(Each question 2 marks)

Question 12 Marks

Solve the following system of equations in R. 3x - 6 > 0, 2x - 5 > 0

Answer

Consider the first inequation, 3x - 6 > 0 3x > 6 ...(i) Consider the secound inequation, 2x - 5 > 0 2x > 5 $\text{x}>\frac{5}{2}...(\text{ii})$ From (i) and (ii), $\Big[\frac{5}{2},\infty\Big]$ is the solution of the simultaneous equations.

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Question 22 Marks

Solve the following linear inequations in R: -(x - 3) + 4 < 5 - 2x

Answer

-(x - 3) + 4 < 5 - 2x ⇒ - x + 3 + 4 < 5 - 2x ⇒ -x + 7 < 5 - 2x ⇒ -x + 2x < 5 - 7 ⇒ x < -2 $(-\infty,-2)$ is the solution set.

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Question 32 Marks

Solve the following system of equations in R. x - 2 > 0, 3x < 18

Answer

Consider the first inequation, x - 2 > 0 x > 2 ...(i) Consider the secound inequation, 3x < 18 x < 6 ...(ii) From (i) and (ii), (2, 6) is the solution of the simultaneous equations.

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Question 42 Marks

Solve the following system of equations in R.2x - 3 < 7, 2x > -4

Answer

Consider the first inequation, 2x - 3 < 7 2x < 7 + 3 2x < 10 x < 5 ...(i) Consider the secound inequation, 2x > -4 $\text{x}>\frac{-4}{2}$ x > -2 ...(ii) From (i) and (ii), $[-2,5]$ is the solution of the simultaneous equations.

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Question 52 Marks

Solve the following linear inequations in R: $2(3-\text{x})\geq\frac{\text{x}}{5}+4$

Answer

$2(3-\text{x})\geq\frac{\text{x}}{5}+4$ $\Rightarrow6-2\text{x}\geq\frac{\text{x}}{5}+4$ $\Rightarrow-2\text{x}-\frac{\text{x}}{5}\geq4-6$ $\Rightarrow\frac{-11\text{x}}{5}\geq-2$ $\Rightarrow\frac{11\text{x}}{5}\leq2$ $\Rightarrow\text{x}\leq\frac{10}{11}$ $\Big(-\infty,\frac{10}{11}\Big]$ is the solution set.

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Question 62 Marks

Solve the following linear inequations in R: 3x - 7 > x + 1

Answer

3x - 7 > x + 1 ⇒ 3x - x > 1 + 7 ⇒ 2x > 8 $\Rightarrow\text{x}>\frac{8}{2}=4$ ⇒ x > 4 $\therefore(4,\infty)$ is the solution set.

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Question 72 Marks

Solve the following system of equations in R. $\frac{2\text{x}+1}{7\text{x}-1}>5,\frac{\text{x}+7}{\text{x}-8}>2$

Answer

Consider the first inequation, $\frac{2\text{x}+1}{7\text{x}-1}>5$ $\frac{2\text{x}+1}{7\text{x}-1}-5>0$ $\frac{2\text{x}+1-5(7\text{x}-1)}{7\text{x}-1}>0$ 2x + 1 - 35x + 5 > 0 -33x + 6 > 0 -33x > -6 $\text{x}<\frac{6}{33},\text{x}>\frac{1}{7}\ ...(\text{i})$ Consider the second inequation, $\frac{\text{x}+7}{\text{x}-8}>2$ $\frac{\text{x}+7}{\text{x}-8}-2>0$ $\frac{\text{x}+7-2(\text{x}-8)}{\text{x}-8}>0$ $\frac{\text{x}+7-2\text{x}+16}{\text{x}-8}>0$ $\text{x}>8,\text{x}<23\ ..(\text{ii})$ From (i) and (ii), There is no solution set of the simultaneous equations.

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Question 82 Marks

Solve the following system of equations in R. $0<\frac{-\text{x}}{2}<3$

Answer

Consider the first inequation, $\frac{\text{x}}{2}<0$ x < 0 ...(i) Consider the second inequation, $\frac{-\text{x}}{2}<3$ $-\text{x}<6$ $-\text{x}>-6\ ..(\text{ii})$From (i) and (ii), $(-6, 0)$ is the solution set of the simultaneous equations.

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Question 92 Marks

Solve the following system of equations in R. $2\text{x} - 7 > 5 -\text{ x}, 11 - 5\text{x} \leq 1$

Answer

Consider the first inequation, 2x - 7 > 5 - x ⇒ 2x + x > 5 + 7 ⇒ 3x > 12 $\Rightarrow\text{x}>\frac{12}{3}$ ⇒ x > 4 ...(i) Consider the secound inequation, $11\text{x} - 5\text{x} \leq1$ $\Rightarrow-5\text{x}\leq-11$ $\Rightarrow-5\text{x}\leq-10$ $\Rightarrow5\text{x}\geq10$ $\Rightarrow\text{x}\geq2\ ..(\text{ii})$ From (i) and (ii), $(4,\infty)$ is the solution of the simultaneous equations.

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Question 102 Marks

Solve the following system of equations in R. 2(x - 6) < 3x - 7, 11 - 2x < 6 - x

Answer

Consider the first inequation, 2(x - 6) < 3x - 7 ⇒ 2x - 12 < 3x - 7 ⇒ -5 < x Consider the second inequation, 11 - 2x < 6 - x -2x + x < 6 - 11 -x < -5 x > 5 ...(ii) From (i) and (ii), $(5,\infty)$ is the solution set of the simultaneous equations.

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Question 112 Marks

Solve the following system of equations in R. $10\leq-5(\text{x}-2)<20$

Answer

Consider the first inequation, $10\leq-5(\text{x}-2)$ $2\leq-(\text{x}-2)$ $2\leq-\text{x}+2$ $2-2\leq-\text{x}$ $0\leq-\text{x}$ $\text{x}\leq0\ ...(\text{i})$ Consider the second inequation, -5 (x - 2) < 20 -5x + 10 < 20 -5x < 20 - 10 -5x < 10 -x < -2 x > -2From (i) and (ii), $(-2, 0)$ is the solution set of the simultaneous equations.

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Question 122 Marks

Solve the following linear inequations in R: $3\text{x}+9\geq-\text{x}+19$

Answer

$3\text{x}+9\geq-\text{x}+19$ $\Rightarrow3\text{x}+\text{x}\geq19-9$ $\Rightarrow4\text{x}\geq10$ $\Rightarrow\text{x}\geq\frac{10}{4}=\frac{5}{2}$ $\therefore\Big[\frac{5}{2},\infty\Big)$ is the solution set.

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Question 132 Marks

Solve the following linear inequations in R: Solve: -4x > 30, when

$\text{x}\in\text{R}$
$\text{x}\in\text{Z}$
$\text{x}\in\text{N}$

Answer

Now, -4x > 30 $\Rightarrow\text{x}<\frac{-30}{4}=\frac{-15}{2}$

If $\text{x}\in\text{R},$ then $\text{x}<\frac{-15}{2}\Rightarrow\text{x}\in\Big(-\infty,-\frac{15}{2}\Big)$
If $\text{x}\in\text{R},$ then $\text{x}<-\frac{15}{2}\Rightarrow\text{x}\in\{...,-10,-9-8\}$
$-4\text{x}>30$

$\Rightarrow-\text{x}>\frac{30}{4}$
$\Rightarrow\text{x}<-\frac{30}{4}$
As $\text{x}\in\text{R},$ so x cannot be less than 1.
$\therefore$ The solution set of the inequality -4x > 30 is null set $\phi$

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Question 142 Marks

Solve the following linear inequations in R: $\frac{1}{\text{x}-1}\leq2$

Answer

$\frac{1}{\text{x}-1}\leq2$ $\frac{1}{\text{x}-1}-2\leq0$ $\frac{1-2(\text{x}-1)}{\text{x}-1}\leq0$ $\frac{1-2\text{x}+2}{\text{x}-1}\leq0$ $\frac{3-2\text{x}}{\text{x}-1}\leq0$ Case 1: $3-2\text{x}\geq0$ and $\text{x}-1<0$ $\Rightarrow\text{x}\leq\frac{3}{2}$ and $\text{x}<1$ Case 2: $3-2\text{x}\leq0$ and $\text{x}-1>0$ $\Rightarrow\text{x}\geq\frac{3}{2}$ and $\text{x}>1$ Hence the solution set is $(-\infty,1)\cup\Big[\frac{3}{2},\infty\Big)$

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Question 152 Marks

Solve the following system of equations in R. $11-5\text{x}>-4, \ 4\text{x}+13\leq-11$

Answer

Consider the first inequation, 11 - 5x > -4 -5x > -4 - 11 5x > -15 x < 3 ...(i) Consider the second inequation, $4\text{x}+13\leq-11$ $4\text{x}\leq-11-13$ $4\text{x}\leq-24$ $\text{x}\leq-6 \ ...(\text{ii})$ From (i) and (ii), $[-\infty,-6]$ is the solution set of the simultaneous equations.

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Question 162 Marks

Solve the following linear inequations in R: $\frac{\text{x}}{5}<\frac{3\text{x}-2}{4}-\frac{5\text{x}-3}{5}$

Answer

$\frac{\text{x}}{5}<\frac{3\text{x}-2}{4}-\frac{5\text{x}-3}{5}$ $\Rightarrow\frac{\text{x}}{5}<\frac{3\text{x}-2}{4}-\frac{(5\text{x}-3)}{5}$ $\Rightarrow\frac{\text{x}}{5}<\frac{5(3\text{x}-2)-4(5\text{x}-3)}{20}$ $\Rightarrow\text{x}<\frac{15\text{x}-10-20\text{x}+12}{4}$ ⇒ 4x < -5x + 2 ⇒ 4x + 5x < 2 ⇒ 9x < 2 $\Rightarrow\text{x}<\frac{2}{9}$ $\therefore$ The solution set is $\Big(-\infty,\frac{2}{9}\Big)$

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Question 172 Marks

Solve the following system of equations in R. x + 3 > 0, 2x < 14.

Answer

Consider the first inequation, x + 3 > 0 x > -3 ...(i) Consider the secound inequation, 2x < 14 $\text{x}<\frac{14}{2}=7$ x < 7 ...(ii) From (i) and (ii), (-3, -7) is the solution of the simultaneous equations.

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Question 182 Marks

Solve the following linear inequations in R: x + 5 > 4x - 10

Answer

x + 5 > 4x - 10 ⇒ x - 4x > -10 - 5 ⇒ -3x > -15 ⇒ 3x < 15 $\Rightarrow\text{x}<\frac{15}{3}=5$ ⇒ x < 5 $\therefore(-\infty,5)$ is the solution set.

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Question 192 Marks

Solve the following system of equations in R. x + 5 > 2(x + 1), 2 - x < 3(x + 2)

Answer

Consider the first inequation, x + 5 > 2(x + 1), x > 2x + 2 - 5 x > 2x - 3 x - 2 > -3 -x > -3 x < 3 ...(i) Consider the second inequation, 2 - x < 3(x + 2) 2 - x < 3x + 6 -x - 3x < 6 - 2 -4x < 4 x > -1 ...(ii) From (i) and (ii), (-1, 3) is the solution set of the simultaneous equations.

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Question 202 Marks

Solve the following system of equations in R. $3\text{x}-1\geq5, \ \text{x}+2>-1$

Answer

Consider the first inequation, $3\text{x}-1\geq5$ $3\text{x}\geq5 + 1$ $3\text{x}\geq6$ $\text{x}\geq2 \ ...(1)$ Consider the first inequation, x + 2 > -1 x > -1 - 2 x > -3 ...(ii) From (i) and (ii), $[2, \infty]$ is the solution set of the simultaneous equations.

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Question 212 Marks

Solve the following system of equations in R. $4\text{x}-1\leq0, \ 3-4\text{x}>0$

Answer

Consider the first inequation, $4\text{x}-1\leq0$ $4\text{x}>-1$ $-5\text{x}\leq-15$ $\text{x}\leq\frac{1}{4} \ ...(\text{i})$ Consider the second inequation, $3-4\text{x}<0$ $-4\text{x}<-3$ $-\text{x}<\frac{-3}{4}$ $\text{x}<\frac{3}{4} \ ...(\text{ii})$ From (i) and (ii), is the solution set of the simultaneous equations.

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Question 222 Marks

Solve the following linear inequations in R: Solve: 12x < 50, when:

$\text{x}\in\text{R}$
$\text{x}\in\text{Z}$
$\text{x}\in\text{N}$

Answer

Now, 12x < 50 $\Rightarrow\text{x}<\frac{50}{12}=\frac{25}{6}$

Since $\text{x}\in\text{R},\text{x}\in\Big(-\infty,\frac{25}{6}\Big)$
Since $\text{x}\in\text{z},\text{x}\in\{...,-3,-2,-1,0,1,2,3,4\}$
Since $\text{x}\in\text{N},\text{x}\in\{1,2,3,4\}$

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Question 232 Marks

Solve the following linear inequations in R: Solve: 4x - 2 < 8, when

$\text{x}\in\text{R}$
$\text{x}\in\text{N}$
$\text{x}\in\text{N}$

Answer

Now, 4x - 2 < 8 ⇒ 4x < 8 + 2 ⇒ 4x < 10 $\Rightarrow\text{x}<\frac{10}{4}=\frac{5}{4}$

If $\text{x}\in\text{R},$ then $\text{x}<\frac{5}{2}\Rightarrow\text{x}\in\Big(-\infty,\frac{5}{2}\Big)$
If $\text{x}\in\text{Z},$ then $\text{x}<\frac{5}{2}\Rightarrow\text{x}\in\{...,-2,-1,0,1,2\}$
If $\text{x}\in\text{N},$ then $\text{x}<\frac{5}{2}\Rightarrow\text{x}\in\{1,2\}$

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Question 242 Marks

Solve the following linear inequations in R: $\frac{4+2\text{x}}{3}\geq\frac{\text{x}}{2}-3$

Answer

$\frac{4+2\text{x}}{3}\geq\frac{\text{x}}{2}-3$ $\frac{4+2\text{x}}{3}\geq\frac{\text{x}}{2}-3$ $2(4+2\text{x})\geq3(\text{x}-6)$ $8+4\text{x}\geq3\text{x}-18$ $4\text{x}-3\text{x}\geq-18-8$ $\text{x}\geq-26$ $\therefore$ The solution set is $[-26,\infty)$

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