Which of the following integers is in the solution set of [tex]\left|1 - 3x\right| < 5[/tex]?

I. -1
II. 1
III. 2

∣1 − 3 x ∣ < 5 x = − 1 ∣1 − 3 ⋅ ( − 1 ) ∣ < 5 ∣1 + 3∣ < 5 ∣4∣ < 5 4 < 5 − t h e t r u e
x = 1 ∣1 − 3 ⋅ 1∣ < 5 ∣1 − 3∣ < 5 ∣ − 2∣ < 5 2 < 5 − t h e t r u e
x = 2 ∣1 − 3 ⋅ 2∣ < 5 ∣1 − 6∣ < 5 ∣ − 5∣ < 5 5 < 5 − t h e f a l se
-5\\\\-3x < 5-1\ \wedge\ -3x > -5-1\\\\-3x < 4\ /:(-3)\ \wedge\ -3x > -6\ /:(-3)\\\\x > -\frac{4}{3}\ \wedge\ x < 2\\\\x\in(-\frac{4}{3};\ 2)\\\\I\ x=-1\in(-\frac{4}{3};\ 2)\\\\II\ x=1\in(-\frac{4}{3};\ 2)\\\\III\ x=2\notin(-\frac{4}{3};\ 2)"> ∣1 − 3 x ∣ < 5 1 − 3 x < 5 ∧ 1 − 3 x > − 5 − 3 x < 5 − 1 ∧ − 3 x > − 5 − 1 − 3 x < 4 / : ( − 3 ) ∧ − 3 x > − 6 / : ( − 3 ) x > − 3 4 ​ ∧ x < 2 x ∈ ( − 3 4 ​ ; 2 ) I x = − 1 ∈ ( − 3 4 ​ ; 2 ) II x = 1 ∈ ( − 3 4 ​ ; 2 ) III x = 2 ∈ / ( − 3 4 ​ ; 2 )

Answered by Anonymous | 2024-06-10

The integers -1 and 1 are in the solution set for the inequality ∣1 − 3 x ∣ < 5 , while 2 is not. Thus, the correct options are I (-1) and II (1). Option III (2) does not satisfy the conditions of the inequality.
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Answered by Anonymous | 2024-12-23