Problem 23 - 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. $$ 4, \quad x, \quad d(A, B) \text { is not greater than } 3 $$

Step-by-Step Solution

Answer

The inequality is expressed as \(|x - 4| \leq 3\), which means \(1 \leq x \leq 7\).

Step 1: Identify the Meaning of the Given Statement

The problem states that the distance \( d(A, B) \) between points \( A \) and \( B \) is not greater than 3. Mathematically, the distance between two points \( A(x_1) \) and \( B(x_2) \) on a coordinate line is given by the absolute value of their difference, \( |x_2 - x_1| \). In our problem, \( A \) is 4 and \( B \) is \( x \).

Step 2: Write the Absolute Value Inequality

Using the concept of absolute value, express the distance \( d(A, B) \) as \( |x - 4| \), where \( x \) is the coordinate of point \( B \) and 4 is the coordinate of point \( A \). The statement that this distance is not greater than 3 can be written as an inequality: \( |x - 4| \leq 3 \).

Step 3: Simplify the Inequality Expression

The inequality \( |x - 4| \leq 3 \) represents all the values of \( x \) that are within 3 units of 4 on the coordinate line. This absolute value inequality can be split into two separate inequalities without the absolute value: \( x - 4 \leq 3 \) and \( x - 4 \geq -3 \).

Step 4: Solve the Split Inequalities

Solving the first inequality \( x - 4 \leq 3 \), we add 4 to both sides to get \( x \leq 7 \). For the second inequality \( x - 4 \geq -3 \), we also add 4 to both sides to get \( x \geq 1 \).

Step 5: Combine the Solutions

From the two inequalities \( x \leq 7 \) and \( x \geq 1 \), we conclude that \( 1 \leq x \leq 7 \). This is the interval notation representation of the inequality involving the distance condition given in the problem statement.

Key Concepts

Absolute Value

The absolute value of a number is a measure of how far the number is from zero on a number line, regardless of its direction. It is denoted with vertical bars, for example, \( |x| \). This concept is crucial when dealing with distances because distance does not have a direction and is always non-negative.
For example, whether 4 units to the left or right, the absolute value remains 4. In the context of inequalities, like \( |x-4| \leq 3 \), it means that the value of \( x \) can move at most 3 units away from 4, either to the left or right on the coordinate line.

Coordinate Line

A coordinate line, often simply a number line, is a straight line on which every point corresponds to a real number. The positions on this line can be positive, negative, or zero.
In problems involving inequalities and distances, such as our example with points \( A \) and \( B \), a coordinate line helps visualize where these points lie relative to one another. When distances and differences between points are assessed, the coordinate line provides a visual framework to interpret absolute value expressions and inequalities geometrically.

Distance Formula

The distance formula is a way to calculate how far apart two points are on a line. On a coordinate line, it's the absolute value of the difference between their coordinates. This formula is given by \( d(A, B) = |x_2 - x_1| \).
In our exercise, this meant looking specifically at \( |x-4| \), which tells how far point \( x \) (where point \( B \) is) is from point \( A \) at position 4. When stated that \( d(A, B) \) is not greater than 3, it translates mathematically to an inequality saying the distance should not exceed the value of 3.

Interval Notation

Interval notation is a compact way to express a range of values between two numbers. In the context of inequalities, it neatly indicates all possible solutions to an equation that can vary between two points.
For instance, the inequality \( 1 \leq x \leq 7 \) can be expressed using interval notation as \([1, 7]\). This indicates that \( x \) is any number within and including 1 and 7. Interval notation is often used in expressions resulting from solving absolute value inequalities, as it clearly conveys the interval of possible solutions.