if-x-3-lt-2-then-prove-that-x-2-9-16-

Question Number 167296 by mkam last updated on 12/Mar/22

i f ∣ x - 3 ∣< 2 t h e n p r o v e t h a t ∣ x^{2} - 9 ∣⩽ 16

Commented by mkam last updated on 12/Mar/22

? ? ? ? ? ?

Commented by cortano1 last updated on 12/Mar/22

\Rightarrow - 2 < x - 3 < 2

\Rightarrow - 2 + 6 < x - 3 + 6 < 2 + 6

\Rightarrow 4 < x + 3 < 8

{\begin{cases} - 2 < x - 3 < 2 \\ 4 < x + 3 < 8 \end{cases} \Rightarrow - 16 < (x - 3) (x + 3) < 16

\Rightarrow - 16 < x^{2} - 9 < 16

\Rightarrow∣ x^{2} - 9 ∣< 16

Commented by MJS_new last updated on 12/Mar/22

I beg your pardon, next time I^{'} ll set my alarm

half an hour earlier so that I can answer your

questions in time .

Answered by MJS_new last updated on 12/Mar/22

∣ x - 3 ∣< 2

\Rightarrow 1 < x < 5

\Rightarrow 1 < x^{2} < 25

\Rightarrow - 8 < x^{2} - 9 < 16

\Rightarrow ∣ x^{2} - 9 ∣< 16

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