Problem 29 - Calculus: One and Several Variables

Question

Indicate on a number line the numbers \(x\) that satisfy the condition. \(|x| \leq 0\).

Step-by-Step Solution

Answer

The only number that satisfies the given condition \(|x| \leq 0\) is \(x = 0\), and this is represented by a single point at 0 on the number line.

Step 1: Understand the Concept of Absolute Value

The absolute value of a number, denoted as \(|x|\), is the distance of the number from zero on the number line, regardless of the direction. Therefore, it is always non-negative. The absolute value of \(x\) is defined as \(x\) if \(x \geq 0\), and \(-x\) if \(x < 0\).

Step 2: Solve the Absolute Value Inequality

Given \(|x| \leq 0\), we want to find all values of \(x\) which when taken their absolute value result in a value that is less or equal to 0. Considering the definition of absolute value, it is clear that the only value that can satisfy this condition is 0 itself, as any other positive or negative value will yield an absolute value greater than 0.

Step 3: Represent the Solution on the Number Line

On a number line, 0 is marked. Because the solution to the inequality is only 0, no other points are marked or intervals indicated.

Key Concepts

Absolute Value

The concept of absolute value is fundamental in understanding how to approach inequalities involving distances from zero on a number line. In essence, the absolute value represents the distance from zero without regard to direction. Mathematically, if you have a number \(x\), the absolute value, noted as \(|x|\), is \(x\) if the number is positive or zero, and it is \(-x\) if the number is negative.

For example, both \(4\) and \(-4\) have an absolute value of \(4\) because they are both four units away from zero on the number line. It's important to remember that absolute values are always zero or positive because distance cannot be negative. This characteristic becomes very useful when solving absolute value inequalities.

Number Line

A number line is a visual representation of numbers laid out in a straight line, where each point on the line corresponds to a number. It is an excellent tool for visualizing and solving problems related to inequalities and absolute values. In number lines, numbers to the right are greater, and numbers to the left are smaller. Zero is typically set as the central point, from which we can easily compare other numbers.

When working with absolute value inequalities, a number line helps you to clearly see which numbers satisfy the inequality. For instance, marking the number that provides the boundary for the inequality can show at a glance all the numbers that are within the solution set. This ease of demonstration makes the number line an invaluable part of solving and understanding equations and inequalities.

Inequality Solution

When dealing with inequalities, you are often searching for a range of values that meet the given condition, rather than a single solution as you would have with an equation. Inequality solutions can be represented with a variety of notations, including set notation, inequality notation, and interval notation, as well as visually with a number line.

For an absolute value inequality like \(|x| \< a\), where \(a\) is a positive number, the solution consists of all numbers \(x\) that are within \(a\) units of zero on the number line. Conversely, for \(|x| > a\), the solution would include all numbers that are more than \(a\) units away from zero. When the inequality uses \(\leq\) or \(\geq\), we also include the endpoints, meaning the points \(a\) and \(-a\) themselves are part of the solution set.

In our example, with the inequality \(|x| \<= 0\), the solution on a number line is simply the point zero because no other number can have an absolute value less than or equal to zero, which would imply a negative distance from zero.