Question

✧ Solve :-
$\\ ➣ \: \: \tt 3 - \frac{x}{4} < 2x - 9 < 12 - \frac{x}{2}, x \in \: Z {}^{ + }$
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Answer 1

✰ Topic ➣ Inequalities.

✯ Your Question :-

• $\\ ➣ \: \: \tt 3 - \frac{x}{4} < 2x - 9 < 12 - \frac{x}{2}, x \in \: Z {}^{ + }$

✯ EXPLANATION :-

❍  Taking up the two inequalities seperately, we get :

ㅤㅤㅤㅤㅤㅤㅤㅤCase 1.

$: \implies \bf \: 3 - \frac{x}{4} < 2x - 9$

$: \implies \tt \:12 - \frac{x}{4} < 2x$

$: \implies \tt \: 12 < 2x + \frac{x}{4}$

$: \implies \tt \:12 < \frac{9x}{4}$

$: \implies \tt \: \frac{48}{9} < x$

$✰\:\: \underline{\boxed{ ➥ \tt \: 5 \frac{1}{3}< x}}$

.°. Solution set of 3-(x/4) < 2x - 9 is A ➠ {6,7,8,9,$...$}.

ㅤㅤㅤㅤㅤㅤㅤㅤCase 2.

$: \implies \bf \: 2x - 9 < 12 - \frac{x}{2}$

$: \implies \tt \:2x < 21 - \frac{x}{2}$

$: \implies \tt \: 2x + \frac{x}{2} < 21$

$: \implies \tt \: \frac{5x}{2} < 21$

$: \implies \tt \: 5x < 42$

$: \implies \tt \: x < \frac{42}{5}$

$✰\:\: \underline{\boxed{➥ \tt \: x < 8 \frac{2}{5}}}$

.°. Solution set of 2x - 9 < 12 - (x/2) is B ➠ {1,2,3,4,5,6,7,8}.

.°. Hence, the solution set for both the given inequations together ➲ A ∩ B ➠ {6,7,8}.

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✯ Hope it helps u ✯