Question

Please give me an explanation, the answer in the book shows :
5\leq x <7$5\leq x \ \textless \ 7$

Answer 1

Solution :-

⇒x² - 3x ≥ 10

⇒x² - 3x - 10 ≥ 0

⇒x² - 5x + 2x - 10 ≥ 0

⇒x ( x - 5 ) + 2 ( x - 5 ) ≥ 0

⇒( x + 2 ) ( x - 5 ) ≥ 0

Therefore critical points are -2 and 5.

Hence x ∈ ( - ∞, - 2 ] U [ 5 , ∞ ) . . . Say eqⁿ(1)

Now,

⇒( x - 5 )² < 4

⇒x² + 25 - 10 x < 4

⇒x² - 10x - 21 < 0

⇒x² - 7x - 3x - 21 < 0

⇒x ( x - 7 ) - 3 ( x - 7 ) < 0

⇒( x - 3 ) ( x - 7 ) < 0

So the critical points are 3 and 7.

Hence x ∈ ( 3, 7 ) . . . Say eqⁿ(2)

Finding intersection of eqⁿ(1) and eqⁿ(2)

According to eqⁿ(1), x ∈ ( -∞, - 2 ] U [ 5 , ∞ )

According to eqⁿ(2), x ∈ ( 3, 7 )

The common solution set of eqⁿ(1) and eqⁿ(2) is given by,

⇒ x ∈ ( 5, 7)

Therefore the solution is,

⇒ 5 < x < 7