Inequalities and Their Applications - Exam Prep Problem Set

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Inequalities and Their Applications - Problem Set

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Inequalities and Their Applications - Problem Set MCQ & Objective Questions

Inequalities and their applications are crucial topics in mathematics that frequently appear in school and competitive exams. Mastering this subject through practice questions can significantly enhance your understanding and boost your scores. Engaging with MCQs and objective questions not only helps in reinforcing concepts but also prepares you effectively for important exams.

What You Will Practise Here

Understanding the types of inequalities: linear, quadratic, and polynomial.
Solving inequalities using graphical methods and number lines.
Application of inequalities in real-life scenarios and word problems.
Key formulas related to inequalities and their derivations.
Identifying and using properties of inequalities in problem-solving.
Common applications of inequalities in optimization problems.
Conceptual clarity on theorems related to inequalities, such as the Cauchy-Schwarz inequality.

Exam Relevance

The topic of inequalities is highly relevant in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect questions that test both theoretical understanding and practical applications. Common question patterns include solving inequalities, graphing solutions, and applying inequalities to real-world problems. Familiarity with these patterns will enhance your exam preparation and confidence.

Common Mistakes Students Make

Misinterpreting the direction of inequalities when multiplying or dividing by negative numbers.
Overlooking the importance of checking solutions in the context of the original inequality.
Confusing the properties of different types of inequalities.
Failing to apply the correct method for solving compound inequalities.
Neglecting to represent solutions accurately on a number line or graph.

FAQs

Question: What are the key types of inequalities I should focus on for exams?
Answer: Focus on linear, quadratic, and polynomial inequalities, as these are commonly tested in exams.

Question: How can I effectively prepare for MCQs on inequalities?
Answer: Regular practice with objective questions and understanding the underlying concepts will help you excel in MCQs.

Start solving practice MCQs today to test your understanding of inequalities and their applications. The more you practice, the better prepared you will be for your exams. Good luck!

Q. Find the solution set for the inequality: x^2 - 4 > 0.

A. (-∞, -2) ∪ (2, ∞)
B. (-2, 2)
C. (2, -2)
D. (-2, ∞)

Solution

Step 1: Factor the inequality: (x - 2)(x + 2) > 0. Step 2: The solution set is (-∞, -2) ∪ (2, ∞).

Correct Answer: A — (-∞, -2) ∪ (2, ∞)

Q. Find the solution to the inequality: 4x + 1 ≥ 2x + 5.

A. x ≥ 2
B. x ≤ 2
C. x ≥ 4
D. x ≤ 4

Solution

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 5. Step 2: Subtract 1: 2x ≥ 4. Step 3: Divide by 2: x ≥ 2.

Correct Answer: A — x ≥ 2

Q. Find the solution to the inequality: 4x - 7 ≤ 9.

A. x ≤ 4
B. x ≥ 4
C. x ≤ 2
D. x ≥ 2

Solution

Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.

Correct Answer: A — x ≤ 4

Q. Find the solution to the inequality: x^2 + 4x < 5.

A. (-5, 1)
B. (1, -5)
C. (1, 5)
D. (-5, 5)

Solution

Step 1: Rearrange: x^2 + 4x - 5 < 0. Step 2: Factor: (x + 5)(x - 1) < 0. Step 3: Test intervals: solution is (-5, 1).

Correct Answer: A — (-5, 1)

Q. If 5x - 2 > 3x + 6, what is the value of x?

A. x < 4
B. x > 4
C. x = 4
D. x = 3

Solution

Step 1: Subtract 3x from both sides: 2x - 2 > 6. Step 2: Add 2: 2x > 8. Step 3: Divide by 2: x > 4.

Correct Answer: B — x > 4

Q. Solve the inequality: -4x + 1 ≤ 9.

A. x ≥ -2
B. x ≤ -2
C. x > -2
D. x < -2

Solution

Step 1: Subtract 1 from both sides: -4x ≤ 8. Step 2: Divide by -4 (reverse inequality): x ≥ -2.

Correct Answer: A — x ≥ -2

Q. Solve the inequality: 4x - 7 > 2x + 5.

A. x > 6
B. x < 6
C. x > 5
D. x < 5

Solution

Step 1: Subtract 2x from both sides: 2x - 7 > 5. Step 2: Add 7: 2x > 12. Step 3: Divide by 2: x > 6.

Correct Answer: A — x > 6

Q. Solve the inequality: 7 - 2x < 1.

A. x > 3
B. x < 3
C. x > 4
D. x < 4

Solution

Step 1: Subtract 7 from both sides: -2x < -6. Step 2: Divide by -2 (reverse the inequality): x > 3.

Correct Answer: B — x < 3

Q. Solve the inequality: x^2 + 2x - 8 < 0.

A. -4 < x < 2
B. x < -4 or x > 2
C. x > -4 and x < 2
D. x < 2

Solution

Step 1: Factor: (x + 4)(x - 2) < 0. Step 2: Critical points are x = -4 and x = 2. Step 3: Test intervals: -4 < x < 2.

Correct Answer: A — -4 < x < 2

Q. Solve the inequality: x^2 + 4x < 12.

A. x < 2 or x > -6
B. x > 2 and x < -6
C. x < -2 or x > 6
D. x > -6 and x < 2

Solution

Step 1: Rearrange: x^2 + 4x - 12 < 0. Step 2: Factor: (x + 6)(x - 2) < 0. Step 3: The solution is -6 < x < 2.

Correct Answer: D — x > -6 and x < 2

Q. Solve the inequality: x^2 - 4 > 0.

A. x < -2 or x > 2
B. x > -2 and x < 2
C. x < 2
D. x > 2

Solution

Step 1: Factor: (x - 2)(x + 2) > 0. Step 2: Test intervals: solution is x < -2 or x > 2.

Correct Answer: A — x < -2 or x > 2

Q. Solve the inequality: x^2 - 4x < 0.

A. x < 0 or x > 4
B. 0 < x < 4
C. x > 0
D. x < 4

Solution

Step 1: Factor the inequality: x(x - 4) < 0. Step 2: The critical points are x = 0 and x = 4. Step 3: Test intervals: (0, 4) is the solution.

Correct Answer: B — 0 < x < 4

Q. Solve the inequality: x^2 - 5x + 6 ≤ 0.

A. x ≤ 2 or x ≥ 3
B. x ≥ 2 and x ≤ 3
C. x < 2 or x > 3
D. x > 2 and x < 3

Solution

Step 1: Factor the quadratic: (x - 2)(x - 3) ≤ 0. Step 2: The solution is 2 ≤ x ≤ 3.

Correct Answer: B — x ≥ 2 and x ≤ 3

Q. What is the solution set for the inequality: -4x + 1 < 3?

A. x > -1/2
B. x < -1/2
C. x > 1/2
D. x < 1/2

Solution

Step 1: Subtract 1 from both sides: -4x < 2. Step 2: Divide by -4 (reverse inequality): x > -1/2.

Correct Answer: B — x < -1/2

Q. What is the solution set for the inequality: x^2 - 5x + 6 < 0?

A. (2, 3)
B. (3, 2)
C. (1, 6)
D. (0, 5)

Solution

Step 1: Factor the quadratic: (x - 2)(x - 3) < 0. Step 2: Test intervals: solution is (2, 3).

Correct Answer: A — (2, 3)

Q. What is the solution to the inequality: 2(x - 1) > 3(x + 2)?

A. x < -1
B. x > -1
C. x < 1
D. x > 1

Solution

Step 1: Distribute: 2x - 2 > 3x + 6. Step 2: Rearrange: -x > 8. Step 3: Divide by -1 (reverse inequality): x < -8.

Correct Answer: A — x < -1

Q. What is the solution to the inequality: 3(x + 2) ≤ 2(x + 5)?

A. x ≤ 1
B. x ≥ 1
C. x ≤ -1
D. x ≥ -1

Solution

Step 1: Distribute: 3x + 6 ≤ 2x + 10. Step 2: Subtract 2x: x + 6 ≤ 10. Step 3: Subtract 6: x ≤ 4.

Correct Answer: A — x ≤ 1

Q. What is the solution to the inequality: 3(x - 2) > 2(x + 1)?

A. x > 1
B. x < 1
C. x > 2
D. x < 2

Solution

Step 1: Distribute: 3x - 6 > 2x + 2. Step 2: Subtract 2x: x - 6 > 2. Step 3: Add 6: x > 8.

Correct Answer: A — x > 1

Q. What is the solution to the inequality: 4x + 1 ≥ 2x + 9?

A. x ≤ 4
B. x ≥ 4
C. x < 4
D. x > 4

Solution

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 9. Step 2: Subtract 1: 2x ≥ 8. Step 3: Divide by 2: x ≥ 4.

Correct Answer: B — x ≥ 4

Q. Which of the following is a solution to the inequality: 2x + 3 ≥ 11?

A. x = 2
B. x = 3
C. x = 4
D. x = 5

Solution

Step 1: Subtract 3 from both sides: 2x ≥ 8. Step 2: Divide by 2: x ≥ 4.

Correct Answer: C — x = 4

Q. Which of the following is a solution to the inequality: 3x + 2 < 11?

A. x = 1
B. x = 2
C. x = 3
D. x = 4

Solution

Step 1: Subtract 2 from both sides: 3x < 9. Step 2: Divide by 3: x < 3.

Correct Answer: B — x = 2

Q. Which of the following is a solution to the inequality: 4x - 1 ≤ 3x + 2?

A. x ≤ 3
B. x ≥ 3
C. x < 3
D. x > 3

Solution

Step 1: Subtract 3x from both sides: x - 1 ≤ 2. Step 2: Add 1: x ≤ 3.

Correct Answer: A — x ≤ 3

Q. Which of the following is a solution to the inequality: 6 - 2x ≤ 4?

A. x ≥ 1
B. x ≤ 1
C. x ≥ 2
D. x ≤ 2

Solution

Step 1: Subtract 6 from both sides: -2x ≤ -2. Step 2: Divide by -2 (reverse the inequality): x ≥ 1.

Correct Answer: B — x ≤ 1

Q. Which of the following represents the solution to the inequality: 4x + 1 ≥ 2x + 9?

A. x ≥ 4
B. x ≤ 4
C. x > 4
D. x < 4

Solution

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 9. Step 2: Subtract 1: 2x ≥ 8. Step 3: Divide by 2: x ≥ 4.

Correct Answer: A — x ≥ 4

Q. Which of the following represents the solution to the inequality: 7 - 2x < 1?

A. x > 3
B. x < 3
C. x > 4
D. x < 4

Solution

Step 1: Subtract 7 from both sides: -2x < -6. Step 2: Divide by -2 (reverse inequality): x > 3.

Correct Answer: A — x > 3

Q. Which of the following values satisfies the inequality: 5 - 2x > 1?

A. x < 2
B. x > 2
C. x < 1
D. x > 1

Solution

Step 1: Subtract 5 from both sides: -2x > -4. Step 2: Divide by -2 (reverse inequality): x < 2.

Correct Answer: A — x < 2

Q. Which of the following values satisfies the inequality: 5 - x > 2?

A. x < 3
B. x > 3
C. x < 5
D. x > 5

Solution

Step 1: Subtract 5 from both sides: -x > -3. Step 2: Multiply by -1 (reverse the inequality): x < 3.

Correct Answer: A — x < 3

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