Problem 22 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Distance Formula on a Number Line

The distance between two points on a number line, say \(x\) and \(4\), is given by the absolute value formula \(|x - 4|\). This represents the positive difference or distance between \(x\) and \(4\).

Step 2: Set Up the Inequality

According to the problem, the distance between the points \(A\) and \(B\), \(d(A, B)\), should be at most 2. Therefore, the absolute value of the difference between these two points can be expressed as an inequality: \(|x - 4| \leq 2\).

Step 3: Interpret the Inequality

The inequality \(|x - 4| \leq 2\) suggests that the distance between \(x\) and \(4\) is 2 or less. This encompasses all the \(x\) values that are within 2 units of the number 4 on the coordinate line.

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on the number line, without considering the direction. This means that whether a number is negative or positive, its absolute value is always positive. For instance, the absolute value of both 3 and -3 is 3. Absolute values are represented using vertical bars, like this: \(|x|\).
For example:

• \(|5| = 5\)
• \(|-5| = 5\)

In our context, the absolute value \(|x - 4|\) indicates how far the number \(x\) is from 4, regardless of whether \(x\) is to the left or right of 4 on the number line.

Distance on a Coordinate Line

Distance on a coordinate line refers to the space between two points, which can be measured simply by finding the difference between their coordinates. This concept extends naturally from our understanding of distance in everyday life: how far one object is from another. On a number line, this distance is quantified using absolute values.
For two points, \(A\) located at \(x\) and \(B\) at 4, their distance is \(|x - 4|\). This formula gives us a non-negative number, reflecting the true distance between the two points on the line.
Understanding this helps in recognizing that distance is always a positive value, even if one of the coordinates is negative.

Number Line

A number line is a straight, horizontal line representing all real numbers at equal intervals. It's a simple yet powerful way to illustrate concepts of distance and position in math.

• The center of the line is typically marked as zero.
• Numbers increase positively to the right and decrease negatively to the left.

Visualizing problems on a number line can make it easier to understand abstract math concepts, such as inequalities and distances. \(|x - 4| \leq 2\) represents all numbers on the number line that lie within 2 units of the point 4. This set includes numbers from 2 to 6.

Distance Formula

The distance formula on a coordinate line is a straightforward application of absolute value. For any two points \(x_1\) and \(x_2\) on a line, the distance \(d\) between them is calculated as \(|x_1 - x_2|\).
This formula simply provides the absolute difference between the coordinates. It reflects the physical distance separating the points without indicating direction.
In our problem, when the statement is that \(d(A, B)\) should be at most 2, mathematically it translates to \(|x - 4| \leq 2\). This tells us how to use the distance formula to create inequalities and solve for ranges of possible values for \(x\). This formula is crucial in many fields, not just math, but also in areas like physics and engineering that require precise measurements.