Question 1: Solve the following inequation and graph the solution on a number line $2x- 5 \leq 5x+4 < 11$, where $x \in I$.     [2011]

$2x-5 \leq 5x+4 < 11$

$2x-5 \leq 5x+4$ or $-9 \leq 3x$  or $-3 \leq x$

$5x+4 < 11$  or $5x < 7$  or  $x < \frac{7}{5}$

$-3 \leq x <\frac{7}{5}$

Therefore $x \in \{-3, -2, -1, 0, 1 \}$

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Question 2: Given that $x \in I$, solve the inequation and graph it on a number line: $3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2$.     [2004]

$3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2$

$18 \geq 3(x-4)+2x \geq 12$

$30 \geq 5x \geq 24$

$6 \geq x \geq 4.8$

Therefore $x \in \{5, 6 \}$

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Question 3: Given $A = \{x: 11x-5 > 7x + 3, x \in R \}$$B = \{x: 18x-9 \geq 15+12x , x \in R \}$. Find the range of the set $A \cap B$ and represent it on a number line.      [2005]

$A: 11x-5 > 7x+3$

$4x >8$  or  $x >2$

$B: 18x-9 \geq 15+12x$

$6x \geq 24$  or  $x \geq 4$

$A \cap B = \{ x: x \geq 4, x \in R \}$

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Question 4: Solve the given inequation and graph it on a number line: $2y-3 < y+1 \leq 4y+7, y \in R$.     [2008]

$2y-3 < y+1 \leq 4y+7$

$2y-3 < y+1$  or  $y < 4$

$y+1 \leq 4y+7$  or  $-6 \leq 3y$  or  $-2 \leq y$

Hence $\{ x: -2 \leq y < 4, x \in R \}$

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Question 5: Solve the given inequation and graph it on a number line: $-3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}, x \in R$.     [2010]

$-3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}$

$-3 < -\frac{1}{2}-\frac{2x}{3}$

$-18 < -3 -4x$

$4x < 15$  or  $x < \frac{15}{4}$

$-\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}$  or  $-3-4x \leq 5$

$-8 \leq 4x$  or  $-2 \leq x$

Therefore $\{ x: -2 \leq x < \frac{15}{4}, x \in R \}$

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Question 6: Solve the given inequation and graph it on a number line: $4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x, x \in R$.     [2012]

$4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x$

$4x-19 < \frac{3x}{5}-2$  or  $20x-95 < 3x-10$  or   $17x < 85$  or   $x < 5$

$\frac{3x}{5}-2 \leq -\frac{2}{5}+x$  or   $3x-10 \leq -2 +5x$  or  $-8 \leq 2x$  or  $-4 \leq x$

Therefore $\{x : -4 \leq x < 5, x \in R \}$

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Question 7: Solve the given inequation and graph it on a number line: $-\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} <\frac{1}{6}.x \in R$.     [2013]

$-\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} < \frac{1}{6}$

$-\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} < \frac{1}{6}$

$-2x \leq 3x-8 < 1$  or $-2x \leq 3x-8$  or  $8 \leq 5x$

$\frac{8}{5} \leq x$  or  $3x-8 < 1$  or  $3x < 9$  or   $x < 3$

Therefore $\{ x : \frac{8}{5} \leq x < 3, x \in R \}$

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Question 8: Find the value of $x$ which satisfies the inequation: $-2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2, x \in W$.     [2014]

$-2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2$

$-\frac{17}{6} < \frac{1}{2} -\frac{2x}{3} \leq 2$

$-17 < 3-4x \leq 12$  or $-17 < 3-4x$  or $4x < 20$  or $x < 5$

$3-4x \leq 12$  or $-9 \leq 4x$  or $-2.25 \leq x$

Therefore $\{x : -2.25 \leq x < 5, x \in W \}$

$x \in \{0, 1, 2, 3, 4\}$

Question 9: Solve the inequation: $3-2x \geq x-12$  given that $x \in N$  [1987]

$3-2x \geq x-12$

$\Rightarrow 3x \leq 15$

$\Rightarrow x \leq 5 \ or \ x \in \{1, 2, 3, 4, 5 \}$

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Question 10: Solve the inequation: $12+1\frac{5}{6}x \leq 5+3x$  and $x \in R$.    [1999]

$12+1\frac{5}{6}x \leq 5+3x$
$\Rightarrow 12+\frac{11}{6}x \leq 5+3x$
$\Rightarrow 7 \leq \frac{7}{6}x$
$\Rightarrow x \geq 6 \ or\ \{x: x\in R \ and \ x \geq 6 \}$
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