < x < 3 and x > 3 ? Are you sure that there's no other solution outside that range?
See Manish Dash's solution for a proper approach.
If your ipothesis is 2<x<3, the solution x=2 is not possible... You're right but with wrong demonstration... :-)
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Oh, you're absolutely right. Thank you Ernesto! :) It was a typo mistake. A better hypothesis could be: " 1<x<3" (from that we notice that "2" is the only integer under that condition, so it works fine with the expression). If 1<x<3 then, |x-1| = x-1 , |x-2| = 0 (since x=2) And |x-3| = -x+3
What is fox theorem
Consider the following cases:
Case 1 If x ≥ 3
x - 1 = (x - 2) + (x - 3)
=> x = 4
Case 2 If 2 ≤ x < 3
x - 1 = (x - 2) - (x - 3)
=> x = 2
Case 3 If 1 ≤ x < 2
x - 1 = - ( x - 2) - ( x - 3)
=> x = 2
Case 4 If x < 1
-(x - 1) = - ( x - 2 ) - ( x - 3 )
=> x = 4
Hence the given equation has 2 solutions: 2 and 4