Problem 16 - A Graphical Approach to Precalculus with Limits

Question

The compound inequality \(a < x < b\) means "a is less than \(x,\) and \(x\) is less than \(b\) ". Which one of the following inequalities is not satisfied by some real number \(x ?\) A. \(-3 < x < 5\) B. \(0 < x < 4\) C. \(-3 < x < -2\) D. \(-7 < x < -10\)

Step-by-Step Solution

Answer

The inequality \(-7 < x < -10\) (Option D) cannot be satisfied by any real number.

Step 1: Understanding Compound Inequalities

A compound inequality involves two separate inequalities joined by 'and'. The inequality \(a < x < b\) signifies all values of \(x\) that are greater than \(a\) and less than \(b\). Our task is to find an inequality that does not hold true for any real numbers \(x\).

Step 2: Analyze Option A

The inequality \(-3 < x < 5\) includes all real numbers between -3 and 5. Therefore, there are indeed real numbers that can satisfy this inequality, such as \(x = 0\). Hence, this option is satisfied by some real numbers.

Step 3: Analyze Option B

The inequality \(0 < x < 4\) includes all real numbers between 0 and 4. There are real numbers that satisfy this condition, such as \(x = 2\). Thus, this option is also satisfied by real numbers.

Step 4: Analyze Option C

The inequality \(-3 < x < -2\) includes all real numbers between -3 and -2. An example of such a number is \(x = -2.5\). This option is, therefore, satisfied by some real numbers.

Step 5: Analyze Option D

The inequality \(-7 < x < -10\) implies \(x\) should be greater than -7 and also less than -10. However, there are no real numbers that can be simultaneously greater than -7 and less than -10. Therefore, this inequality is not satisfied by any real number.

Step 6: Conclusion: Determine the Unsatisfied Inequality

Comparing all options, the inequality in option D \(-7 < x < -10\) is the one that can't be satisfied by any real number.

Key Concepts

Real Numbers

Real numbers comprise a broad collection of numbers that include both rational and irrational numbers. Rational numbers can be written as fractions, such as \( \frac{1}{2} \) or \( -3 \), while irrational numbers are non-repeating, non-terminating decimals like \( \pi \) and \( \sqrt{2} \). Real numbers include:

• Whole numbers like 0, 1, 2
• Integers such as -3, 0, 4
• Fractions and decimals like \( \frac{1}{2} \) and 0.75
• Surds such as \( \sqrt{5} \)

Real numbers are represented on the number line, where every point on the line corresponds to a real number. This characteristic is crucial for understanding inequalities, as it helps us visualize the relative size of numbers at a glance. In the context of the exercise, we are exploring the concept of compound inequalities to check which inequalities can or cannot be satisfied by real numbers.

Inequalities

Inequalities are mathematical expressions that show how numbers relate to each other in terms of greater or lesser value. They are as fundamental as equalities (equations) and come in various forms:

• Strict inequalities, such as \( < \) (less than) or \( > \) (greater than)
• Non-strict inequalities, including \( \leq \) (less than or equal to) and \( \geq \) (greater than or equal to)

In compound inequalities, like the ones in the exercise, two inequalities are combined using 'and' or 'or'. This means finding values that satisfy both conditions simultaneously. For example, \( a < x < b \) is a compound inequality indicating \( x \) is greater than \( a \) and less than \( b \). Understanding this concept helps identify the key differences between achievable and unachievable conditions. From the exercise, we learned how breaking down each inequality reveals whether it holds for any real numbers.

Mathematical Analysis

Mathematical analysis focuses on understanding functions and limits. While it's a broad field, the principles of analysis help us work with inequalities by simplifying complex expressions and finding solution sets. In analyzing compound inequalities, we use a step-by-step logical approach to evaluate if real number solutions exist.
The analysis of each option in the exercise was systematic:

• Assess if any real numbers fit the inequality conditions.
• Use logic to determine realizable values within the set bounds.
• Identify inconsistencies, as demonstrated with option D, where no real number could satisfy \( -7 < x < -10 \).

Such analysis helps students understand not only the outcome but also the process, enabling them to tackle similar problems independently. By practicing analysis through step-by-step assessments, students become proficient in determining when and why certain inequalities hold within the realm of real numbers.