Problem 76 - Calculus with Concepts in Calculus

Question

Find all numbers \(x\) with the property that the distance from \(x\) to 2 is less than twice the distance from \(x\) to 3 .

Step-by-Step Solution

Answer

The solution is all numbers \( x < \frac{8}{3} \).

Step 1: Understanding the Problem

We’re tasked with finding numbers \( x \) such that the distance from \( x \) to 2 is less than twice the distance from \( x \) to 3. We need to use the concept of absolute value to express these distances.

Step 2: Setting Up the Distance Expression

The distance from \( x \) to 2 is expressed as \( |x - 2| \) and the distance from \( x \) to 3 is expressed as \( |x - 3| \). According to the problem, \( |x - 2| < 2|x - 3| \).

Step 3: Breaking Down the Inequality

We need to solve the inequality \( |x - 2| < 2|x - 3| \). This can be split into two cases: when \( x \geq 3 \) and when \( x < 3 \), due to properties of absolute values.

Step 4: Case 1: Solving for \( x \geq 3 \)

For \( x \geq 3 \), \( |x - 3| = x - 3 \). Substituting into the inequality gives \( |x - 2| < 2(x - 3) \). Simplifying further, we consider both \( x - 2 \) and \( 2 - x \) but due to \( x \geq 3 \), \( x - 2 \) is non-negative. Thus, \( x - 2 < 2x - 6 \), leading to \( -4 < x \). However, since \( x \geq 3 \), we have \( x \geq 3 \) as the solution for this case.

Step 5: Case 2: Solving for \( x < 3 \)

For \( x < 3 \), \( |x - 3| = 3 - x \). Substituting into the inequality gives \( |x - 2| < 2(3 - x) \), leading to either \( x - 2 < 6 - 2x \) or \( 2 - x < 6 - 2x \). For \( x - 2 < 6 - 2x \), simplify to find \( 3x < 8 \), thus \( x < \frac{8}{3} \). For \( 2 - x < 6 - 2x \), simplify to get \( x < 4 \). Combined with \( x < 3 \), the solution for this case is \( -\infty < x < \frac{8}{3} \).

Step 6: Combining the Cases

The solution must satisfy both cases. Case 1 (\( x \geq 3 \)) provides no overlap with the results from Case 2 (\( x < \frac{8}{3} \)), so we should take the intersection of solutions only in Case 2, which is simply \( x < \frac{8}{3} \), considering that \( x < 3 \).

Step 7: Final Solution

The complete solution is \( x < \frac{8}{3} \). This ensures the conditions stated in the problem are met across all possible intervals.

Key Concepts

Understanding Absolute Value in Inequalities

The concept of absolute value is crucial when dealing with inequalities in calculus. It represents the distance of a number from zero, irrespective of its direction on the number line. For a real number, say \( x \), the absolute value \( |x| \) is defined as:

• \( x \) if \( x \geq 0 \)
• \(-x \) if \( x < 0 \)

In the context of solving inequalities, the absolute value helps in expressing distances. For example, in our exercise, the distance from \( x \) to 2 is expressed as \( |x - 2| \). The absolute value provides a uniform way of expressing these distances, regardless of whether \( x \) is greater than or less than the reference point, which in turn simplifies solving inequalities that involve them. Recognizing how absolute values alter inequalities depending on the sign of the components within them is key to approaching such problems successfully.

Exploring Distance in Inequalities

Distance in mathematics often translates to an expression involving absolute values. When an exercise mentioned distance from a point, it often denotes the absolute value of the difference between two numbers. For example, the exercise posed addresses the distance from \( x \) to both 2 and 3, which can be expressed as \( |x - 2| \) and \( |x - 3| \), respectively.
This concept of distance becomes especially significant in inequalities when comparing different distances, like in the expression \( |x - 2| < 2|x - 3| \). Here, the inequality states that the distance from \( x \) to 2 is less than twice the distance from \( x \) to 3. Understanding this opens the door to manipulating the inequality further to find the range of \( x \) that satisfies it. The role of distance in inequalities emphasizes understanding relative positions on the number line and intuitively relates to problems involving physical distances or abstract number relationships.

Strategies for Solving Inequalities

Solving inequalities, particularly those involving absolute values, requires strategic steps. Here's a simplified guide to solving such inequalities:

• Understand the inequality: Start by expressing the problem in terms of absolute values. Recognize which quantities need to be compared.
• Break it into cases: Depending on the terms inside the absolute values, split the inequality into multiple cases. This means determining intervals where the terms are positive or negative to ensure the removal of the absolute value signs is accurate.
• Solve each case separately: For each case, solve the inequality as a regular one, keeping track of the conditions specific to that case. This helps find all possible solutions.
• Combine results: Finally, evaluate which solutions overlap or meet the requirements set by the conditions of each case. Draw a conclusion for the range of solutions that satisfy the initial problem.

This methodical approach allows for a clear pathway through potentially complex inequalities and ensures that no solutions are missed. Particularly in calculus, solving inequalities often facilitates understanding the behavior of functions across different intervals.