Question

In Problems 149 and 150, write an inequality using an absolute value to describe each statement. \(x\) is at least 6 units from 4

Short answer

|x - 4| \geq 6

Step-by-step solution

- Understand the Problem

The problem asks to express that the value of \(x\) is at least 6 units away from 4 using an absolute value inequality.

- Define Absolute Value Distance

Recall that the absolute value \(|a - b|\) represents the distance between \(a\) and \(b\) on the number line.

- Set Up the Inequality

Since \(x\) needs to be at least 6 units away from 4, we write this requirement using absolute value as \(|x - 4| \geq 6\). This inequality means the distance between \(x\) and 4 is 6 or more units.

Key concepts

Distance on Number Line

To understand absolute value inequalities, it's crucial to become familiar with the concept of distance on a number line. The number line is a visual representation of numbers in order. By plotting points, we can visually track and measure distance. Distance, in this context, is always non-negative.
For example, if you have two points on a number line, say 3 and 7, the distance between them is simply the difference in their values, without considering direction. Thus, the distance between 3 and 7 is \(|7 - 3| = 4\).
This fundamental idea will help us work through problems involving absolute values, as absolute values provide a measure of distance regardless of direction.

Absolute Value

The absolute value of a number refers to its distance from 0 on the number line, irrespective of direction. The absolute value of a number is always non-negative. We denote absolute value using two vertical bars, e.g., \(|x|\).
Mathematically, for any real number \(a\), the absolute value is defined as: \(|a| = a\) if \(a \geq 0\), and \(|a| = -a\) if \(a < 0\).
Here’s an example: \(|5| = 5\) and \(|-5| = 5\). Both values are 5 units away from 0, showing how absolute values measure distance regardless of direction.
Absolute values are particularly useful in representing distances because they ignore direction, focusing solely on how far apart values are.

Inequality Representation

Inequalities help us express ranges and limits in algebraic terms. When combined with absolute values, they describe sets of values within certain distances from a point.
In this specific case, we seek to describe the condition that a variable \(x\) is at least a certain distance from a particular value. The given problem states that \(x\) is at least 6 units from 4. This can be translated into an absolute value inequality.
Using the absolute value concept, we write \(|x - 4| \geq 6\). This inequality states that the distance between \(x\) and 4 is 6 or more, meaning \(x\) must be either 6 units less than 4 or 6 units more than 4. Visually, \(|x - 4| \geq 6\) defines all the points on the number line that are outside the interval from \(-2\) to \(10\), inclusive of -2 and 10.