The sum of the numerator and the denominator of a fraction is 18 . If 1 is added to both its numerator and denominator, the reduced form of the new fraction so obtained is \( \frac{2}{3} \). Then, the original fraction is ........

aa \( |2 x+3|=|7 x-5| \)

(a) 2023: If \( y \propto x \) and \( y=5 \) when \( x=15 \), Find the value of \( x \) when \( y=3 \) (b) If \( y \) varies inversely with \( x^{2} \) and \( x=3 \) when \( y=4 \), Find: [1] The relation between \( y \) and \( x \) [2] The value of \( x \) when \( y=9 \) (C) If \( a, b, c \) and \( d \) are proportion quantities, then prove that : \( \frac{a}{b+2 c}=\frac{c-a}{d-b} \)

2 Find the solution set in \( R:|2 x-3|+3>7 \)

\( 1 \frac{3}{2} \leqslant x<5 / 2 \) et \( \frac{2}{3} \leqslant y \leqslant \frac{4}{3} \) exarder \( x+y, x-y, x+y, \frac{x-3 y}{x} \)

Esefcizio 10.7 Sia dato \( S \) il sottoinsieme di \( \mathbb{R}[x] \) definito da \[ S=\{f(x) \in \mathbb{R}[x] \mid f(-1)=f(1)=0, \operatorname{deg} f(x) \leqslant 4\} \cup\{0\} \] sottospazio vettoriale di \( \mathbb{R}[x] \). Determinare la dimensione, la codimensione, una base, equazioni cartesiane, equazioni parametriche ed un complemento per \( S \).