Question

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Answer 1

$\: \: \: \: \: \: \: Question \: \: 1 \: a$




$\dfrac{x}{2} + 2\geqslant x - 1$


add 1 on both sides ,


$\dfrac{x}{2} + 2 - 1 \geqslant x - 1 +1 \\ \\ \\ \frac{x}{2} + 1 \geqslant x$


adding - x / 2 from sides,


$\dfrac{x}{2} - \dfrac{x}{2} + 1 \geqslant x - \dfrac{x}{2} \\ \\ \\ 1 \geqslant \frac{2x - x}{2}$


Multiply by 2 on both sides,


$1 \times 2 \geqslant \dfrac{x}{2} \times 2 \\ \\ \\ 2 \geqslant x$



$\textbf{ Therefore,solution set is }[......0 , 1 , 2 ]$


But in the question x € N



$\textbf{So, Solution set is }[ 1 , 2 ]$





$\: \: \: \: \: \: \: \: \: Question \: \: 1 \: \: b$



$2(4 - 3x) \leqslant 4(x - 5) \\ \\ \\ 8 - 6x \leqslant 4x - 20$


Adding 6x and 20 on both sides,


$8 - 6x + 6x + 20 \leqslant 4x - 20 + 6x + 20 \\ \\ 8 + 20 \leqslant 10x \\ \\ 28 \leqslant 10x$


Divide by 10 on both sides,


$\frac{28}{10} \leqslant \frac{10x}{10} \\ \\ \\ 2.8 \leqslant x$


$\textbf{ Therefore solution set is} [ 3 , 4 , 5 ...... ]$




$\: \: \: \: \: \: \: \: Question \: \: 2$



$5(x - 2) > 9x - 3(2x - 4) \\ \\ 5x - 10 > 9x - 6x + 12 \\ \\ 5x -10 > 3x + 12$


Adding - 3x and 10 on both


$5x - 3x - 10 + 10 > 3x - 3x + 12 + 10 \\ \\ 2x > 22$


divide by 2 on both sides,


$\dfrac{2x}{2} > \dfrac{22}{2} \\ \\ \\ x > 1 1$



$\textbf{ Therefore solution set is} [ 1 , 2 , 3 ....... 9 , 10 ]$




Solution is represented graphically in the attached
document .



$\: \: \: \: \: \: \: Question \: \: 3 \: \: a$



$4 < - (x + 2) < 9$


solution the equation in two parts ,


$4 < - (x + 2) \: \: \: \: \: \: \: \: | \: \: \: \: \: \: - (x + 2) < 9 \\ \\ 4 < - x - 2 \: \: \: \: \: \: \: \: \: \: \: | \: \: \: \: \: \: \: - x - 2 < 9 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: | \\ \\ add \: x \: and \: 2 \: on \: \: \: \: | \: \: add \: x \: and \: 2 \: on \: both \: sides \: \\ both \: sides \: \\ \\ 4 + x + 2 < 0 \: \: \: \: \: |0 < 9 + 2 + x \\ \\ \: \: \: 6 + x < 0 \: \: \: \: \: \: \: \: \: | 0 < 11 + x \\ \\ x < - 6 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: | - 11 < x$

- 11 < x < - 6



$\textbf{ Therefore solution set is [ - 10 , - 9 , - 8 , - 7 ]}$





$\: \: \: \: \: \: \: \: Question \: \: 3 \: \: b$


$- 4 \leqslant - 2(x + 8) < 8 \\ \\ - \dfrac{ 4}{2} \leqslant \dfrac{- 2(x + 8)}{2} < \dfrac{8}{2}$


solving the equation in two parts,


- 2 ≤ - x - 8 And - x - 8 < 4

x ≤ - 6 And - 12 < x



$\textbf{ therefore solution set is} [ - 6 , - 7 , - 8 , - 9 , - 10 , - 11 ]$







On Finding any mistake, comment immediately.