2 Marks Questions

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

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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