Question

if |x-2|<1, prove that |x^2-4|<5​

Answer 1

Step-by-step explanation:

|x-2|<1

then,

⟹ -1<(x-2)<1 ...............(1)

⟹ -1+4<(x-2+4)<1+4

⟹ 3<(x+2)<5 ..............(2)

eqn (1)× eqn (2) =>

⟹ (3)(-1)<(x-2) (x+2)<(5)(1)

⟹ -3<(x-2) (x+2)<5

⟹ -3<(x^2 -4)<5 ............(3)

because,

-3<(x^2 -4)

then also,

-5<(x^2 -4) ........(4)

by eqn (3) & (4) =>

⟹ -5<(x^2-4)<5

then,

⟹ |x^2-4|<5