Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-24: The two given numbers are coordinates of points \(\boldsymbol{A}\) and \(\boldsymbol{B}\), respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol. $$ x, \quad 4 ; \quad d(A, B) \text { is at most } 2 $$

Step-by-Step Solution

Answer

The inequality is \(|x - 4| \leq 2\).

Step 1: Understand the Problem

We are given two points, \( A \) and \( B \), on a coordinate line with coordinates \( x \) and \( 4 \), respectively. The problem states that the distance \( d(A, B) \) is at most 2. We need to express this condition as an inequality involving absolute values.

Step 2: Use Distance Formula

The distance between two points \( A \) and \( B \) on a coordinate line is given by \( |x - 4| \). This is the absolute value of the difference between \( x \) and 4.

Step 3: Set Up the Inequality

The problem states that the distance \( d(A, B) \) is at most 2. Hence, we can write the inequality as \( |x - 4| \leq 2 \). This represents that the distance between point \( A \) at coordinate \( x \) and point \( B \) at coordinate 4 is no more than 2 units.

Key Concepts

Understanding Absolute Value

The absolute value of a number is a measure of its distance from zero on a number line, without considering direction. This means that whether a number is positive or negative, its absolute value is always non-negative.

For example, both 3 and -3 have an absolute value of 3, denoted by \(|3| = 3\) and \(|-3| = 3\). Absolute value is essential in expressing distances because it provides a non-negative solution that represents magnitude rather than direction.

• Absolute values find widespread usage in inequalities, which define ranges of numbers.
• While handling coordinates, absolute values help in removing the direction factor, focusing solely on the distance.

The Role of Coordinate Line in Inequalities

A coordinate line is simply a one-dimensional number line where each point corresponds to exactly one real number. The introduction of a coordinate line simplifies the visualization of points and distances in problems involving inequalities.

In our example, points \(A\) and \(B\) have coordinates \(x\) and \(4\), respectively. The coordinate line helps to determine where these points are in relation to each other and to zero.

• On a coordinate line, we can visually measure the distance between two points as a straight line segment.
• The direction of points on this line can indicate whether a number is greater or lesser, but does not affect the distance calculation.

The Distance Formula on a Line

When dealing with points on a coordinate line, the distance between two points can be calculated using the distance formula. This formula on a coordinate line is expressed as the absolute value of the difference between the two points' coordinates.

In the given problem, the formula used is \(|x - 4|\), representing the distance between points \(A\) with coordinate \(x\) and \(B\) at 4.

• By using absolute value, the distance becomes a magnitude, free from any direction.
• The formula \(|x - b|\) works by subtracting the coordinates, ensuring the result is a non-negative distance.

Significance of Points on a Line

Points on a line have specific coordinates, which make it easy to calculate distances and set up inequalities. Each point can clearly be evaluated in terms of its relation to other points through arithmetic operations.

For the inequality \(|x - 4| \leq 2\), the locations of points \(x\) and \(4\) along the coordinate line allow us to interpret the range of \(x\) values.

• This range signifies that point \(A\) can be at most 2 units away from \(B\)'s location.
• Understanding points and their coordinates aids in setting effective boundaries within inequalities.

The distance or the inequality is direct and straightforward due to its computation using these well-defined points.