Answer:
(b) (x + 1 ≤ 1) ∩ (x + 1 ≥ 1) = {x | x = 0}
Step-by-step explanation:
Here, the given expression is : {x | x = 0}
So, the ONLY element in the given set = {0}
Now, take each option and solve the given expression:
(a) x + 1 < -1
Adding -1 BOTH sides, we get:
x + 1 -1 < -1 -1
or, x < - 2 ⇒ x = { -∞ , .... , -4,-3}
Also, x + 1 < 1
Adding -1 BOTH sides, we get:
x + 1 -1 < 1 -1
or, x <0 ⇒ x = { -∞ , .... , -4,-3,-2,-1}
So, (x + 1 < -1) ∩ (x + 1 < 1) = { -∞ , .... , -4,-3}∩ { -∞ , .... , -4,-3,-2,-1}
= { -∞ , .... , -4,-3}
⇒ (x + 1 < -1) ∩ (x + 1 < 1) ≠ {0}
Similarly, solving
(b) (x + 1 ≤ 1) ∩ (x + 1 ≥ 1)
(x + 1 ≤ 1) = x≤ 0 = { -∞ , .... , -4,-3,-2,-1, 0}
(x + 1 ≥ 1) = x ≥ 0 = {0,1,2,3,... ∞}
⇒ (x + 1 ≤ 1) ∩ (x + 1 ≥ 1) = {0}
(b)(x + 1 < 1) ∩ (x + 1 > 1)
(x + 1 < 1) = x < 0 = { -∞ , .... , -4,-3,-2,-1}
(x + 1 > 1) = x > 0 = { 1,2,3,... ∞}
⇒ (x + 1 ≤ 1) ∩ (x + 1 ≥ 1) = Ф
Hence, (x + 1 ≤ 1) ∩ (x + 1 ≥ 1) = {x | x = 0}
