20. x2≥(5x−2)3−(7x−3)5

20. $\frac{x}{2} \geq \frac{(5 x - 2)}{3} - \frac{(7 x - 3)}{5}$

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Payal Gupta | Contributor-Level 10

9 months ago

20. Given,

$\frac{x}{2} \geq \frac{(5 x - 2)}{3} - \frac{(7 x - 3)}{5 .}$

$\Rightarrow \frac{x}{2} \geq \frac{5 (5 x - 2) - 3 (7 x - 3)}{3 * 15}$

15x ≥ 2 [25x –10 –21x + 9].

15x ≥ 2 [4x –1]

15x ≥ 8x –2

15x – 8x ≥ –2

7x ≥ –2

x ≥ $- \frac{2}{7} .$

So, x $[- \frac{2}{7}, \infty)$ ${∴ - \frac{2}{7} = - 0.28}$

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$\begin{array}{l} (i) - 4 x \geq 12 \Rightarrow x \leq - 3 \\ H e n c e, t h e f i l l e r i s (\leq) \\ (i i) I f - \frac{3}{4} x \leq - 3 \Rightarrow x \geq 3 x \times \frac{4}{3} \Rightarrow x \geq 4 \\ H e n c e, t h e f i l l e r i s (\geq) \\ (i i i) I f \frac{2}{x + 2} > 0 \Rightarrow x > - 2 \\ H e n c e, t h e f i l l e r i s (>) \\ (i v) I f x > - 5 \Rightarrow 4 x > - 20 \\ H e n c e, t h e f i l l e r i s (>) \\ (v) I f x > y a n d z < 0, t h e n \\ x z < y z \Rightarrow - x z > - y z \\ H e n c e, t h e f i l l e r i s (>) \\ (v i) I f p > 0 a n d q < 0, t h e n \\ p - q > p \\ H e n c e, t h e f i l l e r i s (>) \\ (v i i) I f | x + 2 | > 5 t h e n \\ x + 2 < - 5 o r x + 2 > 5 \\ \Rightarrow x < - 5 - 2 o r x > 5 - 2 \\ \Rightarrow x < - 7 o r x > 3 \\ S o, x \in (- \infty, - 7) \cup (3, \infty) \\ H e n c e, t h e f i l l e r i s (<) o r (>) \\ (v i i i) I f - 2 x + 1 \geq 9 t h e n \\ \Rightarrow - 2 x \geq 9 - 1 \Rightarrow - 2 x \geq 8 \Rightarrow 2 x \leq - 8 \Rightarrow x \leq - 4 \\ H e n c e, t h e f i l l e r i s (\leq) \end{array}$

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$\begin{array}{l} (i) I f x < y a n d b < 0 \\ \Rightarrow \frac{x}{b} > \frac{y}{b} \\ H e n c e, s t a t e m e n t (i) i s F a l s e . \\ (i i) I f x, y > 0 t h e n x > 0, y > 0 o r x < 0, y < 0 \\ H e n c e, s t a t e m e n t (i i) i s F a l s e . \\ (i i i) I f x y > 0 t h e n x < 0, a n d y < 0 \\ H e n c e, s t a t e m e n t (i i i) i s T r u e . \\ (i v) I f x y < 0 t h e n x < 0, y > 0 o r x > 0, y < 0 \\ H e n c e, s t a t e m e n t (i v) i s F a l s e . \\ (v) I f x < - 5 a n d x < - 2 \Rightarrow x \in (- \infty, - 5) \\ H e n c e, s t a t e m e n t (v) i s T r u e . \\ (v i) I f x < - 5 a n d x > 2 t h e n x h a s n o v a l u e . \\ H e n c e, s t a t e m e n t (v i) i s F a l s e . \\ (v i i) I f x > - 2 a n d x < 9 \Rightarrow x \in (- 2, 9) \\ H e n c e, s t a t e m e n t (v i i) i s T r u e . \\ (v i i i) I f | x | > 5 t h e n x < - 5 o r x > 5 \\ \Rightarrow x \in (- \infty, - 5) \cup (5, \infty) \\ H e n c e, s t a t e m e n t (v i i i) i s F a l s e . \\ (i x) I f | x | \leq 4 t h e n - 4 \leq x \leq 4 \\ \Rightarrow x \in [- 4, 4] \\ H e n c e, s t a t e m e n t (i x) i s T r u e . \\ (x) T h e g i v e n g r a p h r e p r e s e n t s x \leq 3 \\ H e n c e, s t a t e m e n t (x) i s F a l s e . \\ (x i) T h e g i v e n g r a p h r e p r e s e n t s x \geq 0 \\ H e n c e, s t a t e m e n t (x i) i s T r u e . \\ (x i i) T h e g i v e n g r a p h r e p r e s e n t s y \geq 0 \\ H e n c e, s t a t e m e n t (x i i) i s &thins \end{array}$