Linear Inequalities

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Summary

Chapter 5: Linear Inequalities

Summary

Linear inequalities involve expressions with inequality signs: <, >, ≤, ≥.
They can be solved for different sets of numbers: natural numbers, integers, and real numbers.
Graphical representation of solutions is essential for understanding.
Rules for solving inequalities include adding/subtracting equal numbers and multiplying/dividing by positive numbers without changing the inequality sign, but reversing it when multiplying/dividing by negative numbers.
Examples include solving inequalities for average marks, costs, and temperature ranges.

Key Examples

Example 1: Solve 30x < 200 for natural numbers: {1, 2, 3, 4, 5, 6}.
Example 2: Solve 5x - 3 < 3x + 1 for integers: {..., -4, -3, -2, -1, 0, 1}.
Example 3: Solve 4x + 3 < 6x + 7: x > -2.
Example 4: Solve 7x + 3 < 5x + 9: x < 3.

Important Diagrams

Number Line Representation:
- Solid dot indicates included values.
- Open circle indicates excluded values.
- Arrows indicate ranges extending to infinity.

Common Mistakes & Exam Tips

Mistake: Forgetting to reverse the inequality sign when multiplying/dividing by a negative number.
Tip: Always check the solution by substituting values back into the original inequality.

Learning Objectives

Understand the concept of linear inequalities in one and two variables.
Solve linear inequalities for different types of numbers (natural, integers, real).
Graphically represent the solutions of linear inequalities on a number line.
Apply linear inequalities to real-world problems, such as budgeting and averages.
Distinguish between strict and slack inequalities.
Identify and solve systems of linear inequalities.

Detailed Notes

Chapter 5: Linear Inequalities

5.1 Introduction

Study of linear inequalities in one and two variables.
Useful in various fields: science, mathematics, statistics, economics, psychology.

5.2 Inequalities

Definition

Two real numbers or algebraic expressions related by symbols: <, >, ≤, or ≥ form an inequality.
Examples:
- Numerical inequalities: 3 < 5; 7 > 5
- Literal inequalities: x < 5; y > 2; x ≥ 3; y ≤ 4
- Double inequalities: 3 < x < 5

Examples of Inequalities

Strict Inequalities:
- ax + b < 0
- ax + b > 0
Slack Inequalities:
- ax + b ≤ 0
- ax + b ≥ 0
Linear Inequalities in Two Variables:
- ax + by < c
- ax + by > c

Examples of Problem Solving

Example 1: Ravi's rice purchase
- Total amount spent: ₹30x < 200
Example 2: Reshma's purchase of registers and pens
- Total amount spent: 40x + 20y ≤ 120

5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

Solving Inequalities

Example: Solve 4x + 3 < 6x + 7
- Solution: x > -2 (solution set: (-2, ∞))
Example: Solve 5 - 2x ≤ 8
- Solution: x ≥ 8 (solution set: [8, ∞))

Graphical Representation

Fig 5.1: Number line showing solutions of inequalities.
Fig 5.2: Graphical representation of solutions for x < 3.

5.4 Miscellaneous Examples

Example 9: Solve -8 ≤ 5x - 3 < 7
- Solution: -1 ≤ x < 2
Example 10: Solve the system of inequalities: 3x - 7 < 5 + x < 11 - 5x
- Solution: 2 ≤ x < 6

Important Diagrams

Diagram 1: Number line representing x < 3 (open circle at 3).
Diagram 2: Number line representing x ≥ 1 (closed dot at 1).

Conclusion

Linear inequalities provide a framework for solving various real-world problems through algebraic expressions and graphical representations.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips for Linear Inequalities

Common Pitfalls

Misinterpreting Inequalities: Students often confuse the symbols for inequalities (e.g., using > instead of <). Always double-check the direction of the inequality sign.
Ignoring the Domain: When solving inequalities, it's crucial to consider the specified domain (natural numbers, integers, real numbers). Solutions can vary significantly based on this.
Incorrectly Applying Rules: When multiplying or dividing by negative numbers, remember to reverse the inequality sign. This is a common mistake that can lead to incorrect solutions.

Tips for Success

Graphical Representation: Always represent your solutions on a number line. This visual aid can help you understand the range of solutions better and avoid mistakes.
Check Your Solutions: After finding the solutions, substitute them back into the original inequality to verify their correctness.
Practice with Different Domains: Solve inequalities under different conditions (natural numbers, integers, real numbers) to become familiar with how solutions change.
Use Systematic Approaches: Instead of trial and error, apply systematic techniques for solving inequalities to save time and reduce errors.

Practice & Assessment

Multiple Choice Questions

Correct Answer: B

Solution:

Let

x

be the marks obtained in the annual examination. Then,

\frac{62 + 48 + x}{3} \geq 60

. Solving, we get

x \geq 70

Correct Answer: B

Solution:

The inequality is

30x < 200

. Solving for

x

, we get

x < \frac{200}{30} = \frac{20}{3} \approx 6.67

. Since

x

must be a whole number, the maximum number of packets Ravi can buy is 6.

(1, 3)

(3, 5)

(5, 7)

(7, 9)

Correct Answer: D

Solution:

Let the integers be

x

and

x+2

. We have

x + (x + 2) > 11

. Solving gives

2x + 2 > 11

, so

x > 4.5

. The pairs are (5, 7) and (7, 9), but only (7, 9) is smaller than 10.

500 litres

750 litres

1000 litres

1500 litres

Correct Answer: B

Solution:

Let

x

be the litres of water added. The new total volume is

1125 + x

. The new acid concentration is

\frac{0.45 \times 1125}{1125 + x}

. We need

0.25 < \frac{0.45 \times 1125}{1125 + x} < 0.30

. Solving these inequalities gives

750 < x < 2250

. Therefore, 750 litres of water should be added.

Correct Answer: A

Solution:

To find the minimum score, solve

\frac{62 + 48 + x}{3} \geq 60

, which simplifies to

x \geq 70

40x + 20y \leq 120

40x + 20y < 120

40x + 20y \geq 120

40x + 20y > 120

Correct Answer: A

Solution:

The inequality representing her spending limit is

40x + 20y \leq 120

because she cannot spend more than ₹120.

1, 2, 3, 4

1, 2, 3

1, 2, 3, 4, 5

1, 2

Correct Answer: A

Solution:

Dividing both sides by 24, we get

x < \frac{100}{24} \approx 4.17

. Therefore, the natural numbers satisfying the inequality are 1, 2, 3, and 4.

Correct Answer: A

Solution:

After buying 2 registers, Reshma spends ₹80 (2 x 40). She has ₹40 left, which allows her to buy 2 pens (40/20 = 2).

(6, 8), (8, 10)

(6, 8), (8, 10), (10, 12)

(6, 8), (8, 10), (10, 12), (12, 14)

(8, 10), (10, 12)

Correct Answer: A

Solution:

Let the integers be

x

and

x+2

. Then,

x > 5

and

x + (x + 2) < 23

. Solving, we get

6 < x < 10.5

. Thus, the pairs are (6, 8) and (8, 10).

Correct Answer: D

Solution:

The total marks needed for an average of 90 is 450. The sum of the first four scores is 368. Therefore, she needs at least 450 - 368 = 82 in the fifth exam.

x < 2

x > 2

x \leq 2

x \geq 2

Correct Answer: A

Solution:

Rearranging the inequality gives

5x - 3 < 3x + 1 \Rightarrow 2x < 4 \Rightarrow x < 2

. Therefore, the solution is

x < 2

Correct Answer: C

Solution:

Let

x

be the marks obtained in the third test. The average is

\frac{70 + 75 + x}{3} \geq 60

. Solving gives

145 + x \geq 180

, so

x \geq 35

9.6 to 16.8

10 to 14

8 to 15

11 to 17

Correct Answer: A

Solution:

For

80 \leq IQ \leq 140

, we have

80 \leq \frac{MA}{12} \times 100 \leq 140

. Solving,

9.6 \leq MA \leq 16.8

x < -2.5

x > -2.5

x < -3

x > -3

Correct Answer: A

Solution:

Dividing both sides by -12 and reversing the inequality sign, we get

x < -2.5

320 litres

640 litres

960 litres

1280 litres

Correct Answer: B

Solution:

Let x be the litres of 2% solution added. The inequality becomes 4 < (8*640 + 2x)/(640 + x) < 6. Solving gives x = 640 litres.

20°C to 25°C

22°C to 25°C

18°C to 24°C

19°C to 23°C

Correct Answer: A

Solution:

Using the conversion formula

F = \frac{9}{5}C + 32

, we solve

68 \leq \frac{9}{5}C + 32 \leq 77

. Solving these inequalities gives

20 \leq C \leq 25

20 cm

21 cm

22 cm

23 cm

Correct Answer: B

Solution:

Let the shortest length be

x

. Then the second length is

x + 3

and the third length is

2x

. The total length is

x + (x + 3) + 2x \leq 91

. Solving,

4x + 3 \leq 91 \Rightarrow 4x \leq 88 \Rightarrow x \leq 22

. Therefore, the maximum possible length of the shortest board is 22 cm.

5 cm

6 cm

7 cm

8 cm

Correct Answer: C

Solution:

Let the shortest side be

x

. Then the longest side is

3x

and the third side is

3x - 2

. The perimeter is

x + 3x + (3x - 2) \geq 61

. Solving,

7x - 2 \geq 61 \Rightarrow 7x \geq 63 \Rightarrow x \geq 9

. Therefore, the minimum length of the shortest side is 9 cm.

2 \leq x < 4

3 \leq x < 5

4 \leq x < 6

1 \leq x < 3

Correct Answer: A

Solution:

First, solve

6 \leq 3(2x - 4)

6 \leq 6x - 12

, or

18 \leq 6x

, giving

x \geq 3

. Next, solve

3(2x - 4) < 12

6x - 12 < 12

, or

6x < 24

, giving

x < 4

. Combining,

3 \leq x < 4

. Therefore, the solution is

2 \leq x < 4

True or False

Correct Answer: True

Solution:

Solving the inequality

5x - 3 < 3x

gives

x < 2

. Therefore, integer solutions are

x = -4, -3, -2, -1, 0, 1

Correct Answer: False

Solution:

Let the shortest length be

x

. Then, the lengths are

x

x + 3

, and

2x

. The total length is

x + (x + 3) + 2x \leq 91

, simplifying to

4x + 3 \leq 91

, or

4x \leq 88

, giving

x \leq 22

. Additionally, for the third piece to be at least 5 cm longer than the second,

2x \geq x + 3 + 5

, simplifying to

x \geq 8

. Therefore,

x

must be between 8 and 22, so 20 cm is possible.

Correct Answer: True

Solution:

Let the shortest side be

x

. Then the longest side is

3x

and the third side is

3x - 2

. The perimeter is

x + 3x + (3x - 2) = 7x - 2

. Setting

7x - 2 \geq 61

gives

7x \geq 63

, so

x \geq 9

Correct Answer: False

Solution:

Solving the inequality

3x - 7 < 5 + x

gives

2x < 12

, so

x < 6

. The solution set does not include

x \geq 6

Correct Answer: True

Solution:

By adding a 2% solution to the 8% solution, the concentration can be adjusted to be more than 4% but less than 6%.

Correct Answer: False

Solution:

Solving

-12x > 30

gives

x < -2.5

. However, there are no natural numbers that satisfy this inequality.

Correct Answer: True

Solution:

To achieve a boric acid content between 4% and 6%, the amount of 2% solution added must satisfy the inequality for the resulting mixture's concentration. Solving this inequality confirms the possibility.

Correct Answer: True

Solution:

When both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed to maintain the truth of the statement.

Correct Answer: True

Solution:

Using the formula

IQ = \frac{MA}{CA} \times 100

, for CA = 12, the mental age MA ranges from 9.6 to 16.8 years.

Correct Answer: True

Solution:

Solving

-12x > 30

gives

x < -2.5

. Since natural numbers are positive integers, there are no natural numbers that satisfy this inequality.

Correct Answer: True

Solution:

Solving

30x < 200

gives

x < \frac{200}{30} = \frac{20}{3}

. The integer solutions are

x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6

Correct Answer: False

Solution:

The average of five exams must be at least 90. So,

\frac{87 + 92 + 94 + 95 + x}{5} \geq 90

. Solving gives

x \geq 92

. However, the correct calculation shows

x \geq 92

is needed, making the statement true.

Correct Answer: True

Solution:

The IQ formula is

IQ = \frac{MA}{CA} \times 100

. For a 12-year-old,

80 \leq \frac{MA}{12} \times 100 \leq 140

simplifies to

9.6 \leq MA \leq 16.8

Correct Answer: False

Solution:

For the inequality

30x < 200

, the natural number solutions are 1 through 6. For

x = 7

30 \times 7 = 210

, which is not less than 200.

Correct Answer: True

Solution:

For the inequality

30x < 200

, when

x

is an integer, the values that satisfy the inequality are

-3, -2, -1, 0, 1, 2, 3, 4, 5, 6

. This is because

x

must be less than

20/3

Correct Answer: False

Solution:

Solving

-12x > 30

gives

x < -2.5

. Since natural numbers are positive, there are no natural number solutions.

Correct Answer: False

Solution:

Let the shortest piece be

x

. Then the second piece is

x + 3

and the third is

2x

. The condition

x + (x + 3) + 2x \leq 91

gives

4x + 3 \leq 91

, so

x \leq 22

. Also,

2x \geq (x + 3) + 5

gives

x \geq 8

. Thus,

8 \leq x \leq 22

, so the shortest piece can be as small as 8 cm.

Correct Answer: True

Solution:

Solving the inequality

3x - 7 < 5 + x

gives

2x < 12

, which simplifies to

x < 6

. Thus, the solution is

x < 6

Correct Answer: False

Solution:

When both sides of an inequality are multiplied (or divided) by a negative number, the inequality sign is reversed.

Correct Answer: True

Solution:

Using the formula

\text{IQ} = \frac{\text{MA}}{\text{CA}} \times 100

, where CA is 12, the range for MA is calculated as

\frac{80 \times 12}{100} \leq \text{MA} \leq \frac{140 \times 12}{100}

, resulting in

9.6 \leq \text{MA} \leq 16.8

Correct Answer: True

Solution:

Using the formula

\text{IQ} = \frac{\text{MA}}{\text{CA}} \times 100

, where MA is mental age and CA is chronological age, we have

80 \leq \frac{\text{MA}}{12} \times 100 \leq 140

. Solving for MA gives

9.6 \leq \text{MA} \leq 16.8

Correct Answer: True

Solution:

According to Rule 1, equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality.

Correct Answer: True

Solution:

Using the conversion formula

F = \frac{9}{5}C + 32

, we can convert 68°F to 20°C and 77°F to 25°C.

Correct Answer: False

Solution:

The integer solutions for the inequality 30x < 200 are -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6.

Correct Answer: True

Solution:

Solving

-12x > 30

gives

x < -2.5

. The integer solutions are all integers less than -2, including -3.

Correct Answer: True

Solution:

The IQ formula is given by

IQ = \frac{MA}{CA} \times 100

. For a chronological age (CA) of 12, the mental age (MA) must satisfy

80 \leq \frac{MA}{12} \times 100 \leq 140

. Solving gives

9.6 \leq MA \leq 16.8

Correct Answer: False

Solution:

The inequality

-12x > 30

simplifies to

x < -\frac{5}{2}

. Since natural numbers are positive, there are no natural number solutions.

Correct Answer: True

Solution:

The inequality is

30x < 200

. Solving for

x

, we get

x < \frac{200}{30} = \frac{20}{3} \approx 6.67

. Since

x

must be a whole number, Ravi can buy at most 6 packets.

Correct Answer: True

Solution:

The student needs to score at least 70 in the third exam to achieve an average of 60, as calculated by

\frac{62 + 48 + x}{3} \geq 60

, which simplifies to

x \geq 70

Correct Answer: True

Solution:

Ravi has ₹200 and each packet costs ₹30. The inequality

30x < 200

correctly represents the scenario where

x

is the number of packets he can buy.

Correct Answer: True

Solution:

Solving

3x - 7 < 5 + x

gives

3x - x < 5 + 7

, leading to

2x < 12

, so

x < 6

Correct Answer: True

Solution:

When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign is reversed to maintain the truth of the statement.

Correct Answer: True

Solution:

Using the conversion formula

F = \frac{9}{5}C + 32

, converting 68°F gives

C = \frac{5}{9}(68 - 32) = 20°C

, and 77°F gives

C = \frac{5}{9}(77 - 32) = 25°C

. Thus, the range is 20°C to 25°C.

Correct Answer: True

Solution:

The inequality

120 < x < 300

directly describes the condition that the solution has more than 120 liters but less than 300 liters.

Correct Answer: False

Solution:

Solving the inequality gives

2x < 4

x < 2

. Therefore,

x = 2

is not included in the solution set.

Correct Answer: True

Solution:

When solving

30x < 200

for natural numbers, dividing both sides by 30 gives

x < \frac{200}{30} = \frac{20}{3}

. The natural numbers less than

\frac{20}{3}

are

1, 2, 3, 4, 5, 6

Correct Answer: True

Solution:

Dividing both sides by 30 gives

x < \frac{200}{30}

, which simplifies to

x < 6.67

. Therefore, the natural numbers satisfying this inequality are 1, 2, 3, 4, 5, and 6.

Correct Answer: True

Solution:

According to Rule 1, equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of the inequality.

Correct Answer: True

Solution:

Solving

-12x > 30

gives

x < -\frac{30}{12} = -2.5

. Thus,

x

can be any integer less than -2.5, such as -3, -4, etc.

Correct Answer: True

Solution:

Ravi can buy rice packets such that

30x < 200

. Solving gives

x < \frac{200}{30} = \frac{20}{3}

. Thus, he can buy at most 6 packets.

Correct Answer: False

Solution:

Solving

3x - 7 < 5 + x

gives

2x < 12

, hence

x < 6

. The solution does not include

x \geq 2

Correct Answer: True

Solution:

The solution to the inequality 5x - 3 < 3x + 1 is x < 2, which includes all real numbers less than 2.

Correct Answer: True

Solution:

If x is the smaller odd number, then x + (x + 2) < 40. Solving gives 10 < x < 19, so pairs like (11, 13) and (13, 15) satisfy the condition.

Correct Answer: True

Solution:

Solving

2x + 2 < 40

gives

x < 19

. If

x

is an odd number greater than 10, the possible values are 11, 13, 15, and 17.

Correct Answer: True

Solution:

The inequality is

\frac{62 + 48 + x}{3} \geq 60

. Solving gives

x \geq 70

Correct Answer: True

Solution:

Using the conversion formula

F = \frac{9}{5}C + 32

, convert 68°F and 77°F to Celsius:

C = \frac{5}{9}(68 - 32) = 20

and

C = \frac{5}{9}(77 - 32) = 25

. Thus, the temperature range in Celsius is 20°C to 25°C.

Correct Answer: True

Solution:

According to Rule 1, equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality.

Correct Answer: False

Solution:

When x is a natural number, the solution set is {1, 2, 3, 4, 5, 6}. When x is an integer, the solution set is {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

Correct Answer: True

Solution:

The student needs at least 70 because

\frac{62 + 48 + x}{3} \geq 60

simplifies to

110 + x \geq 180

, leading to

x \geq 70

Correct Answer: False

Solution:

Let the shortest side be

x

. Then the longest side is

3x

and the third side is

3x - 2

. The perimeter is

x + 3x + (3x - 2) = 7x - 2

. Solving

7x - 2 \geq 61

gives

x \geq 9

Correct Answer: True

Solution:

Solving the inequality

5x - 3 < 3x + 1

gives

2x < 4

, or

x < 2

. The integer solutions include

x = 0, 1

Correct Answer: True

Solution:

Dividing both sides by 30, we get

x < \frac{200}{30} = \frac{20}{3}

. The natural numbers less than

\frac{20}{3}

are 1, 2, 3, 4, 5, and 6.

Correct Answer: False

Solution:

When both sides of an inequality are multiplied (or divided) by a negative number, the inequality sign is reversed.

Correct Answer: True

Solution:

Solving the inequality

5x - 3 < 3x + 1

gives

2x < 4

, which simplifies to

x < 2

. Therefore, the solution for

x

when

x

is a real number is

x < 2

Correct Answer: False

Solution:

Rearranging the inequality gives

4x - 5x < 7 - 3

, which simplifies to

-x < 4

. Multiplying both sides by -1 reverses the inequality, resulting in

x > -4

Correct Answer: True

Solution:

To find the minimum score in the annual exam, we set up the inequality:

\frac{62 + 48 + x}{3} \geq 60

. Solving gives

110 + x \geq 180

, thus

x \geq 70

Correct Answer: True

Solution:

Rule 2 states that when both sides of an inequality are multiplied or divided by a negative number, the sign of inequality is reversed.

Correct Answer: True

Solution:

Let

x

be the litres of 2% solution added to 640 litres of 8% solution. The inequality is

\frac{8 \times 640 + 2x}{640 + x}

must be between 4% and 6%. Solving gives the range of

x

that satisfies the condition.

Correct Answer: True

Solution:

The inequality is

\frac{62 + 48 + x}{3} \geq 60

, which simplifies to

110 + x \geq 180

, giving

x \geq 70

Correct Answer: False

Solution:

For the inequality

5x - 3 < 3x

, solving gives

2x < 3

, which means

x < 1.5

. Therefore, the integer solutions are

x = 0, 1

Correct Answer: True

Solution:

To achieve an acid content between 25% and 30%, the resulting mixture must be diluted with a certain amount of water. Solving the inequality for the amount of water added shows that it must be between 600 and 900 liters.

Correct Answer: True

Solution:

If the longest side is 3 times the shortest side and the third side is 2 cm shorter than the longest, the perimeter can be at least 61 cm if the shortest side is at least 12 cm.

Correct Answer: False

Solution:

The inequality is

40x + 20y \leq 120

. If she buys 2 registers,

40(2) + 20y \leq 120

simplifies to

80 + 20y \leq 120

, which gives

y \leq 2

. Thus, she can buy 2 registers and 2 pens.

Linear Inequalities

AI Learning Assistant

Summary

Chapter 5: Linear Inequalities

Summary

Key Examples

Important Diagrams

Common Mistakes & Exam Tips

Learning Objectives

Learning Objectives

Detailed Notes

Chapter 5: Linear Inequalities

5.1 Introduction

5.2 Inequalities

Definition

Examples of Inequalities

Examples of Problem Solving

5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

Solving Inequalities

Graphical Representation

5.4 Miscellaneous Examples

Important Diagrams

Conclusion

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips for Linear Inequalities

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.What is the minimum marks a student should get in the annual examination to have an average of at least 60 marks, if the marks obtained in the first two exams are 62 and 48?

Correct Answer: B

Q2.Ravi goes to the market with ₹200 to buy rice, which is available in packets of 1 kg. The price of one packet of rice is ₹30. What is the maximum number of packets Ravi can buy?

Correct Answer: B

Q3.Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

Correct Answer: D

Q4.How many litres of water should be added to 1125 litres of a 45% acid solution to make the acid content between 25% and 30%?

Correct Answer: B

Q5.A student scored 62 and 48 marks in the first and second terminal exams. What is the minimum score needed in the annual exam to have an average of at least 60?

Correct Answer: A

Q6.Reshma has ₹120 and wants to buy some registers and pens. The cost of one register is ₹40 and that of a pen is ₹20. If xxx denotes the number of registers and yyy the number of pens, which inequality represents her spending limit?

Correct Answer: A

Q7.Solve the inequality: 24x<10024x < 10024x<100 when xxx is a natural number.

Correct Answer: A

Q8.Reshma has ₹120 and wants to buy some registers and pens. The cost of one register is ₹40 and that of a pen is ₹20. If she buys 2 registers, how many pens can she buy with the remaining money?

Correct Answer: A

Q9.Find all pairs of consecutive even positive integers, both larger than 5, such that their sum is less than 23.

Correct Answer: A

Q10.Sunita needs an average of 90 marks in five exams to receive Grade 'A'. If her scores in the first four exams are 87, 92, 94, and 95, what is the minimum score she needs in the fifth exam?

Correct Answer: D

Q11.Solve the inequality 5x−3<3x+15x - 3 < 3x + 15x−3<3x+1 for xxx when xxx is an integer.

Correct Answer: A

Q12.Ravi obtained 70 and 75 marks in the first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Correct Answer: C

Q13.If the IQ of a person is given by the formula IQ=MACA×100IQ = \frac{MA}{CA} \times 100IQ=CAMA​×100, where MAMAMA is mental age and CACACA is chronological age, what is the range of mental age for a 12-year-old child with an IQ between 80 and 140?

Correct Answer: A

Q14.Solve the inequality: −12x>30-12x > 30−12x>30 when xxx is an integer.

Correct Answer: A

Q15.A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

Correct Answer: B

Q16.A solution is to be kept between 68°F and 77°F. What is the range in temperature in degree Celsius (C) if the Celsius/Fahrenheit conversion formula is given by F=95C+32F = \frac{9}{5}C + 32F=59​C+32?

Correct Answer: A

Q17.A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What is the maximum possible length of the shortest board?

Correct Answer: B

Q18.What is the minimum length of the shortest side of a triangle if the longest side is 3 times the shortest side, the third side is 2 cm shorter than the longest side, and the perimeter is at least 61 cm?

Correct Answer: C

Q19.Solve the inequality 6≤3(2x−4)<126 \leq 3(2x - 4) < 126≤3(2x−4)<12 for xxx as a real number.

Correct Answer: A

True or False

Q1.The inequality 5x−3<3x5x - 3 < 3x5x−3<3x has integer solutions that are less than 2.

Correct Answer: True

Q2.A man wants to cut three lengths from a 91 cm board: the second length is 3 cm longer than the shortest, and the third is twice the shortest. The shortest length can be 20 cm.

Correct Answer: False

Q3.For a triangle with the longest side being 3 times the shortest side and the third side 2 cm shorter than the longest, if the perimeter is at least 61 cm, the shortest side is at least 11 cm.

Correct Answer: True

Q4.For the inequality 3x−7<5+x3x - 7 < 5 + x3x−7<5+x, the solution set includes x≥6x \geq 6x≥6.

Correct Answer: False

Q5.A solution of 8% boric acid can be diluted with a 2% solution to achieve a concentration between 4% and 6%.

Correct Answer: True

Q6.To solve the inequality −12x>30-12x > 30−12x>30 for natural numbers, the solution is x<−2.5x < -2.5x<−2.5.

Correct Answer: False