## Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 6 Linear Inequalities

**Short Answer** **Type Questions **

**(Score 3)**

Question 1.

Solve the inequality 37 – (3x + 5) > 9x – (x-3).

a. (-4, ∞)

b. (4, ∞)

c. (- ∞, 2]

d. (-4,4)

Answer:

c. (- ∞, 2],

37 – (x + 5) ≥ 9x – 8 (x-3)

37-3x-5 ≥ 9x-8x + 24

– 3x – x ≥ 24-32

-4x ≥ -8

x ≤ 2

solutions are x = (- ∞ ,2)

Question 2.

A value of the variable satisfying the inequation is called solution of the inequation.

i. Solve the inequality 6 – 3x ≥ 2 – 2x

ii. Solution of the inequality.

a. x∈(-23,2)

b. x∈(-23,2)

c. x∈(23,2)

d. x∈(-2,23)

Answer:

Question 3.

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content.

a. between 562 and 900 litres

b. between 590 and 1000 litres

c. between 100 and 200 litres

d. none of these

Answer:

Question 4.

i. Solve 4x – 7 < 0

ii. Represent the solution graphically.

Answer:

i. 4x – 7 < 0,4x < 7, x < 7/4

i.e, x ∈ (-∞, 7/4]

Question 5.

i. Solve x – 4 < 0 and 4x + 5 ≤ 0.

ii. Draw the graphical representation of the solution.

Answer:

i. x-4<0 ⇒ x<4……………..(1)

4x + 5≤0,4x≤-5

⇒ x≤ …………………….. (2)

Solution is the intersection of (1) and (2)

x ∈ (-∞, -5/4]

ii.

.

Question 6.

If x satisfy the inequalities x + 7 < 2x + 3 and 2x + 4 < 5x + 3, then x lies in the interval

a. (-∞,3)

b. (1,3)

c.(4,∞)

d. (-∞,1)

Answer:

c.(4, ∞).

x + 7 <2x +3

7-3 < 2x-x

4<x……………… (1)

2x + 4 < 5x + 3

4-3 < 5x-2x

1 < 3x

1/3 < x ………………..(2)

Question 7.

a Identify the figure in which the line has a positive slope.

b. Find the x and y intercepts of the line 3x+4y-12 = 0.

Answer:

a. iii.x<0,y>0

b. -2<x<2 -1≤y≤1

**Short Answer** **Type Questions **

**(Score 4)**

Question 1.

If 7x – 2 < 4 – 3x and 3x -1 < 2 + 5x, then x lies

a.

b.

c.

d.

Answer:

b.

7x – 2 < 4 – 3x ⇒7x + 3x<4 + 2

10x <6 ⇒ x <

x< ……………..(1)

3x -1 < 2 + 5x ⇒ 3x – 5x <2+1

-2x<3 ⇒ x> ……………..(2)

combining eqn (1) and (2)

we get x∈

Question 2.

i. Draw the graph of the equations:

x + 2y = 4, x + y = 3, 2x-3y = 6

ii. Solve graphically, the in equations:

x + 2y≥4, x + y≤3, 2x-3y≤6

Answer:

(0,0) is not on x+2y ≥ 4

So the half plane not containig (0,0)

(4, 0) is not on x + y ≤3

so the half plane not containing (4, 0)

(3,0) is not on (x+2y) ≥ 4

so the half plane not containing (3, 0)

so the shaded region is the solution.

The shaded part gives the solution of the given inequations.

Question 3.

Solve the following system graphically

x+ 2y ≤ 10,3x+ y ≤10, x ≤1,y ≤ 3, x, y ≥ 0.

Answer:

x, y, ≥ 0 lies on the plane so the shaded area is the solution.

Question 4.

A wire of length 91cm is to be cut into three pieces. The second piece is 3cm longer than the shortest and third length is twice as long as the shortest. The third piece is to be atleast 5cm longer than the second.

i. If x is the length of the shortest piece, find the length of others.

ii. Construct inequations.

iii. What are the possible lengths of the shortest wire.

Answer:

i. x is the length of shortest piece. Second piece of length x + 3.

Length of third piece = 2x

x + (x + 3) + 2x ≤91

i.e.,4x + 3≤91. …………………..(1)

Also 2x≥ (x+3) + 5 i.e,2x≥ x+8.

From(1), 4x≤88, x≤22 From(2), x≥ 8,8≤x ≤22

The shortest piece is at least 8 cm long but not more than 22 cm.

Question 5.

Solve 24x ≤ 100, when

i. x is a natural number.

ii. x is an integer.

Answer:

We have, 24x <100

⇒

⇒

i. When x is a natural number, then solutions of the inequality are given by i.e., all natural numbers x which are less than

In this case, the following values of x make the statement true, x = 1,2,3,4. Hence, the solution set of inequality is {1,2,3,4}.

ii. When x is an integer.In this case, solutions of given inequality are……..-4,-3,-2,-1,0,1,2,3,4. Hence, the solution set of inequality is { …………… ,-4,-3,-2,-1,0,1,2,3,4}

Question 6.

a. Solve the linear inequality

b. Solve the linear inequality

Answer:

**Long Answer Type Questions**

**(Score – 6)**

Question 1.

Solve the inequation |x-2|+|x+2|<4

Answer:

Question 2.

Solve the inequation

b. Draw the graph of the inequation x+2y≤ 5,2x+y≤ 4,x≥ 0,y≥ 0 amd make the corresponding

Answer:

Question 3.

i. Solve

ii. solve

iii. Solve the system of inequations (i) and (ii)

Answer:

Question 4.

i. Split the inequation -3 ≤ 3 – 2x < 9, x ∈ R into two inequations.

ii. Solve each of the inequations and represent the solution set on the real line.

iii. Split the inequation -2 ≤ 6x – 1 < 2 into two inequations.

iv. Solve each inequation separately and hence find the solution of -2 ≤ 6x -1 < 2

Answer:

Question 5.

i. Solve

ii. Draw the graph of the solution set.

iii. Find graphically, the solution set of the following system of linear inequations.

x + 3y ≤ 60,3x + 5y ≤ 165,4x + 3y ≤144, x, y ≥ 0

Answer:

Question 6.

Solve graphically the following system of inequalities.

x + 2y ≤ 3

3x + 4y ≥ 12

x ≥ 0

y ≥ 1

Answer:

we have,

Shaded region for inequality x + 2y ≤3 contains origin.

Shaded region for inequality 3x + 4y≥12 does not contains origin,

x ≥ 0 represent the region in first quadrant.

y ≥ i

corresponding linear equation is y = 1, which is a line parallel to X axis at distance of i unit If from X-axis and this shaded does not contain origin.

We observe that, there is no common region represented by these inequalities. We thus conclude that, there is no solution for given system of inequalities.

Question 7.

a. Solve the inequality and show the solution graphically on the number line. 4x+3 ≤ 5x + 7

b. Solve the system of inequalities graphically.

2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6

Answer:

Hence, shade the common region which gives required solution set.

**NCERT Question and Answers**

Question 1.

Solve the inequality 5x – 3 < 7, when

i. x is an integer

ii. x is a real number

Answer:

We have, 5x – 3 < 7

5x-3 + 3<7 + 3 (adding 3 on both sides)

5x<10

5x<2

Thus any number less than 2 satisfies given inequality.

i. When x is an integer, then the solution of the given inequality is {……… -4, -3, -2, -1,0,1}

ii. When x is a real number, then the solution of the inequality is given by x < 2, i.e., all real numbers x, which are less than 2. Hence, solution set is (-∞,2)

Question 2.

Solve the inequality 5x – 3 < 3x +1, when x is an integer.

Answer:

We have, 5x – 3 < 3x + 1

5x-3 + 3 <3x+1 +3

5x<3x+4

5x -3x<3x+4-3x

2x<4

x<2

When x is an integer, then the solution of the given inequality is {…., -4, -3, -2, -1,0,1}

Question 3.

Solve the inequality , showing graph of solution on number line.

Answer:

Question 4.

The marks obtained by a student of Class XI in first and second terminal examination are 62 and 48 respectively. Find the minimum. Marks he should get in the annual examination to have an average of at least 60 marks.

Answer:

Let x be the marks obtained by student in the annual examination.

Then

or 110 + x ≥ 180 or x≥ 70

Thus, the student must obtain a minimum of 70 marks to get an average of at least 60 marks.

Question 5.

To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87,92,94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

Answer:

Let Sunitha obtains x marks in her fifth exam.

368+ x ≥ 450

368+ x-368 ≥ 450-368

x≥ 82

Thus Sunitha must obtain a minimum of 82 marks to get grade A in the course.

Question 6.

A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must he added to it so that acid content in the resulting mixture will be more than 15% but less than 18%?

Answer:

Let x litres of 30% acid solution is required to be added. Then Total mixture = (x + 600) litres Therefore 30% x + 12% of 600 > 15% of (x + 600) and 30% x + 12% of 600 < 18% of (x+600) (x + 600)

or

or 30x + 7200 >15x +9000

and 30x + 7200 < 18x +10800

or 15x> 1800 and 12x<3600

or x> 120 and x<300, i.e. 120<x<300

Thus, the number of litres of the 30% solution of acid will have to be more than 120 litres but less than 300 litres.