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24 Feb 2024, 13:50
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
33% (03:06) correct
67%
(02:46)
wrong
based on 85
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Date
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Not Attempted Yet
Find the number of Integral values of x that satisfy
\(|x^2 + 3x -1| < 2|x| +5\)
A. 1
B. 3
C. 7
D. 5
E. 0
\(|x^2 + 3x -1| < 2|x| +5\)
A. 1
B. 3
C. 7
D. 5
E. 0
25 Feb 2024, 04:44
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any solution?
25 Feb 2024, 05:27
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TaranjeetKaur
Bit tricky due to multiple possible cases. Will try to keep it short. Option C is correct answer
We have |\(x^{2}\)+3x-1| < 2|x|+5
First, notice that 2|x|+5 is always greater or equal to 5. This is an observation.
Now, we will roughly find out the roots of the quadratic inside that is \(x^{2}\)+3x-1 - which are
a=\(\frac{(-3 + \sqrt{3^{2}+4})}{2}\) and b=\(\frac{(-3 - \sqrt{3^{2}+4})}{2}\)
Which comes out to be \(a\approx{0.3}\) and \(b\approx{-3.3}\)
Case 1:
x=0
put in the euation
-1<5 -> always true. 0 is a solution.
Case 2:
0<x<0.3
LHS < 0 and RHS = 2x+5
so we can write
-\(x^{2}\)-3x+1 < 2x+5
=> \(x^{2}\)+5x+4 > 0
We see that x ∈ (-4,-1)
We can't take any integer in this range because our case says 0<x<0.3
No integrl values in this range
Case 3:
-3.3<x<0
In this range LHS < 0 and RHS = -2x+5
So, -\(x^{2}\)-3x+1 < -2x+5
=> \(x^{2}\)+x+4 > 0
Notice, that this is always positive becasue \(x^{2}\) has positive coefficient and roots are complex.
So, we have 3 values in thios range -3.3<x<0 - which are -1, -2, -3
Case 4:
x<-3.3
Notice the quadratic in this case will have positive values and RHS = -2x +5 [ because x<0]
So, we can write \(x^{2}\)+3x-1 < -2x+5
=> \(x^{2}\) + 5x -6 < 0
=> (x+6)(x-1) < 0
so, here we have 2 more values -> -4,-5
Lase Case
Case 5:
x>0.3
LHS>0 and RHS=2x+5
so, \(x^{2}\)+3x-1 < 2x+5
=>\(x^{2}\)+x-6 < 0
=> (x+3)(x-2) <0
So just 1 value in this range which is 1
Hence, total we have 7 integral values of x which are -5, -4, -3, -2, -1, 0, 1
Option C
Although it looks like a lengthy solution, it can be done within 3 minutes if you have enough practice and clarity of concepts
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