Question

Solve the equation. $$|n|=5$$

Short answer

The solutions of the equation \(|n|=5\) are \(n = 5\) and \(n = -5\).

Step-by-step solution

Understand Absolute Value

Absolute value of a number is its distance from 0 on the number line. It is always positive or zero. It is represented as \( |x| \). When \( |x| = a \) is defined, it means x could be a distance a on the right or left of 0 on the number line. Hence, two scenarios can come out of it i.e. \( x = a \) or \( x = -a \).

Solve for Positive Case

Apply the above scenario by replacing \( a \) with given number '5'. The first scenario indicates positive distance from zero i.e., \( n = 5 \). This is the first solution.

Solve for Negative Case

The second scenario indicates negative distance from zero i.e., \( n = -5 \). This is the second solution.

Key concepts

Absolute Value

When it comes to understanding absolute value, picture it as a measure of how far a number is from zero on the number line, regardless of direction. Imagine you're standing on a straight path at point zero, and absolute value tells you the number of steps you'd take to reach a certain point, without caring if you move forward or backward. For example, if we are working with the equation |n|=5, it signifies that n is either 5 steps to the right or 5 steps to the left of zero. It shows the distance, not the direction. Therefore, the absolute value of a number is always non-negative.

In the context of our problem, this concept leads us to understand that there are two potential spots on the path (number line) that are five steps away from zero, giving us the foundation to find our two solutions.

Number Line

The number line is a visual representation that allows us to see where numbers fall in relation to each other, especially when dealing with absolute values. It's like having a ruler for numbers, where you can clearly see zero as the starting point and then see equal spaces stretching out in both the positive and negative directions. By using this tool, we can comprehend that any number has two counterparts with the same absolute value: one positive and one negative. When solving the equation |n|=5, plotting the absolute value on the number line ensures that you can visually check to find the numerical values that are 5 units away from zero in both directions — positive and negative.

Positive and Negative Solutions

Absolute value equations can often lead to two distinct solutions: one positive and one negative. This pair of solutions reflects the concept that absolute value represents a distance that can be measured in two opposite directions on the number line. For the equation |n|=5, we can interpret this to mean that n is either 5 or -5, since both numbers have an absolute value of 5. This shows the dual nature of absolute value solutions—each absolute value equation has a positive root and a negative root since both numbers lie the same distance from zero on the number line but in opposite directions. Understanding this duality is crucial for grasping the full set of potential solutions.

Algebraic Solution Steps

Approaching absolute value equations algebraically involves a methodical process. First, understand that the term within the absolute value symbols can take on two potential values: one positive and one negative. To solve an equation like |n|=5, we set up two separate equations based on the definition of absolute value: n = 5 and n = -5. We then solve these equations as we would any other simple algebraic equation. By separating the equation into its two possible cases, we are able to find all possible solutions to the absolute value equation. Remember that both solutions are valid and should be considered when analyzing the full set of answers.