Which of the following has a solution set of x?

(x + 1 < - 1) ∩ (x + 1 < 1)

(x + 1 ≤ 1) ∩ (x + 1 ≥ 1)

(x + 1 1)

(b) (x + 1 ≤ 1) ∩ (x + 1 ≥ 1) = x

Step-by-step explanation:

Here, the given expression is : 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 < 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 + 1 ≤ 1) ∩ (x + 1 ≥ 1) = x