Problem 31 - Calculus: One and Several Variables

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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 \(x \geq 2\).

Step 4: Combining the results

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

Key Concepts

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—larger 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.