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

Question 1 : Represent the following inequalities on real number lines:

i) $2x-1 < 5$

$2x < 6 \text{ or } x < 3$

ii) $3x+1 \geq -5$

$3x \geq -5 \text{ or } 3x \geq -6 \text{ or } x \geq -2$

iii) $2 (2x-3) \leq 6$

$2x-3 \leq 3 \text{ or } 2x \leq 6 \text{ or } x \leq 3$

iv) $-4 < x < 4$

v) $-2 \leq x < 5$

vi) $8 \geq x > -3$

vii) $-5 < x \leq -1$

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Question 2: For each graph given alongside, write an inequation taking $x$ as the variable:

i) $x \leq -1$

ii) $x \geq 2$

iii) $-4 \leq x < 3$

iv) $5 \geq x > -1$

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Question 3: For the following inequations, graph the solution set on the real number line:

i) $-4 \leq 3x-1 < 8$

$-3 \leq 3x < 9 \text{ or } -1 \leq x < 3$

ii) $x-1 < 3-x \leq 5$

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

$3-x \leq 5 \text{ or } -2 \leq x$

Hence $-2 \leq x < 2$

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Question 4: Represent the solution of each of the following inequalities on a real number line:

i) $4x-1 > x + 11$

$3x > 12 \text{ or } x > 4$

ii) $7-x \leq 2-6x$

$5x \leq -5 \text{ or } x \leq -1$

iii) $x+3 \leq 2x+9$

$-6 \leq x$

iv) $2-3x > 7-5x$

$2x > 5 \text{ or } \displaystyle x > \frac{5}{2}$

v) $1+x \geq 5x - 11$

$12 \geq 4x \text{ or } 3 \geq x$

vi) $\displaystyle \frac{2x+5}{3} > 3x-3$

$2x+5 > 9x-9 \text{ or } 14 > 7x \text{ or } 2 > x$

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Question 5: $x \in \{real \ numbers \} \ and \ -1 < 3-2x \leq 7$ , evaluate $x$ and represent it on a number line.

Answer:

$-1 < 3-2x \leq 7$

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

$3-2x \leq 7 \text{ or } -3+2x \geq -7 \text{ or } 2x \geq -4 \text{ or } x \geq -2$

$2 \leq x < 2$

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Question 6: List the elements of the solution set of the inequation $-3 <x-2 \leq 9-2x, x \in N$ .

Answer:

$-3 <x-2 \leq 9-2x$

$-3 < x-2 \text{ or } -1 < x$

$x-2 \leq 9-2x \text{ or } 3x \leq 11 \text{ or } \displaystyle x \leq x \frac{11}{3}$

Hence $x \in \{1, 2, 3 \}$

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Question 7: Find the range of values of $x$ which satisfies $\displaystyle -2 \frac{2}{3} \leq x+ \frac{1}{3} < 3 \frac{1}{3} , \ x \in R$

Answer:

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

$\displaystyle - \frac{8}{3} \leq x+ \frac{1}{3} < \frac{10}{3}$

$-8 \leq 3x+1 < 10$

$-8 \leq 3x+1 \text{ or } -9 \leq 3x \text{ or } -3 \leq x$

$3x+1 < 10 \text{ or } 3x < 9 \text{ or } x < 3$

Therefore $-3 \leq x < 3$

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Question 8: Find the range of values of $x$ which satisfies $\displaystyle -2 \leq \frac{1}{2} - \frac{2x}{3} \leq 1 \frac{5}{6} , \ x \in N$ . Graph the solution on a number line.

Answer:

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

$\displaystyle -2 \leq \frac{1}{2} - \frac{2x}{3} \leq \frac{11}{6}$

$\displaystyle -12 \leq 3-4x \leq 11 \text{ or } -15 \leq -4x \text{ or } -4x \leq 8$

$4x \leq 15 \text{ or } 4x \geq -8 \text{ or } x \geq -2$

Therefore

$\displaystyle -2 \leq x \leq \frac{15}{4}$

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Question 9: Given $x \in \{ real \ number \}$ , find the range of the values of $x$ for which $-5 \leq 2x-3 < x+2$ and represent it on number line.

Answer:

$-5 \leq 2x-3 < x+2$

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

$2x-3 < x+2 \text{ or } x < 5$

Therefore $-1 \leq x < 5$

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Question 10: If $5x-3 \leq 5+3x \leq 4x+2$ , express it as $a \leq x \leq b$ and state the values of $a \ and \ b$ .

Answer:

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

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

$5+3x \leq 4x+2 \text{ or } 3 \leq x$

Therefore $3 \leq x \leq 4$

Hence $a = 3 \ and \ b = 4$

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Question 11: Solve the following inequation and graph the solution on a number line: $2x-3 < x+2 \leq 3x+5; x \in R$

Answer:

$2x-3 < x+2 \leq 3x+5$

$2x-3 < x+2 \text{ or } x < 5$

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

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

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Question 12: Solve and graph the solution set of the following:

Answer:

i) $2x-9 < 7 \ and \ 3x+9 \leq 25; x \in R$

$2x-9 < 7 \text{ or } 2x < 16 \text{ or } x < 8$

$\displaystyle 3x+9 \leq 25 \text{ or } 3x \leq 16 \text{ or } x \leq \frac{16}{3} = 5 \frac{1}{3}$

Therefore $\displaystyle x \leq 5 \frac{1}{3}$

ii) $2x-9 \leq 7 \ and \ 3x+9 > 25; x \in I$

$2x-9 \leq 7 \text{ or } 2x \leq 16 \text{ or } x \leq 8$

$3x+9 > 25 \text{ or } 3x > 16 \text{ or } \displaystyle x >5 \frac{1}{3}$

iii) $x+5 \geq 4(x-1) \ and \ 3-2x <-7; x \in R$

$x+5 \geq 4(x-1) \text{ or } 9 \geq 3x \text{ or } 3 \geq x$

$3-2x <-7 \text{ or } 2x > 10 \text{ or } x > 5$

Therefore solution set is Empty Set.

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Question 13: Solve and graph the solution set of:

Answer:

i) $3x-2 > 19 \text{ or } 3-2x \geq -7; x \in R$

$3x-2 > 19 \text{ or } 3x > 21 \text{ or } x > 7$

$3-2x \geq -7 \text{ or } 2x \leq 10 \text{ or } x \leq 5$

Therefore $x \leq 5 \text{ or } x > 7$

ii) $5 > p-1 > 2 \text{ or } 7 \leq 2p-1 \leq 17; p \in R$

$5 > p-1 > 2 \text{ or } 6 > p > 3$

$7 \leq 2p-1 \leq 17 \text{ or } 8 \leq 2p \leq 8 \text{ or } 4 \leq p \leq 8$

Therefore $3 < p \leq 8$

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Question 14: Given $A = \{ x \in R: -2 \leq x < 5 \} \ and \ B = \{x \in R: -4 \leq x < 3 \}$ . Represent $A \cap B$ and $A \cap B^{'}$ on two different number lines.