Problem 24 - 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. $$-2, x$$ \(d(A, B)\) is not less than 2

Step-by-Step Solution

Answer

\(|x + 2| \geq 2\)

Step 1: Define the Distance Formula

The distance between two points on a coordinate line, say point \(A\) with coordinate \(-2\) and point \(B\) with coordinate \(x\), is given by the absolute value of their difference: \(d(A, B) = |x - (-2)|\) which simplifies to \(|x + 2|\).

Step 2: Understand the Given Condition

The problem states that the distance \(d(A, B)\) is not less than 2. This implies that \(|x + 2|\) should be greater than or equal to 2.

Step 3: Write the Inequality

Based on the interpretation of the condition provided, the inequality can be expressed as: \(|x + 2| \geq 2\). This inequality states that the distance between point \(A\) at \(-2\) and point \(B\) at \(x\) is at least 2.

Key Concepts

Distance Formula

The distance formula is a fundamental concept in mathematics, especially when working with coordinate lines. It allows us to find the distance between two points by measuring the 'gap' between them. On a coordinate line, if we have two points, let's call them point \( A \) with a coordinate \( -2 \) and point \( B \) with a coordinate \( x \), the distance \( d(A, B) \) between them is simply the absolute value of their coordinate difference. So, this is expressed as:

• \( d(A, B) = |x - (-2)| \),
• which simplifies to \( |x + 2| \).

The use of absolute value ensures that the distance is always a non-negative number, which makes sense because distance cannot be negative. This formula is compact and powerful for solving many real-world problems involving measurements on a line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use the coordinate system to understand geometric shapes and properties. Points on a plane or line are identified using coordinates. In this example, the points \( A \) and \( B \) lie on a coordinate line. For this line:

• Point \( A \) is at \( -2 \),
• Point \( B \) is at \( x \).

This coordinate system allows us to apply formulas, such as the distance formula, in a straightforward manner. Using coordinates simplifies the process of visualizing and working with geometric interpretations, easing the solution of complex problems like finding distances or defining regions on a line or plane. With this understanding, solving inequalities and measuring distances becomes a systematic process.

Inequalities Involving Absolute Value

Inequalities using absolute value tell us about the range or region that a variable can take, based on certain constraints. They are essential for defining boundaries and limits in coordinate geometry.The exercise gives us an inequality \( |x + 2| \geq 2 \). This describes a region on the coordinate line where point \( B \) can be located with respect to point \( A \). The condition \(|x + 2| \geq 2\) means we are looking for all the points \( x \) whose distance from \(-2\) is at least 2 units. The inequality can play out in two scenarios:

• The expression is greater than or equal to \(2\), \( x + 2 \geq 2 \), leading to \( x \geq 0 \),
• or the expression is less than or equal to \(-2\), \( x + 2 \leq -2 \), leading to \( x \leq -4 \).

Combining these gives a solution set for \( x \) which lies in two separate intervals, highlighting how absolute values split scenarios. They show us all the possible positions for \( x \) such that the distance is at least 2 units apart from \(-2\). This approach is very useful in determining ranges and ensuring solutions fulfill certain spatial conditions.