Problem 29 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The solution set is \( n \geq 0 \). Graphically, it's the shaded area from 0 to the positive direction on the number line.

Step 1: Understanding Absolute Value

The expression \(|n|\) refers to the absolute value of \(n\), which means the distance of \(n\) from zero on the number line without considering direction. Thus, \(|n| \geq n\) if \(n\) is negative because absolute value is always non-negative.

Step 2: Rewrite the Inequality

The inequality given is \(|n| \leq n\). This statement can be split into two cases: \(n \geq 0\) and \(n < 0\).

Step 3: Case 1: Non-negative Values

For \(n \geq 0\), we have \(|n| = n\). This satisfies the inequality \(|n| \leq n\) since it simplifies to \(n \leq n\), which is always true.

Step 4: Case 2: Negative Values

For \(n < 0\), \(|n| = -n\). The inequality becomes \(-n \leq n\) which simplifies to \(n \geq \-n\). This depicts \(n > 0\), which contradicts \(n < 0\). Thus, no negative \(n\) satisfies this inequality.

Step 5: Combine Cases

The inequality is only satisfied when \(n \geq 0\). All negative values of \(n\) do not satisfy \(|n| \leq n\).

Step 6: Graph the Solution Set

On a number line, shade the region from 0 to the right, including 0, to indicate all values \(n \geq 0\) as possible solutions. Close the circle at 0 to show it is included in the set.

Key Concepts

Understanding Absolute Value

The concept of absolute value is fundamental when solving inequalities involving absolute values. Absolute value, denoted by the symbol \(|n|\), represents the distance of a number \(n\) from zero on the number line. It is always a non-negative number because distance cannot be negative. For example, both \(|3|\) and \(|-3|\) evaluate to \(3\), since each is exactly 3 units away from zero, regardless of direction. In inequalities, absolute values can greatly influence the form and solution because they affect how both negative and positive numbers are treated. Remember, the absolute value of any number is its positive version or zero.

Number Line Graphing

Graphing solutions on a number line helps visualize inequality solution sets clearly. For the inequality \(|n| \leq n\), we graph the solution set \(n \geq 0\) on a number line.
To do this:

• Draw a straight horizontal line to represent the number line.
• Mark the critical point, which is 0 in this case, and indicate this point with a filled circle to show it's included in the solution set (since \(n\geq0\)).
• Shade the region extending to the right from 0, as all numbers equal to or greater than 0 satisfy the inequality.

Visualizing inequalities with a number line is an effective way to understand which numbers make an inequality true and to clearly show the range of solutions.

The Inequality Solution Set

An inequality solution set refers to all the values of a variable that satisfy the inequality condition. In our inequality \(|n| \leq n\), the solution set is determined after evaluating both non-negative and negative cases.
For non-negative values \(n \geq 0\), since \(|n| = n\), the inequality \(|n| \leq n\) always stands true. Thus, any \(n\) in this range belongs to the solution set.
On the other hand, for negative values \(n < 0\), the condition does not hold because \(|n| = -n\) and does not satisfy \(-n \leq n\). After evaluating the inequality, we conclude that only non-negative numbers meet the criteria. Thus, our solution set is all non-negative numbers starting from zero.

Negative and Non-negative Numbers

Negative and non-negative numbers play distinct roles in solving inequalities, especially those involving absolute values.
Negative numbers are those less than zero, such as -1, -2, and so forth. They do not satisfy the inequality \(|n| \leq n\) because their absolute values are positive which contradicts \(-n \leq n\).
Non-negative numbers, however, include all positive numbers and zero. These numbers satisfy the inequality because for any \(n \geq 0\), the absolute value \(|n|\) equals \(n\) and the inequality holds true. Understanding the difference between these types of numbers is key in determining which parts fulfill the requirements of an absolute value inequality.