Problem 19 - 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. $$x, \quad 7$$ \(d(A, B)\) is less than 5

Step-by-Step Solution

Answer

\(|x - 7| < 5\)

Step 1: Understand the Problem

The problem states that the distance between the points with coordinates \(x\) and 7 is less than 5. We need to translate this statement into an inequality involving absolute value.

Step 2: Write the Distance Formula

On a coordinate line, the distance \(d\) between two points \(A(x)\) and \(B(7)\) is given by the absolute value of their difference: \(d(A, B) = |x - 7|\).

Step 3: Formulate the Inequality

According to the problem statement, the distance \(d(A, B)\) is less than 5. We can write this as an inequality: \(|x - 7| < 5\).

Key Concepts

Coordinate Plane

The coordinate plane is essentially a flat surface made up of two intersecting lines called axes. These axes are the x-axis (horizontal) and the y-axis (vertical). The point where they intersect is called the origin, which is at (0, 0). Whenever we place a point on this plane, we use an ordered pair of numbers known as coordinates. The first number indicates the position on the x-axis, and the second number shows the position on the y-axis.

These coordinates help us locate points accurately in space. For instance, a point like (3, 5) would mean moving 3 units along the x-axis and 5 units up on the y-axis. This two-dimensional space allows us to visually interpret mathematical equations and easily identify relationships between variables. The coordinate system provides the foundation to graph equations and understand the geometry of shapes.

Distance Formula

The distance formula is a crucial tool for calculating the distance between two points in a coordinate plane. If you have two points, say \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance between them is determined by the formula: \[d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula comes from the Pythagorean theorem, which relates the sides of a right triangle. However, in one-dimensional space, such as the coordinate line mentioned in the exercise, the formula simplifies. Here, the distance \(d(A, B)\) is just the absolute value of the difference of their x-coordinates: \[d(A, B) = |x - 7|\]The absolute value is used because distance is always a positive value or zero. This simple version of the distance formula is perfect for problems involving coordinates on a single line.

Inequality Translation

Inequality translation involves expressing a verbal statement, often about limits or boundaries, as a mathematical inequality. In the context of the exercise, we started with a verbal statement: "The distance between points A and B is less than 5."

To translate this idea into a mathematically precise form, we use absolute value inequalities. The absolute value of \(x - 7\) describes the distance on a coordinate line, and we express this distance being less than 5 as: \[|x - 7| < 5\]This inequality captures the concept that the point \(x\) can be no more than 5 units away from 7 on the number line. The translation process requires understanding both the concept of absolute value and how inequalities represent ranges of allowable values. It's a fundamental skill in math that helps frame real-world constraints in more abstract mathematical forms.