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Chapters
1: Goods and Services Tax (G.S.T.)
2: Banking
3: Shares and Dividends
UNIT 2: ALGEBRA
▶ 4: Linear Inequations
5: Quadratic Equation
6: Problems on Quadratic Equations
7: Ratio and Proportion
Chapter 8: Remainder Theorem and Factor Theorem
9: Matrices
Chapter 10: Arithmetic Progression
Chapter 11: Geometric Progression
12: Reflection
Chapter 13: Section and Mid-point Formulae
Chapter 14: Equation of a Straight Line
UNIT 3: GEOMETRY
Chapter 15: Similarity (As a Size Transformation)
Chapter 16: Similarity of Triangles
17: Loci
Chapter 18: Angle and Cyclic Properties of a Circle
Chapter 19: Tangent Properties of Circles
20: Constructions
UNIT 4: MENSURATION
Chapter 21: Volume and Surface Area of Solids (Cylinder, Cone and Sphere)
UNIT 5: TRIGONOMETRY
Chapter 22: Trigonometrical Identities
Chapter 23: Heights and Distances
UNIT 6: STATISTICS
Chapter 24: Graphical Representation of Statistical Data
Chapter 25: Measures of Central Tendency (Mean)
Chapter 26: Median, Quartiles and Mode
UNIT 7: PROBABILITY
Chapter 27: Probability
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Solutions for Chapter 4: Linear Inequations
Below listed, you can find solutions for Chapter 4 of CISCE R.S. Aggarwal for Mathematics [English] Class 10 ICSE.
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE 4 Linear Inequations EXERCISE 4 [Pages 45 - 46]
Solve the inequation given below and represent its solution set on a number line:
2x – 7 < 4, x ∈ {1, 2, 3, 4, 5, 6, 7}
Solve the inequation given below and represent its solution set on a number line:
2x – 3 > 3, x ∈ {1, 2, 3, 4, 5, 6}
Solve the inequation given below and represent its solution set on a number line:
9 ≤ 1 – 2x, x ∈ {–3, –4, –5, –6}
Solve the inequation given below and represent its solution set on a number line:
`(3x - 5)/6 > 1/2, x ∈ {0, 1, 2, 3, 4, 5, 6}`
Solve the inequation given below and represent its solution set on a number line:
7x – 4(3 – x) ≥ 3 (2x – 5), x ∈ {–3, –2, –1, 0, 1, 2, 3}
Solve the inequation given below and represent its solution set on a number line:
4 – 3x ≥ 3x – 14, x ∈ N
Solve the inequation given below and represent its solution set on a number line:
6 – 5x > 3 – 4x, x ∈ W
Solve the inequation given below and represent its solution set on a number line:
`3/5x - (2x - 1)/3 > 1, x ∈ I`
Solve the inequation given below and represent its solution set on a number line:
`2x + 7/2 > (5x)/3 + 3, x ∈ I`
Solve the inequation given below and represent its solution set on a number line:
2x + 3 ≤ 3x + 1, x ∈ R
Solve the inequation given below and represent its solution set on a number line:
`(5x - 8)/3 ≥ (4x - 7)/2, x ∈ R`
Solve the inequation given below. Write the solution set and represent it on the number line:
–3 < 2x – 1 < x + 4, x ∈ I
Solve the inequation given below. Write the solution set and represent it on the number line:
2 + 4x < 2x – 5 < 3x, x ∈ I
Solve the inequation given below. Write the solution set and represent it on the number line:
2 ≤ 2x – 3 ≤ 5, x ∈ R
Solve the inequation given below. Write the solution set and represent it on the number line:
–1 ≤ 3 + 4x < 23, x ∈ R
Solve the inequation given below. Write the solution set and represent it on the number line:
`-2 ≤ 1/2 - (2x)/3 < 1 5/6, x ∈ I`
Solve the inequation given below. Write the solution set and represent it on the number line:
`-2/3 < 1 + x/3 ≤ 2/3, x ∈ R`
Solve the inequation given below. Write the solution set and represent it on the number line:
2x – 5 ≤ 5x + 4 < 11, x ∈ R
Solve the inequation given below. Write the solution set and represent it on the number line:
1 ≥ 15 – 7x > 2х – 27, x ∈ N
Solve the inequation given below. Write the solution set and represent it on the number line:
`- 8 1/2 < - 1/2 - 4x ≤ 7 1/2, x ∈ I`
Solve the inequation given below. Write the solution set and represent it on the number line:
`-2 2/3 ≤ x + 1/3 < 3 1/3, x ∈ R`
Solve the inequation given below. Write the solution set and represent it on the number line:
$$3x - 16 < \frac{2x}{5} - 3 \leq -\frac{3}{5} + 2x ; x \in \mathrm{R}$$
Solve the inequation given below. Write the solution set and represent it on the number line:
`2x - 1 ≥ x + (7 - x)/3 > 2, x ∈ R`
Solve the following inequation, write down the solution set and represent it on the number line.
`-3 + x <= 7x/2 + 2 < 8+2x, x∈l`
Solve the inequation given below. Write the solution set and represent it on the number line:
`-2 5/6 < 1/2 - (2x)/3 ≤ 2, x ∈ W`
Solve the inequation given below. Write the solution set and represent it on the number line:
–5(x – 9) ≥ 17 – 9x > x + 2, x ∈ R
Solve the inequation given below. Write the solution set and represent it on the number line:
`(-x)/3 ≤ x/2 - 1 1/3 < 1/6, x ∈ R`
Solve the inequation given below. Write the solution set and represent it on the number line:
`4x - 19 < (3x)/5 - 2 ≤ (-2)/5 + x, x ∈ R`
Solve the inequation given below. Write the solution set and represent it on the number line:
2y – 3 < y + 1 ≤ 4y + 7, y ∈ R
Solve the inequation given below. Write the solution set and represent it on the number line:
–2 + 10x ≤ 13x + 10 < 24 + 10x, x ∈ Z
Solve the following inequation, write the solution set and represent it on the real number line.
`2x − 5/3 < "3x"/5 + 10 ≤ "4x"/5 + 11; x ∈ R`
Solve the following inequation, write the solution set and represent it on the real number line.
`5x - 21 < (5x)/7 - 6 ≤ -3 3/7 + x, x ∈ R`
Solve the inequation given below. Write the solution set and represent it on the number line:
$$11x - 4 < 15x + 4 \leq 13x + 14, x \in \mathrm{W}$$
Given : P = {x : 5 < 2x – 1 ≤ 11, x ∈ R} and Q = {x : –1 ≤ 3 + 4x < 23, x ∈ I}.
Represent P and Q on the number line. Find P ∩ Q.
[Hint: P = {x ∈ R : 3 < x ≤ 6} and Q = {x ∈ I : –1 ≤ x < 5} = {–1, 0, 1, 2, 3, 4} ∴ P ∩ Q = {4}.]
Let A = {x ∈ R : 11x – 5 > 7x + 3} and B = {x ∈ R : 8x – 9 ≥ 15 + 2x}.
Find A ∩ B and represent it on the number line.
[Hint: A = {x ∈ R : x > 2} and B = {x ∈ R : x ≥ 4). So, A ∩ B = {x ∈ R : x ≥ 4}.]
What is the solution set for the inequation represented by the following number line?
{x ∈ R : – 3 < x ≤ 4)
{x ∈ R : – 3 < x < 4}
{x ∈ R : – 3 ≤ x < 4}
{x ∈ R : – 3 ≤ x ≤ 4}
The solution set of the inequation x – 3 ≥ – 5, x ∈ R is ______.
{x : x > – 2, x ∈ R}
{x : x ≤ – 2, x ∈ R}
{x : x ≥ – 2, x ∈ R}
{–2, –1, 0, 1, 2}
Find the greatest integer which is such that if 7 is added to its double, then the resulting number is greater than three times the integer.
7
8
6
none of these
The solution set for the inequation 2x + 4 ≤ 14, x ∈ W is ______.
{1, 2, 3, 4, 5}
{0, 1, 2, 3, 4, 5}
{1, 2, 3, 4}
{0, 1, 2, 3, 4}
CASE STUDY BASED QUESTIONS
| Shivam’s father is a building contractor. One day, Shivam got his father’s measuring tape. He used it to find the dimensions of the kitchen garden in his home. He found that the length of the garden is one metre more than twice its breadth. He told his friend Akhil that the perimeter of the garden is more than or equal to 110 m and is less than or equal to 140 m. |
Based on this information, answer the following questions:
- If breadth of the garden is x m, then algebraic representation of the given information is:
- 140 ≤ 6x + 2 ≤ 110, x ∈ R
- 110 ≤ 6x + 2 ≤ 140, x ∈ R
- 110 ≤ 4x + 2 ≤ 140, x ∈ R
- 110 ≤ 2x + 1 ≤ 140, x ∈ R
- The solution set for the breadth of the garden is:
- {x ∈ R: 18 ≤ x ≤ 23}
- {x ∈ R : 16 ≤ x ≤ 24}
- {x ∈ R : 18 ≤ x ≤ 24}
- {x ∈ R : 20 ≤ x ≤ 28}
- The greatest possible value of the breadth of the garden is
- 18 m
- 20 m
- 22 m
- 23 m
- What is the least possible length of the garden?
- 34 m
- 36 m
- 37 m
- none of these
- What is the greatest possible length of the garden?
- 47 m
- 51 m
- 46 m
- none of these
| A few countries such as the USA officially use Fahrenheit as a unit for measuring temperature. Other countries prefer Celsius over Fahrenheit. The two different scales are related by the linear equation `(F - 32)/9 = C/5`. A scientist wants to store an experimental solution between a temperature range of 68°F and 77°F. |
Based on the above information, answer the following questions:
1. The algebraic representation of the given information in degree Celsius is:
(a) `68 < 5/9 C + 32 ≤ 77, C ∈ R`
(b) `68 ≤ 5/9 C - 32 ≤ 77, C ∈ R`
(c) `68 < 9/5 C - 32 < 77, C ∈ R`
(d) `68 < 9/5 C + 32 < 77, C ∈ R`
2. The solution set for the temperature in degree Celsius is:
(a) {C ∈ R : 18 C < 23}
(b) {C ∈ R : 20 C < 25}
(c) {C ∈ R : 22 C < 27}
(d) {C ∈ R : 25 C < 30}
3. What is the range of the temperature in degree Celsius?
(a) between 20°C and 25°C
(b) between 25°C and 30°C
(c) between 18°C and 23°С
(d) between 22°C and 27°C
4. Which of the following is the graphical representation of the temperature in degree Celsius?
(a)
(b)
(c)
(d)
5. If the minimum temperature that can be maintained in a particular refrigerator is 0°C, what is the possible temperature range of the refrigerator on a Fahrenheit scale?
(a) `F > 160/9`
(b) `F < 160/9`
(c) F > 32
(d) F < 32
| In drilling world’s deepest hole, the Kola Superdeep Borehole, the deepest man made hole on the earth, 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) and 3 ≤ x ≤ 15. If the temperature lies between 180° C to 330° C, then based on this information, answer the following questions. |
1. The linear inequation for the depth of the hole is:
(a) 180 < 30 + 25 (x – 3) < 330
(b) 180 ≤ 30 + 25 (x – 3) ≤ 330
(c) 330 < 30 + 25 (x – 3) ≤ 180
(d) 330 < 30 + 25 (x – 3) < 180
2. The solution set for the depth is:
(a) {x ∈ R : 6 ≤ x ≤ 12}
(b) {x ∈ R : 9 ≤ x ≤ 12}
(c) {x ∈ R : 3 ≤ x ≤ 15}
(d) {x ∈ R : 9 ≤ x ≤ 15}
3. The minimum possible depth of the hole for the given temperature range is:
(a) 3 km
(b) 6 km
(c) 9 km
(d) cannot be determined
4. The maximum possible depth of the hole for the given temperature range is:
(a) 9 km
(b) 12 km
(c) 15 km
(d) None of these
5. Which of the following is the graphical representation of the solution set for the depth of the hole for the given temperature range?
(a)
(b)
(c)
(d)
ASSERTION-REASON QUESTIONS Directions: In each of the following, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:
Assertion (A): If 8 < 5(x + 1) – 2 ≤ 18, x ∈ R, then the smallest integer value of x is 0.
Reason (R): Multiplying each side of an inequation by the same integer does not change the inequality.
A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false
Assertion (A): For the inequation –12 < 3 – 4x ≤ 11, x ∈ N, the solution set is {1, 2, 3, 4}.
Reason (R): The set of all those values of x from the replacement set which satisfy the given inequation is called the solution set of the inequation.
A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false
Assertion (A): If 2x – 5 ≤ 5x + 4 < 11, x ∈ I, then the greatest value of x is 1.
Reason (R): Adding or subtracting a negative integer to each side of an inequation does not change the inequality.
A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE 4 Linear Inequations COMPETENCY-FOCUSED QUESTIONS [Pages 46 - 48]
I. Multiple Choice Questions Choose the correct alternative:
Which of the following is not a linear inequation?
3x – 8 > 5 + 2x
`5/2x - 3 ≤ 9/4x + 12`
`4x - 7/3 ≥ 13x + 8`
`6x + 13 < 4/x - 3`
Which of the following is a linear inequation?
x2 + 3 ≥ 2x – 7
`7/(x - 1) < 3x + 4`
`7/3x - 9 ≤ 5 + 4/7x`
`5 - 9/x > 3x + 4`
Which of the following is not a general form of a linear inequation?
ax + b > c
`a/x + b ≥ c`
аx + b ≤ c
ax2 + b < c
Which of the following is reducible to a linear inequation?
7 – x2 ≤ 5x + 3
`3/x - 8 > 4`
`4 - 2/x ≥ 1/x^2 - 6`
11 – x ≤ 5x2 + 6
Which of the following is not reducible to a linear inequation?
`3 - 7/x < 5`
`8 + 3/x ≥ 5/x - 4`
`6/x - 8 > 4/x + 3x`
`3/(x - 1) - 7 < 9`
Which of the following is not true for the linear inequations?
Adding the same number to each side of an inequation does not change the inequality.
Multiplying each side of an inequation by the same positive number reverses the inequality.
Multiplying each side of an inequation by the same negative number reverses the inequality.
Dividing each side of an inequation by the same positive number does not change the inequality.
Which of the following is not true?
p > q ⇔ q > p
p < q ⇔ q > p
p ≥ q ⇔ q ≤ p
p ≤ q ⇔ q ≥ p
If – 4x > 8y, then
x > 2y
x > –2y
x < –2y
x < 2y
Which of the following is the solution set of x ≤ 7, when the replacement set is the set of natural numbers?
{1, 2, 3, 4, 5, 6, 7}
{0, 1, 2, 3, 4, 5, 6, 7}
{1, 2, 3, 4, 5, 6}
{0, 1, 2, 3, 4, 5, 6}
Which of the following is the solution set of x < 0, when the replacement set is the set of whole numbers?
{0}
{......, –3, –2, –1}
{............., –3, –2, –1, 0}
Null set
Which of the following is the solution set of x ≤ 3, when the replacement set is the set of integers?
{0, 1, 2, 3}
{.........., –2, –1, 0, 1, 2}
{......, –2, –1, 0, 1, 2, 3}
{......, –2, –1, 1, 2, 3}
Which of the following statements is not true? (r is a positive integer)
If p < q, then p – r < q – r
If p ≥ q, then – pr ≥ – qr
If p > q, then `p/r > q/r`
If p ≤ q, then p + r ≤ q + r
Which of the following statements is true? (r is a positive integer)
If p ≥ q, then `(-p)/r ≤ (-q)/r`
If p ≤ q, then pr ≥ qr
If p > q, then – pr > – qr
If p < q, then p – r > q – r
Graphical representation of the following inequation on the number line is {x : –3 < x < 2, x ∈ I}
Which of the following is the correct graphical representation of {x : x < 1, x ∈ I} on the number line?
Which of the following is the correct graphical representation of {x : – 4 < x ≤ 2, x ∈ R} on the number line?
The solution set of 4x – 3 ≤ 5, where x ∈ N, is ______.
{1}
{1, 2}
{0, 1, 3}
none of these
The solution set of 3x + 1 > 5x – 7, where x ∈ R, is ______.
{x : x < 4, x ∈ R}
{x : x > 4, x ∈ R}
{x : x < 8, x ∈ R}
{x : x > – 4, x ∈ R}
The solution set of 4x – 9 ≥ 7, where x ∈ {1, 2, 3, 4, 5, 6, 7, 8}, is ______.
{1, 2, 3, 4}
{4, 5, 6, 7, 8}
{1, 2, 3}
{5, 6, 7, 8}
Find the values of x in the inequation 3x – 2 > 9x – 16, where x ∈ I.
{–2, –1, 0 1, 2}
{2, 3, 4, 5, ...........}
{........, –2, –1, 0, 1, 2}
{........, –2, –1, 0, 1, 2, 3}
The largest value of x for which, 3(x – 2) ≤ 6 – x, where x ∈ W, is ______.
3
4
6
none of these
If x is a negative integer, then find the solution set of 3 + 2 (x + 1) > – 1.
{–3, –2, 1}
{–2, –1}
{–1}
none of these
Find the smallest value of x in the following inequation.
3(x + 4) ≤ 5(x – 1) + 4 and x ≤ N
5
6
7
8
What is the smallest value of x in the following inequation?
20 – 5x < 5(x + 8) and x ∈ I
–1
0
1
cannot be determined
If – 3 ≤ – 4x + 5 and x ε W, then the solution set is ______.
{.... –3, –2, –1, 0, 1, 2, 3...}
{1, 2}
{0, 1, 2}
{2, 3, 4, 5}
Given, `x + 2 ≤ x/3 + 3` and x is a prime number. The solution set for x is ______.
∅
{0}
{1}
{0, 1}
If 2x – 15 > 4x + 9, then:
x < – 12
x < 12
x > – 12
x > 12
The solution set for `0 < − x/3 < 2, x ∈ z` is ______.
{−5, −4, −3, −2, −1}
{−6, −5, −4, −3, −2, 1}
{−5, −4, −3, −2, −1, −0}
{−6, −5, −4, −3, −2, −1, 0}
What is the solution set for the inequation represented by the following number line?
{x ∈ R : – 3 < x ≤ 4)
{x ∈ R : – 3 < x < 4}
{x ∈ R : – 3 ≤ x < 4}
{x ∈ R : – 3 ≤ x ≤ 4}
The solution set of the inequation x – 3 ≥ – 5, x ∈ R is ______.
{x : x > – 2, x ∈ R}
{x : x ≤ – 2, x ∈ R}
{x : x ≥ – 2, x ∈ R}
{–2, –1, 0, 1, 2}
Find the greatest integer which is such that if 7 is added to its double, then the resulting number is greater than three times the integer.
7
8
6
none of these
The solution set for the inequation 2x + 4 ≤ 14, x ∈ W is ______.
{1, 2, 3, 4, 5}
{0, 1, 2, 3, 4, 5}
{1, 2, 3, 4}
{0, 1, 2, 3, 4}
Solutions for 4: Linear Inequations
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE chapter 4 - Linear Inequations
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Concepts covered in Mathematics [English] Class 10 ICSE chapter 4 Linear Inequations are Linear Inequations, Representation of Inequalities, Combining Inequalities, Product of Two Linear Expressions, Method of Solving a Linear Inequality, Linear Inequations, Representation of Inequalities, Combining Inequalities, Product of Two Linear Expressions, Method of Solving a Linear Inequality.
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