Question

Solve the inequality. Then graph the solution set on the real number line. \(|x-5|<0\)

Short answer

The given inequality \(|x-5|<0\) has no solution because absolute values cannot be less than 0.

Step-by-step solution

Understanding absolute values

To understand this problem, you'll first need to understand what an absolute value is. The absolute value of a number is its distance from zero on the number line, so by definition, it can never be negative.

Creating the inequality for solution

The inequality is \(|x-5|<0\), which means the expression inside the absolute value braces, i.e. \(x - 5\), must be less than 0.

Further understanding the impossibility of the condition

However, an absolute value can never be less than 0, which makes no sense mathematically. This suggests that there are no possible solutions for the given inequality.

Drawing the number line graph

A number line graph is inconsequential here because there is no solution for this inequality. A graph would be meaningful only when there are solutions to depict on it.

Key concepts

Inequalities

Inequalities are mathematical expressions that describe the relationship between two values, indicating that one value is larger or smaller than the other. They use symbols like <, >, ≤, and ≥.

When dealing with inequalities involving absolute values, it's crucial to consider the definition of absolute value. The absolute value of a number represents its distance from zero and is never negative. Thus, an expression like \(|x-5|<0\) suggests a logical impossibility because an absolute value cannot be less than zero. Therefore, inequalities of this type would have no valid solution set.

Real Number Line

The real number line is a visual representation of all real numbers arranged in order. It's like a straight line where each point corresponds to a real number.

When we solve inequalities, we often use the real number line to represent the solution set. Each solution is marked as a point or a range of points on the line, showing visually how the solutions relate to each other.

However, in this example of \(|x-5|<0\), since there is no numerical solution, the real number line remains unchanged; there are no points to be plotted.

Graphing Solutions

Graphing solutions to inequalities helps us visualize the range of values that satisfy the inequality. It's like drawing a map that guides us to understand which values make the inequality true.

To graph the solution of \(|x-5|<0\), we must recognize that it is an inconsistency. Given that absolute values cannot be negative, the graph does not hold any actual value or position for solutions. Typically, when there are solutions, we'd use open or closed dots and shading to express the possible range of x-values on the real number line. However, in cases where the inequality is impossible, as seen here, there's simply nothing to graph.