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Chapter 1: Problem 31

Indicate on a number line the numbers \(x\) that satisfy the condition. \(|x-4| \leq 2\).

Short Answer

Expert verified
The numbers \(x\) that satisfy the condition \(|x-4| \leq 2\) are those between 2 and 6, inclusive. That is, \(2 \leq x \leq 6\).

Step by step solution

01

Understanding absolute value

An absolute value function |x-a| provides the distance of x from a on the number line. The absolute value of a number is non-negative. So, for the given inequality \(|x - 4| \leq 2\), it represents all numbers \(x\) whose distance from 4 on a number line is less than or equal to 2 units. In simpler terms, it means all numbers \(x\) that are at most 2 units away from 4. This can be broken down further into two conditions.
02

Breaking down the absolute value inequality

We split the inequality \(|x - 4| \leq 2\) based on whether \(x - 4\) is negative or non-negative. This yields two inequalities: \(x - 4 \leq 2\) when \(x - 4\) is non-negative and \(-(x - 4) \leq 2\) when \(x - 4\) is negative.
03

Solving the two inequalities

Solving the first inequality, we add 4 to both sides of the equation resulting in \(x \leq 6\). Solving the second inequality, we distribute the negative sign to yield \(4 - x \leq 2\) and after subtracting 4 from both sides we get \(x \geq 2\).
04

Combining the results

Combining the two inequalities \(x \leq 6\) and \(x \geq 2\), we get that \(2 \leq x \leq 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number Line
The number line is a powerful tool for visualizing inequalities, like the one given here. Imagine it as a straight, horizontal line extending infinitely in both directions. Every point on this line represents a real number.
Using a number line makes it easy to understand where numbers lie in relation to each other鈥攍arger numbers are to the right, and smaller numbers to the left.
  • It helps us see whether numbers are positive or negative.
  • It illustrates the concept of distance between numbers, which is essential for understanding absolute value.
In our exercise, we want to graphically represent solutions to the inequality \(|x-4| \leq 2\). By generally marking numbers on this line, like 2 and 6, we grasp which values of \(x\) fulfill the condition.
Inequality Solving
Solving absolute value inequalities requires breaking them down into manageable parts. Remember, the inequality \(|x-4| \leq 2\) asks us to find all \(x\) that are at most 2 units from 4. To solve this inequality, we need to consider both the positive and negative cases!
First, assume the expression inside the absolute value is non-negative: \(x - 4 \).
  • This leads to the inequality \(x - 4 \leq 2\). Solving this yields \(x \leq 6\).
Next, handle the case where the expression inside is negative: \(-(x - 4)\).
  • This gives us \(4 - x \leq 2\), simplifying further to \(x \geq 2\).
By putting these two inequalities together, we find the complete solution set: \(2 \leq x \leq 6\). This means all values from 2 through 6, inclusive, satisfy the original condition.
Distance on a Number Line
The concept of distance on a number line is central to understanding absolute value inequalities. The absolute value \(|x-4|\) specifically tells us how far \(x\) is from the number 4. Always measured as a non-negative quantity, it signifies the "distance" rather than direction.
  • When solving \(|x-4| \leq 2\), we're finding points that are within 2 units away from 4.
  • This translates visually on a number line as the segment between 2 and 6.
Imagine the number line, each point 2 units away from 4 will mark the boundary.
So, every number \(x\) between 2 and 6 is part of the solution. It clarifies that the distances to the left and right of 4 are both considered equally, accounting for every possible number within that range.

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