Question 18: $\displaystyle P$ is the solution set of $\displaystyle 7x-2 > 4x+1$ and $\displaystyle Q$ is the solutions et of $\displaystyle 9x-45 \leq 5(x-5)$ ; where $\displaystyle x \in R$ . Represent i) $\displaystyle P \cap Q$ ii) $\displaystyle P-Q$ iii) $\displaystyle P \cap Q^{'}$ on different number lines.

$\displaystyle P: 7x-2 > 4x+1$

$\displaystyle 3x > 3 \text{ or } x > 1$

$\displaystyle Q: 9x-45 \geq 5(x-5)$

$\displaystyle 9x-45 \geq 5x-25 \text{ or } 4x \geq 20 \text{ or } x \geq 5$

$\displaystyle P \cap Q: x \geq 5$

$P-Q: 1 < x < 5$

$P \cap Q^{'}: 1 < x < 5$

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Question 19: If $P = \{x:7x-4 > 5x+2, x \in R \}$ and $Q = \{x : x-19 \geq 1-3x, x \in R \}$ ; find the range of the set P \cap Q and represent it on number line.

$P: 7x-4 > 5x+2 \text{ or } 2x > 6 \text{ or } x > 3$

$Q: x-19 \geq 1-3x \text{ or } 4x \geq 20 \text{ or } x \geq 5$

$P \cap Q: x \geq 5$

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Question 20: Find the range of values of x, which satisfy: $\displaystyle -\frac{1}{3} \leq \frac{x}{2} +1\frac{2}{3} < 5\frac{1}{6}$ Graph in each of the following cases for the values of x for each of the following cases: i)  $x \in W$    ii) $x \in Z$    iii)  $x \in R$

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

$-2 \leq 3x+10 < 31 \text{ or } -12 \leq 3x < 21 \text{ or } -4 \leq x < 7$

i)  $x \in W$   : $0 \leq x < 7$

ii) $x \in Z$ :   $-4 \leq x < 7$

iii)  $x \in R$: $-4 \leq x < 7$

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Question 21: Given $A = \{x: -8 < 5x+2 \leq 17, x \in I \}$$B = \{x: -2 \leq 7+3x < 17, x \in R \}$ where $R$ is real numbers and $I$ is integers. Represent $A \ and \ B$ on two different number lines. Also write down the elements of $A \cap B$.

$A: -8 < 5x+2 \leq 17 \text{ or } -10 <5x \leq 15 \text{ or } -2 < x \leq 3$

$\displaystyle B: -2 \leq 7+3x < 17 \text{ or } -9 \leq 3x < 10 \text{ or } x < \frac{10}{3}$

$A \cap B: -2 \leq x \leq 3$

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Question 22: 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 $-9 \leq 3x \text{ or } -3 \leq x$

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

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

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

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Question 23: Given that $x \in I$, solve the inequation and graph it on a number line: $\displaystyle 3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2$.     [2004]

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

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

$30 \geq 5x \geq 24$

$6 \geq x \geq 4.8$

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

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Question 24: 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]

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

$4x >8 \text{ or } x >2$

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

$6x \geq 24 \text{ or } x \geq 4$

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

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Question 25: Find the set of values of $x$, satisfying: $7x+3 \geq 3x-5$ and $\displaystyle \frac{x}{4}-5 \leq \frac{5}{4}-x$, where $x \in N$.

$7x+3 \geq 3x-5$

$4x \geq -8 \text{ or } x \geq -2$

$\displaystyle \frac{x}{4}-5 \leq \frac{5}{4}-x \text{ or } \frac{5}{4} x \leq \frac{25}{4} \text{ or } x \leq 5$

Therefore $x \in \{1, 2, 3, 4, 5\}$

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Question 26: Solve  i)   $\displaystyle \frac{x}{2} +5 \leq \frac{x}{3}+6$, where $x$ is a positive odd integer. ii) $\displaystyle \frac{2x+3}{3} \geq \frac{3x-1}{4}$, where  $x$ is a positive even integer.

i)   $\displaystyle \frac{x}{2} +5 \leq \frac{x}{3}+6$

$\displaystyle \frac{1}{6} x \leq 1 \text{ or } x \leq 6$

$x \in \{1, 3, 5 \}$

ii) $\displaystyle \frac{2x+3}{3} \geq \frac{3x-1}{4}$

$8x+12 \geq 9x-3 \text{ or } 15 \geq x$

$x \in \{2, 4, 6, 8, 10, 12, 14 \}$

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Question 27: Solve the inequation: $\displaystyle -2\frac{1}{2}+2x \leq \frac{4x}{5} \leq \frac{4}{3}+2x, x \in W$. Graph the solution on a number line.

$\displaystyle -2\frac{1}{2}+2x \leq \frac{4x}{5} \leq \frac{4}{3}+2x$

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

$\displaystyle -\frac{5}{4}+2x \leq \frac{4x}{5}$

$\displaystyle -25+20x \leq 8x \text{ or } 12x \leq 25 \text{ or } x \leq \frac{25}{12}$

$\displaystyle \frac{4x}{5} \leq \frac{4}{3}+2x$

$\displaystyle -\frac{4}{3} \leq 2x-\frac{4x}{5} \text{ or } -\frac{4}{3} \leq \frac{6}{5}x \text{ or } -\frac{20}{18} \leq x$

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

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Question 28: Find three consecutive largest positive integers such that the sum of one third of the first, one fourth of the second and one fifth of the third is 20.

Let the three numbers be $x, (x+1) and (x+2)$

Therefore

$\displaystyle \frac{1}{3}x+\frac{1}{4}(x+1)+\frac{1}{5}(x+2) \leq 20$

$\displaystyle \frac{20+15+12}{60}x+\frac{13}{20} \leq 20$

$47x+39 \leq 1200$

$x \leq 24.70$

Hence $x = 24$. Therefore the numbers are $24, 25, and 26$.

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Question 29: Solve the given inequation and graph it on a number line: $2y-3 < y+1 \leq 4y+7, y \in R$.     [2008]

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

$2y-3 < y+1 \text{ or } y < 4$

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

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

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Question 30: Solve the inequation: $3z-5 \leq z+3 < 5z-9, z \in R$. Graph the solution set on a number line.

$3z-5 \leq z+3 < 5z-9$

$3z-5 \leq z+3 \text{ or } 2z \leq 8$  or   $z \leq 4$

$z+3 < 5z-9 \text{ or } 12 < 4z \text{ or } 3 < z$

Hence $\{ z: 3 < z \leq 4, z \in R \}$

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Question 31: Solve the given inequation and graph it on a number line: $\displaystyle -3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}, x \in R$.     [2010]

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

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

$-18 < -3 -4x$

$\displaystyle 4x < 15 \text{ or } x < \frac{15}{4}$

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

$-8 \leq 4x \text{ or } -2 \leq x$

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

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Question 32: Solve the given inequation and graph it on a number line: $\displaystyle 4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x, x \in R$.     [2012]

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

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

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

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

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Question 33: Solve the given inequation and graph it on a number line: $\displaystyle -\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} <\frac{1}{6}.x \in R$.     [2013]

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

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

$\displaystyle -2x \leq 3x-8 < 1 \text{ or } -2x \leq 3x-8 \text{ or } 8 \leq 5x$

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

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

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Question 34: Find the value of $x$ which satisfies the inequation: $\displaystyle -2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2, x \in W$.     [2014]

$\displaystyle -2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2$
$\displaystyle -\frac{17}{6} < \frac{1}{2} -\frac{2x}{3} \leq 2$
$-17 < 3-4x \leq 12 \text{ or } -17 < 3-4x \text{ or } 4x < 20 \text{ or } x < 5$
$3-4x \leq 12 \text{ or } -9 \leq 4x \text{ or } -2.25 \leq x$
Therefore $\{x : -2.25 \leq x < 5, x \in W \}$
$x \in \{0, 1, 2, 3, 4\}$