Class 10: Linear Inequations – Exercise 6(bx) – ICSE / ISC / CBSE Mathematics Portal for K12 Students

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.

Answer:

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

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

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

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

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

$P-Q: 1 < x < 5$

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

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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.

Answer:

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

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

$P \cap Q: x \geq 5$

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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) $x \in R$

Answer:

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

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

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 $A \cap B$ .

Answer:

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

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

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

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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]

Answer:

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

$2x-5 \leq 5x+4$ 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]

Answer:

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

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

$30 \geq 5x \geq 24$

$6 \geq x \geq 4.8$

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]

Answer:

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

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

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

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

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

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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 $x \in N$ .

Answer:

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

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

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

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.

Answer:

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.

Answer:

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

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

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

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

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

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

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.

Answer:

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]

Answer:

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

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

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

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.

Answer:

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

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

Answer:

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

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

$-18 < -3 -4x$

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

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

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

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]

Answer:

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

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

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

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]

Answer:

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

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

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