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The absolute value of a number is a key concept used in solving inequalities like the one given: \(5 + |x| \leq 2\). Absolute value represents the distance of a number from zero on the number line, without considering its direction. Therefore,

• \(|x|\) is always non-negative.
• It is denoted by two vertical lines, such as \(|x|\).

When dealing with absolute inequalities, it's important to recognize that the absolute value, combined with other terms, sets a specific upper or lower limit. Here, because \(|x|\) starts from zero and only increases, it implies the sum \(5 + |x|\) cannot be less than 5. This inherent non-negativity and starting point of the expression provides crucial information about whether the inequality has a solution.

In the context of inequalities, a solution set is the collection of real numbers that satisfy the given inequality equation. For the exercise \(5 + |x| \leq 2\), the solution set tells us all possible values of \(x\) that make \(5 + |x|\) less than or equal to 2.
After evaluating the inequality, we see that there are no such values of \(x\) that can satisfy the equation because the minimum possible value of \(5 + |x|\) is 5, which already exceeds 2.
Thus, this particular equation has an empty solution set, meaning no real value of \(x\) fits this condition. In mathematical notation, it is represented as \(\emptyset\), indicating there is no number that can be plugged into \(x\) to make the inequality true.

Interval notation is a method used to describe sets of numbers on a number line. It depicts the start and end points of the intervals and uses parentheses or brackets to indicate whether these endpoints are included or excluded. In cases where there is no solution, like our current inequality, it simplifies the concept even further.

• An empty solution set is written as \(\emptyset\), indicating no parts of the number line satisfy the inequality.
• If there was a valid interval where \(x\) satisfied the inequality, it would be marked as something like \([a, b]\) or \((a, b)\), depending on whether \(a\) or \(b\) were included.

Using interval notation allows us to succinctly convey which parts of the number line meet the conditions of the inequality. However, in this case, the lack of a solution makes interval notation straightforward with \(\emptyset\).

Graphing inequalities involves representing the range of values that satisfy a particular condition on a number line. For an inequality with a solution, this means shading parts of the line or using symbols to show which numbers meet the conditions specified by the inequality.
For the inequality \(5 + |x| \leq 2\), we've determined there is no solution. This results in an empty graph, where no portion of the number line gets shaded or marked, since no value of \(x\) satisfies the inequality.
Understanding how to graph inequalities aids in visualizing real-number solutions, but for cases like this with no solution, the concept of a blank graph solidifies the understanding that no viable solution set exists. Graphing becomes an intuitive way to reflect the outcome of the inequality analysis.