Question

5  + x  ≤  2x < x − 2, x ∈ R.

Statement (1): There is no value of x ∈ R that satisfies the given inequation. 

Statement (2): 5 + x − x ≤ 2x − x < x − 2 − x 

⇒ 5 ≤ x < −2

Options

• Both the statements are true.

• Both the statements are false.

• Statement i is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation: 

Given,

Equation: 5 + x ≤ 2x < x − 2

⇒ 5 + x ≤ 2x

⇒ 5 ≤ 2x − x

⇒ 5 ≤ x .......... (1)

And, 2x < x − 2

⇒ 2x − x < −2

⇒ x < −2 .......... (2)

There can be no real number which is greater than or equal to 5 and less than −2 at the same time.

∴ Both the statements are true.