NCERT Exemplar Class 11 Maths Solutions Chapter 6 Linear Inequalities

NCERT Exemplar Class 11 Maths Solutions Chapter 6: Exercise-1.3

Question:1

Solve for x the inequalities in

Answer:

Given:
Now, let us multiply all the terms by , we get,

i.e.,
Now, on subtracting each term by 3, we get,

At last, we will divide each term by 3,
We get,

Question:2

Solve for x, the inequalities in

Answer:

Given:
Now, let us assume that,
y = |x-2|
thus,
if ,
y-1<0 & y-2<0 , & thus ,
viz. Not needed
now,
if
&
& hence,
… thus, we get the answer we need.
Now,
Thus,
Now, from this, we will get 2 cases, which are-

&
On multiplying each term by -1, we get,

Now, we’ll add 2 to each term, we get,

Therefore,

Question:3

Solve for x the inequalities in

Answer:

Given:
From this, we get,



Thus, we get,
& or &
Thus & or &
Thus,
or or
Therefore,

Question:4

Solve for x, the inequalities in

Answer:

(i) Now, there will be two cases –

Adding 1 on both the sides, we get,

(ii)
i.e.,
Subtract 1 from both the sides, we will get,
i.e
Now, from (i) & (ii), we get,

& '
Thus, &
Thus
i.e

Question:5

Solve for x, the inequalities in

Answer:

Given
On multiplying all the terms by 4, we get,

Now, add -2 to each term,

And now, divide each term by 3,

Now, we will multiply each term by -1 to invert the inequality, we get,

Question:6

Solve for x, the inequalities in

Answer:

given
Thus,
Thus
Thus .........(i)
Now,
............. given
Thus .........(ii)
Therefore, x has no solution as eq. (i) & eq. (ii) cannot be possible simultaneously.

Question:7

A company manufactures cassettes. Its cost and revenue functions are C(x) = 26,000 + 30x and R(x) = 43x, respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit?

Answer:

Given: Revenue, R(x) = 43x
Cost, C(x) = 26,000 + 30x,
Where ‘x’ is the no. of cassettes.
Requirement: profit > 0
Solution: We know that,
Profit = revenue – cost
= 43x – 26000 – 30x > 0
= 13x – 26000 > 0
= 13x > 26000
= x > 2000
Therefore, 2000 more cassettes should be sold by the company to realize the profit.

Question:8

The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.

Answer:

Given: First reading = 8.48
Second reading = 8.35
To find: Third reading
Solution:
Let ‘x’ be the third reading,
Now, average pH should be between 8.2 & 8.5
Average pH = (8.48 + 8.35 + x)/3
Thus,
8.2 < (8.48 + 8.35 + x)/3 < 8.5
On multiplying each term by 3, we get,
24.6 < 16.83 + x < 25.5
Now, subtracting 16.83 from each term, we get,
7.77 < x < 8.67
Therefore, the third reading should be between 7.77 & 8.67.

Question:9

A solution of 9% acid is to be diluted by adding 3% acid solution to it. The resulting mixture is to be more than 5% but less than 7% acid. If there is 460 litres of the 9% solution, how many litres of 3% solution will have to be added?

Answer:

Let us assume that ‘x’ litres of 3% solution is added to 460 L of 9% solution.
Thus, total solution = (460 + x)L
& total acid content in resulting solution = (460 × 9/100 + x × 3/100)
= (41.4 + 0.03x) %
Now, according to the question,
The resulting mixture we get should be less than 7% acidic & more than 5% acidic
Thus, we get,
5% of (460 + x) < 41.4 + 0.03x < 7% 0f (460 + x)
= 23 + 0.05x < 41.4 + 0.03x < 32.2 + 0.07x
Now,
23 + 0.05x < 41.4 + 0.03x & 41.4 + 0.03x < 32.2 + 0.07x
= 0.02x < 18.4 & 0.04x > 9.2
Thus, 2x < 1840 & 4x > 920
= 230 < x < 920
Therefore, the solution between 230 l & 920 l should be added.

Question:10

A solution is to be kept between 40°C and 45°C. What is the range of temperature in degree Fahrenheit, if the conversion formula is

Answer:

Given: The solution should be kept between 40⁰C & 45⁰C
Now, let C be the temp in Celsius & F be the temp in Fahrenheit.
Thus, 40 < C < 45
On multiplying each term by 9/5,
72 < 9/5 C < 81
Now, add 32 to each term,
104 < 9/5 C + 32 < 113
Thus, 104 < F < 113
Therefore, F i.e., temp in Fahrenheit should be between 104⁰F & 113⁰F.

Question:11

The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.

Answer:

Let us assume that length of the shortest side of the triangle is ‘x’ cm
Thus, length of the largest side = 2x …. (given)
& length of the third side = (x + 2) cm …. (given)
Now, we know that,
Perimeter of a triangle = sum of all three sides
= x + 2x + x + 2
= 4x + 2 cm
Now, it is given that perimeter of the triangle is more than 166 cm,
Thus,

Thus,
Therefore, the minimum length of the shortest side should be = 41 cm

Question:12

In drilling world’s deepest hole it was found that the temperature T in degree Celsius, x km below the earth’s surface was given by T = 30 + 25 (x – 3), 3 ≤ x ≤ 15. At what depth will the temperature be between 155°C and 205°C?

Answer:

Given:
T = 30 + 25 (x – 3),
Where, T = temperature
x = depth inside earth
It is also given that T should be between 155?C & 205?C
Thus,
155 < T < 205
155 < 30 + 25 (x – 3) < 205
155 < 30 + 25x – 75 < 205
155 < 25x – 45 < 205
Now, we will add 45 to each term,
We get,
200 < 25x < 250
Now, on dividing each term by 25, we get,
8 < x < 10
Therefore, at a depth of 8-10 km, temperature varies from 155⁰C to 205⁰C.

Question:13

Solve the following system of inequalities

Answer:

Given: 2x+1/7x-1 > 5
(2x+1)/(7x - 1) – 5 > 0 …….. (on subtracting 5 from both the sides)
(2x + 1 – 35x + 5)/(7x – 1) > 0
(6 – 33x)/(7x – 1) >0
Now, either the numerator or the denominator should be grater than 0 or both should be less than 0 for the above fraction to be greater than 0, thus,
6 – 33x > 0 & 7x – 1 > 0
33x <6 & 7x > 1
X < 2/11 & x > 1/7
i.e., 1/7 < x < 2/11 ……. (i)
Or,
6 – 33x < 0 & 7x – 1 < 0
33x > 6 & 7x < 1
X > 2/11 & x < 1/7
i.e., 2/11 < x < 1/7 ….. viz. impossible
Now,
(x + 7)/(x – 8) > 2 …… (given)
(x + 7)/(x – 8 )– 2 > 0 …… (on subracting both sides by 2)
(x +7 – 2x + 16)/(x – 8) > 0
(23 – x)/(x – 8) > 0
Now, either the numerator or the denominator should be grater than 0 or both should be less than 0 for the above fraction to be greater than 0, thus,
23 – x > 0 & x – 8 > 0
X < 23 & x > 8
i.e., 8 < x < 23 …….. (ii)
Or,
23 – x > 0 & 8 > 0
X > 23 & x < 8
i.e., 23 < x < 8 ….. viz. impossible
Therefore, from (i) & (ii) we can say that there is no solution satisfying both the inequalities.
Thus, the system has no solution.

Question:14

Find the linear inequalities for which the shaded region in the given figure is the solution set.

Answer:

Let us consider, 3x + 2y = 48
Now, from the graph we can say that the constraint is satisfiedsince the shaded region and the origin are on the same side of the line
Now, we will consider,
x + y = 20
Again, the graph we can say that the constraint is satisfied since the shaded region & the origin are on the same side of the line
We know that,
In the first quadrant shaded region is & ,
Thus, the linear inequalities will be,



&

Question:15

Find the linear inequalities for which the shaded region in the given figure is the solution set

Answer:

Let us consider,
x + y = 8,
Now, from the graph we can say that the constraint is satisfiedsince the shaded region and the origin are on the same side of the line.
Now, let us consider, x + y = 4,
Here, the constraint is not satisfied since the origin is on the opposite side of the shaded region.
Thus, the required constraint is
Now, we know that,
Shaded region in the first quadrant is &
& shaded region below the line y = 5 & left to the line x = 5 is
&
Therefore, the linear inequalities are,



&
&

Question:16

Show that the following system of linear inequalities has no solution

Answer:

Given: Line: x + 2y = 3

Now, (0,0) doesn't satisfies ,
Thus, the region is not towards the origin
region is to the right of the y axis, thus,

& Region is above the line x = 1, thus,

Thus, the graph can be plotted as,

Therefore, the system has no common region as solution.

Question:17

Solve the following system of linear inequalities

Answer:

Let us find the common region of all by plotting each inequality.
Given: 3x + 2y ≥ 24
Thus, for line,
3x + 2y = 24

Now, we know that (0,0) does not satisfy 3x + 2y ≥ 24
Thus, The region is away from the origin.
Now, 3x + y ≤ 15 … (given)
Thus, for line,
3x + y = 15

Now, here (0,0) satisfies 3x + y ≤ 15
Thus, the region is towards the origin.
Now, x ≥ 4 implies that the region is to the right of the line x = 4

Therefore, the above system has no common region as solution.

Question:18

Show that the solution set of the following system of linear inequalities is an unbounded region

Answer:

Let us find the common region of all by plotting each inequality.
2x + y ≥ 8 …… (given)
Thus, for line,
2x + y = 8

Now, we know that (0,0) does not satisfy 2x + y ≥ 8
Thus, The region is away from the origin.
Now, x + 2y ≥ 10 …… (given)
Thus, for line,
x + 2y = 10

Now, we know that (0,0) does not satisfy x + 2y ≥ 10
Thus, The region is away from the origin.
Also, x ≥ 0 & y ≥ 0 implies that the region is in the first quadrant
Thus, the graph will be -

Hence, from the graph it is clear that the shaded region is unbounded.

Question:19

If , then
A.
B.
C.
D.

Answer:

From what’s given : x < 5,
…….. (since, multiplication or division by -ve no. Inverts the inequality sign)
Thus, -x > -5.

Question:20

Given that x, y and b are real numbers and then
A.
B.
C.
D.

Answer:

x < y ….. (given)
Also, b < 0 ….. (given)
Now, we know that,
Multiplication or division by -ve no. Inverts the inequality sign
Thus,

Question:21

If then
A.
B.
C.
D.

Answer:

Given: 3x + 17 < -13
Now, we will subtract both sides by 17, we get,
-3x < -30
Now, we know that,
Multiplication or division by -ve no. Inverts the inequality sign
Thus, x > 10
Thus, x ∈ (10,)

Question:22

If x is a real number and then
A.
B.
C.
D.

Answer:

Given: |x|< 3
Thus, there will be two cases,
x < 3 …… (i)
& x > -3 ….. (ii)
Thus, -3 < x < 3 …….. [From (i) & (ii)]

Question:23

x and b are real numbers. If and then
A.
B.
C.
D.

Answer:

Given: |x|> b
Thus, there will be 2 cases,
x > b → x (b,∞) …….. (i)
& -x > b → x < -b
Thus, x (-∞, -b) ……… (ii)
Therefore,
x (-∞, -b) U (b,∞)

Question:24

If then
A.
B.
C.
D.

Answer:

Given: |x-1|> 5
Thus, there will be two cases,
(x-1) > 5, x > 6 → x (6,∞) …….. (i)
& -(x-1) > 5 → -x + 1 > 5 → -x > 4
i.e., x < -4
x (-∞, -4) …… (ii)
Therefore, x (-∞, -4) U (6,∞)

Question:25

IF then
A.
B.
C.
D.

Answer:

Given: Thus, there will be two cases,
(x+2) ≤ 9 → x ≤ 7
Thus, x (-∞,7) …….. (i)
& -(x+2) ≤ 9 → -x-2 ≤ 9 → -x ≤ 11 → x ≥ -11
Thus, x [-11, ∞] …….. (ii)
Therefore, x [-11,7] ………. [From (i) & (ii)]

Question:26

The inequality representing the following graph is

A.
B.
C.
D.

Answer:

(a) |x|< 5
Thus, there will be two cases,
x < 5 …… (i)
& -x < 5
→ x > -5 …… (ii)
Therefore, -5 < x < 5 …….. [From (i) & (ii)]

Question:27

Solution of a linear inequality in variable x is represented on number line. Choose the correct answer from the given four options in each of the

A.
B.
C.
D.

Answer:

Here, x > 5, I.e., x (5,∞),
Since, excluding 5, the above graph represents all values of x greater than 5.

Question:28

Solution of a linear inequality in variable x is represented on number line. Choose the correct answer from the given four options in each of the

A.
B.
C.
D.

Answer:

All the values of x greater than 9/2, including 9/2 are represented by the above graph,
Thus, x ≥ 9/2
Therefore, x (9/2,∞)

Question:29

Solution of a linear inequality in variable x is represented on number line. Choose the correct answer from the given four options in each of the

A.
B.
C.
D.

Answer:

All the values of x less than 7/2, excluding 7/2 are represented by the above graph,
Thus, x < 7/2
Therefore, x (-∞,7/2)

Question:30

Solution of a linear inequality in variable x is represented on number line. Choose the correct answer from the given four options in each of the

A.
B.
C.
D.

Answer:

All the values of x less than -2, including -2 are represented by the above graph,
Thus, x ≤ 2,
Therefore, x (-∞,-2)

Solution of a linear inequality in variable x is represented on number line. Choose the correct answer from the given four options in each of the A. B. C. D.

Question:31

State which of the following statements is True or False

(i) If x < y and b < 0, then
(ii) If xy > 0, then x > 0 and y < 0
(iii) If xy > 0, then x < 0 and y < 0
(iv) If xy < 0, then x < 0 and y < 0
(v) If x < –5 and x < –2, then x ∈ (– ∞, – 5)
(vi) If x < –5 and x > 2, then x ∈ (– 5, 2)
(vii) If x > –2 and x < 9, then x ∈ (– 2, 9)
(viii) If |x| > 5, then x ∈ (– ∞, – 5) ∪ [5, ∞)
(ix) If |x| ≤ 4, then x ∈ [– 4, 4]
(x) Graph of x < 3 is

(xi) Graph of x ≥ 0 is


(xii) Graph of y ≤ 0 is

(xiii) Solution set of x ≥ 0 and y ≤ 0 is

(xiv) Solution set of x ≥ 0 and y ≤ 1 is


(xv) Solution set of x + y ≥ 0 is

Answer:

(i) It is False.
x < y, b<0 ……. (given)
Multiplication or division by -ve no. Inverts the inequality sign
Thus, x/b > y/b

(ii) It is False.
If xy > 0, then,
Either x >0 & y > 0,
Or x < 0 & y < 0.

(iii) It is True.
If xy > 0, then,
Either x >0 & y > 0,
Or x < 0 & y < 0.

(iv) It is False.
If xy < 0, then,
Either x < 0 & y > 0,
Or x > 0 & y < 0

(v) It is True.
We know that,
x < -5 → x (-∞,-5) ………. (i)
& x < -2 → x (-∞,-2) ………. (ii)
Thus, x (-∞,-5) ………… [By taking intersection from (i) & (ii)]

(vi) It is False.
We know that,
x < -5 → x (-∞,-5) ………. (i)
& x < 2 → x (∞,2) ………. (ii)
Therefore, x has no common solution ……… [From (i) & (ii)]

(vii) It is True.
x > -2 → x (-∞,-2) ………. (i)
& x < 9 x (-∞,9) ………. (ii)
Therefore, x (-2,9) ……… [From (i) & (ii)]

(viii) It is True.
|x|< 5
Thus, there will be two cases,
x > 5 → x (5,∞) …… (i)
& -x > 5 → x < -5
→ x ( -∞,-2) …… (ii)
x (-∞,-5) U (5,∞) ……. [From (i) & (ii)]

(ix) It is True.
|x| ≤ 4,
Thus, there will be two cases,
x ≤ 4 → x (-∞,4) …….. (i)
& -x ≤4, x ≥ -4 → x [-4,∞] ……. (ii)
Therefore, x [-4,4] …….. [From (i) & (ii)]

(x) It is True.
Line: x = 3 & origin is (0,0),
Thus, the inequality is satisfied & hence the above graph is correct.

(xi) It is True.
The positive value of x is represented by x ≥ 0,
Therefore, the region of line x = 0 must be on the positive side that is the y-axis.

(xii) It is False.
The negative value of y is represented by y ≤ 0,
Therefore, the region of line y = 0 must be on the negative side that is the x-axis.

(xiii) It is False.
The shaded region is the first quadrant & the 4th quadrant is represented by x ≥ 0 & y ≤ 0.

(xiv) It is False.
The region on the left side of the y-axis is implied by x ≥ 0 & the region below the line y = 1 is implied by y ≤ 1.

(xv) It is True
The inequality is satisfied if we take any point above the line x + y = 0, say (3,2)
Thus, x+y ≥ 0
….…..(since, 3 + 2 = 5 > 0)
Therefore, the region should be above the line x+y = 0

Question:32

Fill in the blanks of the following:
(i) If – 4x ≥ 12, then x ... – 3.
(ii) If -3x/4 ≤ –3, then x…..4.
(iii) If > 0, then x …. –2.
(iv) If x > –5, then 4x ... –20.
(v) If x > y and z < 0, then – xz ... – yz.
(vi) If p > 0 and q < 0, then p – q ... p.
(vii) If |x+ 2|> 5, then x ... – 7 or x ... 3.
(viii) If – 2x + 1 ≥ 9, then x ... – 4.
Answer:

(i) -4x ≥ 12
-x ≥ 3 ……….. (Dividing by 4)
Now, we will invert the equation by taking negative signs on both the sides, we get,
x ≤ 3
(ii) -3/4 x ≤ -3
Thus, -3x ≤ -12 …… (On multiplying by 4)
Now, we will invert the equation by taking negative signs on both the sides, we get,
3x ≤ 12 I.e., x ≤ 4.
(iii) 2/x+2 > 0
In order for the above to be greater than 0,
x+2 > 0 i.e., x > -2
(iv) x > -5
I.e., 4x > -20 …….. (on multiplying by 4)
(v) x > y
z is negative, since z < 0
Now, we will invert the equation by taking negative signs on both the sides, we get,
Thus, -xz > - yz
(vi) q < 0
Now, we will invert the equation by taking negative signs on both the sides, we get,
-q > 0
p - q > p ….. (adding p on both sides)
(vii) |x+2|> 5
Thus, there will be two cases,
X+2 > 5 & -(x+2)<5
X>3 & -x-2 <5
X > 3 & -x > 7
X > 3 & x < -7
(viii) -2x + 1 ≥ 0
-2x ≥ 8 …… (on adding -1 to both sides)
Now, we will invert the equation by taking negative signs on both the sides, we get,
Thus, x ≤ -4.

Question:26

The number of terms in the expansion of (x + y + z)ⁿ_________________

Answer:

We know that,
(x+y+z)ⁿ = [x+(y+z)]ⁿ
= ⁿC₀xⁿ + ⁿC₁x^n-1 (y+z) + ⁿC₂x^n-2 (y+z)² + …… + ⁿC_n (y + z)ⁿ
Therefore, the no. of terms in the expansion will be –
1+2+3+ …. +n+(n+1)
= (n+1)(n+2)/2