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10 Nov 2023, 11:02
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C
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Difficulty:
45%
(medium)
Question Stats:
65% (01:35) correct
35%
(01:32)
wrong
based on 504
sessions
History
Date
Time
Result
Not Attempted Yet
How many real numbers \(x\) satisfy \(||x-3|-4|=4\)?
A. 1
B. 2
C. 3
D. 4
E. 5
A. 1
B. 2
C. 3
D. 4
E. 5
10 Nov 2023, 20:01
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Bunuel
OA is C
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General Discussion
10 Nov 2023, 22:23
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The equation ∣∣x−3∣−4∣ = 4 can be broken down into two separate cases based on the definition of absolute value:
1. ∣x−3∣−4 = 4
2. ∣x−3∣−4 = −4
Let's solve each case separately.
Case 1:
∣x−3∣−4 = 4, i.e., ∣x−3∣ = 8
This further breaks down into two sub-cases:
a. x−3 = 8, i.e., x = 11
b. x−3 = −8, i.e., x = −5
Case 2:
∣x−3∣−4 = −4, i.e., ∣x−3∣ = 0, i.e., x−3 = 0 (since 0 can't be positive or negative), i.e., x = 3
Thus, the real numbers x that satisfy ∣∣x−3∣−4∣ = 4 are x = 11, x = −5, and x = 3.
Therefore, there are three real numbers that satisfy the given equation.
1. ∣x−3∣−4 = 4
2. ∣x−3∣−4 = −4
Let's solve each case separately.
Case 1:
∣x−3∣−4 = 4, i.e., ∣x−3∣ = 8
This further breaks down into two sub-cases:
a. x−3 = 8, i.e., x = 11
b. x−3 = −8, i.e., x = −5
Case 2:
∣x−3∣−4 = −4, i.e., ∣x−3∣ = 0, i.e., x−3 = 0 (since 0 can't be positive or negative), i.e., x = 3
Thus, the real numbers x that satisfy ∣∣x−3∣−4∣ = 4 are x = 11, x = −5, and x = 3.
Therefore, there are three real numbers that satisfy the given equation.
