Question

Solve ( x + 1 < 5) ∩ ( x - 4 > -3).

{all real numbers}
x
x < 4 or x > 1

Answer 1

The answer is: all those real numbers x

Explanation:

We have to find the common region for the inequalities:

x+1<5 and x-4>-3 ; where x is a real number.

i.e we need to find the region of ( x + 1 < 5) ∩ ( x - 4 > -3).

let us find the region for : x+1<5

⇒ x<5-1 (subtracting both side by 1)

⇒ x<4

the region is (-∞,4)

in set-builder definition form it could be written as: -∞<x<4

now calculating the region for the second inequality: x-4>-3

⇒ x>-3+4 (Adding 4 on both the sides of the inequality)

⇒ x>1

Hence, the region is (1,∞)

in set-builder definition form it could be written as: 1<x<∞.

So, the common region in (-∞,4) and (1,∞) i.e. (-∞,4)∩(1,∞)=(1,4).

Hence the answer is: all those real numbers x .