Which Of The Following Inequalities Is True

Which of the Following Inequalities is True? A Deep Dive into Comparison and Solution Strategies

Determining which inequality holds true amongst a set of options requires a systematic approach. This isn't simply about plugging numbers into formulas; it's about understanding the underlying principles of inequalities, employing various problem-solving strategies, and developing a strong intuition for numerical relationships. This article will explore various techniques and examples to help you confidently tackle inequality problems of increasing complexity.

Understanding the Fundamentals of Inequalities

Before diving into specific examples, let's solidify our understanding of inequality symbols and their meanings:

• > (Greater than): a > b means 'a' is strictly larger than 'b'.
• < (Less than): a < b means 'a' is strictly smaller than 'b'.
• ≥ (Greater than or equal to): a ≥ b means 'a' is larger than or equal to 'b'.
• ≤ (Less than or equal to): a ≤ b means 'a' is smaller than or equal to 'b'.

Key Properties of Inequalities:

• Addition/Subtraction Property: Adding or subtracting the same number to both sides of an inequality maintains the inequality's direction. If a > b, then a + c > b + c and a - c > b - c.
• Multiplication/Division Property: Multiplying or dividing both sides by a positive number maintains the inequality's direction. If a > b and c > 0, then ac > bc and a/c > b/c. However, multiplying or dividing by a negative number reverses the inequality's direction. If a > b and c < 0, then ac < bc and a/c < b/c.
• Transitive Property: If a > b and b > c, then a > c. This property applies to all inequality symbols.

Solving Inequalities: A Step-by-Step Approach

Let's illustrate these principles with example problems, demonstrating various solution strategies.

Example 1: Simple Linear Inequalities

Which of the following inequalities is true?

A. 3x + 5 > 14 if x = 3 B. 2x - 7 < 1 if x = 4 C. 5x + 2 ≥ 17 if x = 2 D. -2x + 10 ≤ 6 if x = 2

Solution:

A. Substitute x = 3 into 3x + 5 > 14: 3(3) + 5 > 14 => 14 > 14 (False) B. Substitute x = 4 into 2x - 7 < 1: 2(4) - 7 < 1 => 1 < 1 (False) C. Substitute x = 2 into 5x + 2 ≥ 17: 5(2) + 2 ≥ 17 => 12 ≥ 17 (False) D. Substitute x = 2 into -2x + 10 ≤ 6: -2(2) + 10 ≤ 6 => 6 ≤ 6 (True)

Therefore, the correct answer is D.

Example 2: Inequalities Involving Fractions

Which inequality is true?

A. (3/4)x + 1 > 3 if x = 3 B. (2/5)x - 2 < 0 if x = 4 C. -(1/2)x + 5 ≥ 4 if x = 2

Solution:

A. (3/4)(3) + 1 > 3 => (9/4) + 1 > 3 => (13/4) > 3 => 3.25 > 3 (True) B. (2/5)(4) - 2 < 0 => (8/5) - 2 < 0 => (8/5) - (10/5) < 0 => (-2/5) < 0 (True) C. -(1/2)(2) + 5 ≥ 4 => -1 + 5 ≥ 4 => 4 ≥ 4 (True)

In this case, A, B, and C are all true. The question needs to be rephrased to have a single correct answer, perhaps by providing more complex inequalities with overlapping ranges or using different values of 'x'.

Example 3: Inequalities with Absolute Values

Which inequality holds true?

A. |x - 3| < 2 if x = 1 B. |2x + 1| ≥ 5 if x = 2 C. |x + 4| ≤ 1 if x = -3

Solution:

A. |1 - 3| < 2 => |-2| < 2 => 2 < 2 (False) B. |2(2) + 1| ≥ 5 => |5| ≥ 5 => 5 ≥ 5 (True) C. |-3 + 4| ≤ 1 => |1| ≤ 1 => 1 ≤ 1 (True)

Therefore, options B and C are true in this example.

Advanced Inequality Strategies

1. Graphical Representation: Visualizing inequalities on a number line or coordinate plane can significantly aid in understanding and solving complex problems. For example, shading regions representing solution sets can reveal overlapping ranges and relationships between inequalities.

2. Interval Notation: Expressing solution sets using interval notation (e.g., (a, b), [a, b], (a, b], [a, b)) provides a concise and efficient way to represent the range of values satisfying an inequality.

3. Systems of Inequalities: Problems often involve multiple inequalities simultaneously. Solving these systems requires finding the region that satisfies all inequalities in the system. Graphing is particularly useful in these situations.

4. Inequalities Involving Polynomials and Quadratic Expressions: Solving inequalities involving higher-order polynomials or quadratic expressions requires factoring and analyzing the signs of the expression across different intervals defined by the roots.

Example 4: Quadratic Inequality

Solve the inequality x² - 4x + 3 < 0.

Solution:

1. Factor the quadratic: (x - 1)(x - 3) < 0
2. Find the roots: x = 1 and x = 3
3. Analyze the intervals:
   • x < 1: (negative)(negative) > 0
   • 1 < x < 3: (positive)(negative) < 0
   • x > 3: (positive)(positive) > 0

Therefore, the solution to the inequality is 1 < x < 3, or in interval notation (1, 3).

Applying Inequalities in Real-World Scenarios

Inequalities are not just abstract mathematical concepts; they have numerous applications in various fields:

• Optimization Problems: Finding maximum or minimum values subject to constraints often involves solving systems of inequalities (linear programming).
• Engineering and Physics: Many physical laws and constraints are expressed as inequalities (e.g., stress limits in materials science).
• Economics and Finance: Inequalities are used to model resource allocation, budget constraints, and profit maximization.
• Statistics: Confidence intervals and hypothesis testing frequently rely on inequality-based methods.

Conclusion

Mastering inequalities requires a combination of theoretical understanding and practical problem-solving skills. By systematically applying the properties of inequalities, employing various solution techniques, and understanding their practical applications, you can confidently tackle a wide range of challenges involving inequalities, from simple comparisons to complex systems of equations. Remember to visualize, analyze, and always double-check your solutions to ensure accuracy. The more you practice, the more intuitive and proficient you will become in solving inequality problems.