Question

The solution set of the equation \(|x|=5\) is {_____________}.

Short answer

{5, -5}

Step-by-step solution

Understand the Absolute Value

The absolute value of a number refers to its distance from 0 on the number line, without considering direction. Therefore, \(|x| = 5\) means that the distance of x from 0 is 5 units.

Interpret the Equation

Given \(|x| = 5\), x can be either 5 or -5 because both 5 and -5 are 5 units away from 0.

State the Solution Set

Based on the above understanding, the solution set for the equation \(|x| = 5\) is \{5, -5\}.

Key concepts

absolute value

The absolute value of a number represents its distance from zero on a number line, regardless of the direction. This means that it only matters how far a number is from zero, not which side it is on. For example, the absolute value of both -3 and 3 is 3.

The notation used for absolute value is two vertical bars, like this: \(|x|\). So, if we want to find the absolute value of -7, we write \(|-7| = 7\).

Remember, absolute value is always non-negative because it measures distance. When solving absolute value equations, we consider both the positive and negative scenarios. For instance, \(|x| = 5\) implies that x can be both 5 and -5.

solution set

In algebra, a solution set is the group of values that satisfy an equation.

When we solved \(|x| = 5\), we considered the definition of absolute value and found that both 5 and -5 satisfy the equation. This means our solution set for this equation is \{5, -5\}.

Always remember: when dealing with absolute values, your solution set will often have two values – one positive and one negative – since absolute value equations take both directions from zero into account.

algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is essential for solving equations like \(|x| = 5\) because it provides the foundation for how we interpret and manipulate expressions to find solutions.

In the given problem, we used basic algebraic principles to understand and solve the absolute value equation. We first interpreted what \(|x| = 5\) means and then applied algebraic reasoning to find that the solutions must be x = 5 or x = -5.

Algebra helps us systematically approach and solve equations, ensuring that we find all possible values that satisfy the given conditions.