Problem 28 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ |n| \geq n $$

Step-by-Step Solution

Answer

The solution set is all real numbers, \(n \in \mathbb{R}\).

Step 1: Understand the absolute value inequality

The given inequality is \(|n| \geq n\). This inequality involves an absolute value. Absolute value represents the distance of a number from zero on the number line, so \(|n|\) is always non-negative.

Step 2: Analyze the inequality for non-negative n

If \(n\) is non-negative (i.e., \(n \geq 0\)), then \(|n| = n\). The inequality \(|n| \geq n\) becomes \(n \geq n\), which is true for all non-negative \(n\).

Step 3: Analyze the inequality for negative n

If \(n\) is negative (i.e., \(n < 0\)), then \(|n| = -n\). The inequality \(|n| \geq n\) becomes \(-n \geq n\), which is the same as \(-2n \geq 0\). This simplifies to \(n \leq 0\). Thus, for negative \(n\), the inequality holds true as well.

Step 4: Combine solution sets

From Step 2 and Step 3, we know for non-negative \(n\), \(|n| \geq n\) always holds, and for negative \(n\), \(|n| \geq n\) also holds true. Hence, the inequality is true for all \(n\).

Step 5: Graph the solution set on a number line

To represent the solution \(n\) on a number line, mark all real numbers. This means every point on the number line is part of the solution set.

Key Concepts

Number Line Graphing

Number line graphing is a visual way to represent solutions of inequalities. Imagine a straight line labeled with numbers, where each point represents a real number.
This helps us easily see which numbers satisfy our inequality. In this exercise, the inequality \(|n| \geq n\) is true for all numbers, both positive and negative. To graph this, you would mark the entire number line, showing that every point represents a solution.
This reinforces the result that the inequality holds for any real number, whether it's zero, positive, or negative. Using a number line can make understanding inequalities like these more intuitive:

• Draw a horizontal line.
• Label it with evenly spaced numbers, both positive and negative.
• Shade the entire line to indicate all numbers are solutions.

Solving Inequalities

Solving inequalities involves finding the set of values that makes an inequality true. Unlike equations, inequalities show a relationship where one side can be larger or smaller than the other.
In this case, we are dealing with an absolute value inequality \(|n| \geq n\). To solve this, consider the definition of absolute value, which is the distance from zero and is always positive. This means you will analyze the inequality in two parts:

• For non-negative numbers (\(n \geq 0\)), \(|n|\) and \(n\) are equal, and \(n \geq n\) holds true.
• For negative numbers (\(n < 0\)), \(|n|\) becomes \(-n\), and \(-n \geq n\) simplifies to \(n \leq 0\).

Combining these, we see the solution set includes all real numbers.
This approach ensures we don't miss any part of the number line where the inequality might be true.

Real Numbers

Real numbers include everything you find on the number line, from negative numbers to zero to positive numbers. This broad category encompasses all rational numbers (like fractions and integers) and irrational numbers (like roots and pi).
In the context of this inequality, appreciating the concept of real numbers helps us understand why the entire number line is a solution. Real numbers are continuous, meaning there are no gaps on the line.
So, when we say \(|n| \geq n\) is true for all numbers, we're talking about every conceivable number you can plot on the line. Understanding real numbers is crucial:

• They form the basis of most basic mathematics and are used in almost all calculations.
• Their continuous nature ensures that solutions of inequalities can span entire ranges.

Thus, whenever you encounter a real number as a solution, you are encompassing a broad set of values, ensuring inclusivity of both positive, negative, and even streams of fractions.