KC Sinha Mathematics Solution Class 11 Chapter 12 रैखिक असमिकाएँ (Linear Inequalities) Exercise 12.2

Exercise 12.2

Solve the following inequations :

Question 1

-2≤6x-1<2
Sol :
On Adding 1

-2+1≤6x-1+1<2+1

-1≤6x<3

Dividing by 6

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

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

[]$=\left[\frac{-1}{6}, \frac{1}{2}\right)$

Question 2

-3≤4-7x<18
Sol :
On subtracting 4

-3-4≤4-7x-4<18-4

-7≤-7x<14

Dividing with -7

$\frac{-7}{-7} \geq \frac{-7 x}{-7}>\frac{14}{-7}$

1x≥x-2

or -2<x≤1

[]=(-2,1]

Question 3

$0<-\frac{x}{3}<1$
Sol :
Multiplying -3

$0 \times(-3)>-\dfrac{x}{3} \times(-3)>1 \times(-3)$

0>x>-3

or -3<x<0

[]=(-3,0)

Question 4

-7<2x-3<7
Sol :
Adding 3
-7+3<2x-3+3<7+3

-4<2x<10

Dividing with 2

$-\frac{4}{2}<\frac{2 x}{2}<\frac{10}{2}$

-2<x<5

[]=(-2,5)

Question 7

$-3 \leq \frac{4-7 x}{2} \leq 18$
Sol :
Multiplying by 2

$-3 \times 2 \leq \frac{4-7 x}{2} \times 2 \leq 18 \times 2$

-6≤4-7x≤36

Subtracting 4

-6-4≤4-7x-4≤36-4

-10≤-7x≤32

Dividing by -7

$\frac{-10}{-7} \geq-\frac{7 x}{-7} \geq \frac{32}{-7}$

$\frac{10}{7} \geq x \geq-\frac{32}{7}$

or $-\frac{32}{7} \leq x \leq \frac{10}{7}$

[]=$=\left[\frac{-32}{7}, \frac{10}{7}\right]$

Question 8

$-12 \leq \frac{4-3 x}{-5}<2$
Sol :
Multiplying by -5

$-12 \times(-5) \geq \frac{4-3 x}{-5} \times(-5) \geq 2 \times(-5)$

60-4≥4-3x-4>-10-4

56≥-32>-14

Dividing with -3

$\frac{56}{-3} \leq-\frac{3 x}{-3}<\frac{-14}{-3}$

$-\frac{56}{3} \leq x<\frac{14}{3}$

[]$=\left[\frac{-56}{3}, \frac{14}{3}\right)$

Question 9

$-\frac{1}{5} \leq \frac{3 x}{10}+1<\frac{2}{5}$
Represent the solution on number line
Sol :
Subtracting 1

$-\frac{1}{5}-1 \leq \frac{3 x}{10}+1-1<\frac{2}{5}-1$

$\frac{-1-5}{5} \leq \frac{3 x}{10}<\frac{2-5}{5}$

$-\frac{6}{5} \leq \frac{3 x}{10}<\frac{-3}{5}$

Multiplying by $\frac{10}{3}$

$\frac{-6}{-5} \times \frac{10}{3} \leq \frac{3 x}{10} \times \frac{10}{3}<\frac{-3}{5} \times \frac{10}{3}$

-4≤x<-2

[]$=[-4,-2)$

Question 10

Find the value of x , which satisfy the following inequations

$-2 \leq \frac{1}{2}-\frac{2 x}{3} \leq 1 \frac{5}{6}, x \in N$
(Represent the solution set on the number line)

Sol :

Subtracting $\dfrac{1}{2}$

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

$\frac{-4-1}{2} \leq-\frac{2 x}{3} \leq \frac{11-3}{6}$

$-\frac{5}{2} \leq \frac{-2}{3} x \leq \frac{8}{6}$

Multiplying by $-\frac{3}{2}$

$-\frac{5}{2} \times\left(-\frac{3}{2}\right) \geq \frac{-2}{3} x \times\left(-\frac{3}{2}\right) \geq \frac{4}{3} \times\left(-\frac{3}{2}\right)$

$\frac{15}{4} \geq x \geq-2$

or $-2 \leq x \leq \frac{15}{4}$

or $-2 \leq x \leq 3 \frac{3}{4}, x \in N$

x={1,2,3}

[]

Question 11

(Solve the following system of inequations)
2x+5≤0, x-3≤0
Sol :
Subtracting 5 both sides

2x+5-5≤0-5

Dividing with 2 both sides

$\frac{2x}{2} \leq-\frac{5}{2}$

x \leq \frac{-5}{2}

x-3≤0

Adding both sides 3

x-3+3≤0+3

x≤3

[]


[]$=\left(-\infty,-\frac{5}{2}\right]$

Question 12

x-2>0, 3x<18
Sol :
x-2>0

Adding both sides 2

x-2+2>0+2

x>2

3x<18

Dividing both sides with 3

$\frac{3 x}{3}<\frac{18}{3}$

x<6

[]

∴ 2<x<6 , []=(2,6)

Question 15

5x+1>-24, 5x-1<24
Sol :
5x+1>-24

Subtracting both sides with 1

5x+1-1>-24-1

5x>-25

Dividing both sides with 5

$\frac{5 x}{5}>-\frac{25}{5}$

x>-5

5x-1<24

Adding both side with 1

5x-1+1<24+1

5x<25

Dividing both sides by 5

$\frac{5 x}{5}<\frac{25}{5}$

x<5

[]

Question 21

x+2≤5, 3x-4>-2+x
Sol :
x+2≤5

Subtracting both sides 2

x+2-2≤5-2

x≤3


3x-4>-2+x

Subtracting both side with x

3x-4-x>-2+x-x

2x-4>-2

Adding both side 4

2x-4+4>-2+4

2x>2

Dividing both side by 2

$\frac{2 x}{2}>\frac{2}{2}$

x>1

[]

∴1<x≤3 , []=(1.3]

Question 22

4x+3≥2x+17 , 3x-5<-2
Sol :
4x+3≥2x+17

Subtracting both sides 2x

4x+3-2x≥2x+17-2x

2x+3≥17

Subtracting both sides 3

2x+3-3≥17-3

Dividing both sides with 2

$\frac{2 x}{2} \geq \frac{14}{2}$

x≥7


3x-5<-2

Adding both sides 5

3x-5+5<-2+5

Dividing both side by 3

$\frac{3 x}{3}<\frac{3}{3}$

x<1

[]

Question 23

$7 x-8<4 x+7,-\frac{x}{2}>4$
Sol :
7x-8<4x+7

Subtracting both sides 4x

7x-8-4x<4x+7-4x

3x-8<7

Adding both sides 8

3x-8+8<7+8

3x<15

Dividing both sides by 3

$\frac{3 x}{3}<\frac{15}{3}$

x<5

$-\frac{x}{2}>4$

Multiplying both sides by -2

$-\dfrac{x}{2} \times (-2)<4 x(-2)$

x<-8

[]

[]=(-∞,-8)

Question 24

3x-7>2(x-6) , 6-x>11-2x
Sol :
3x-7>2(x-6)

3x-7>2x-12

Subtracting both sides 2x

3x-7-2x>2x-12-2x

x-7>-12

Adding both sides 7

x-7+7>-12+7

x>-5

6-x>11-2x

Adding both sides 2x

6-x+2x>11-2x+2x

6+x>11

Subtracting both sides 6

6+x-6>11-6

x>5

[]

[]=(5,∞)

Question 26

$\frac{4 x}{3}-\frac{9}{4}<x+\frac{3}{4}, \frac{7 x-1}{3}-\frac{7 x+2}{6}>x$
Sol :
$\frac{4 x}{3}-\frac{9}{4}<x+\frac{3}{4}$

Adding $\frac{9}{4}$ both sides

$\frac{4 x}{3}-\frac{9}{4}+\frac{9}{4} < x+\frac{3}{4}+\frac{9}{4}$

$\frac{4 x}{3}<x+\frac{3+9}{4}$

$\frac{4 x}{3}<x+\frac{12}{4}$

Subtracting both sides with x

$\frac{4 x}{3}-x<x+3-x$

$\frac{4 x-3 x}{3}<3$

$\frac{x}{3}<3$

Multiplying both sides with 3

$\frac{x}{3} \times 3<3 \times 3$

x<9


$\frac{7 x-1}{3}-\frac{7 x+2}{6}>x$

$\frac{2(7 x-1)-(7 x+2)}{6}>x$

$\frac{14 x-2-7 x-2}{6}>x$

$\frac{7 x-4}{6}>x$

Multiplying both sides by 6

$\left(\frac{7 x-4}{6}\right) \times 6>6 x$

7x-4>6x

Subtracting both sides with 6x

7x-4-6x>6x-6x

x-4>0

Adding both sides 4

x-4+4>0+4

x>4

[]

[]=(4,9)

Question 27

(i) |x|≤5
Sol :
x≤5

-x≤5

Multiplying both sides -1

-x×(-1)≥5×(-1)

x≥-5

[]

∴-5≤x≤5


(ii) |x|>5
Sol :
x>5

-x>5

Multiplying both side -1

-x×(-1)<5×(-1)

x<-5

[]

[]=(-∞,-5)⋃(5,∞)

Question 28

(i) |x-2|≤5
Sol :
x-2≤5

Adding both sides 2

x-2+2≤5+2

x≤7


-(x-2)≤5

Multiplying both sides by -1

x-2≥-5

Adding both sides 2

x-2+2≥-5+2

x≥-3

[]

-3≤x≤7

[]=[-3,7]


(ii) |x+1|≥3
Sol :

Question 29

$|3 x-2| \leq \frac{1}{2}$
Sol :