Problem 70 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

All real numbers are the solution for this inequality. So, the graph would be a number line stretching out infinitely in both directions.

Step 1: Understand the nature of Absolute Value

The first step is to understand an absolute value. The absolute value of a number represents its distance from zero on the number line, hence it is always positive or zero.

Step 2: Analyze Given Inequality

The inequality is \(|x-5| \geq 0\). Since an absolute value is always nonnegative, every real number will satisfy this inequality.

Step 3: Graph the solution set

In this case, since all real numbers are accepted, the graph will be a number line extending infinitely in both directions.

Key Concepts

Absolute Value

To form a solid foundation in solving absolute value inequalities, one must firmly grasp the concept of absolute value itself. Imagine standing on a number line: the distance from where you stand (a specific number) to the origin (zero) is the absolute value of that number. In more rigorous terms, the absolute value of a number, often denoted as `|a|`, is strictly nonnegative and represents that number's magnitude irrespective of its sign.

An important property to remember is that the absolute value function creates a 'V' shape when graphed, with the vertex at the origin. This shape reflects the idea that as you go further left or right from zero on the number line, the distance (and hence the absolute value) gets larger.

Understanding this concept is crucial while analyzing inequalities that include absolute values, as it dictates that the outputs are always zero or positive. Consequently, when we encounter an inequality such as `|x - a| ≥ 0`, it is true for all `x` because subtracting any number `a` from `x` and taking the absolute value cannot result in a negative number.

Inequalities

Branching out from understanding absolute value, we encounter inequalities which are mathematical expressions comparing two values. They come in various forms: less than `<`, greater than `>`, less than or equal to `≤`, and greater than or equal to `≥`. When we solve inequalities involving absolute values, we're seeking all the possible solutions that satisfy the condition. For example, the inequality `|x - 5| ≥ 0` is rather straightforward because, as mentioned before, absolute values are always nonnegative.

In more challenging scenarios, solving the inequality requires splitting it into separate cases—usually one where the expression inside the absolute value is nonnegative and another where it is negative. Each case is then solved as a standard inequality. However, for the inequality `|x - 5| ≥ 0`, we see that it will hold true for every real number because whatever value `x` takes, the inequality symbol `≥` encapsulates the equal to `0` aspect as well.

Number Line Graphing

When we graph solutions to inequalities on a number line, we are visualizing the set of all possible values that satisfy the inequality. It's a straightforward, yet powerful tool in understanding and communicating solutions. For absolute value inequalities, such graphs help to discern the ranges of solutions. In the context of `|x - 5| ≥ 0`, graphing is simplified because every real number satisfies the inequality; thus, every point on the number line is a part of the solution set.

To graphically represent this, one would draw a number line, with an arrow extending to the left and to the right from the point `5`, signifying all numbers less than or equal to `5` and all numbers greater than or equal to `5` satisfy the inequality. This can be interpreted as the entire number line being shaded or marked, illustrating that we are not excluding any real numbers from our solution.

Algebraic Reasoning

Algebraic reasoning is the process of using symbols and rules of algebra to understand and solve mathematical problems. In the context of absolute value inequalities, this form of reasoning permits us to transform complex-looking problems into simpler pieces that are easier to solve. With `|x - 5| ≥ 0`, algebraic reasoning helps us understand that since the output of an absolute value cannot be negative, any input value for `x` will result in a solution that meets the inequality's criteria.

Further, algebraic reasoning would guide us to realize that attempting to solve `|x - 5| ≥ 0` by splitting into cases or isolating `x` (common methods for more complex inequalities) is not necessary, as the solution set includes all real numbers. Recognizing the nature of the problem presented and applying appropriate algebraic rules or properties can often lead to an immediate solution without intensive calculation.