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Understanding the concept of absolute value is crucial in solving inequalities that contain them. The absolute value of a number is the distance of that number from zero on the number line, regardless of direction. For instance, both 3 and -3 have an absolute value of 3 because they are both exactly three units away from zero on the number line.

When solving the inequality like the one given, \( |4-x|<5 \) , we interpret the expression inside the absolute value, \( 4-x \) , as having a distance from zero that’s less than 5. This will apply in both directions on the number line, thus, we split the inequality into two parts: one for the positive direction and one for the negative. Hence, we maintain the 'distance' but remove the absolute value symbols, leading to two separate inequalities that need to be solved.

Solving inequalities often involves rewriting them in different forms to isolate the variable of interest. Inequality notation is a shorthand way of representing the relationship between variables and numeric values. The symbols '<' and '>' are used to show that something is less than or greater than something else, respectively.

In the given exercise, the inequality \( -5<4-x<5 \) uses inequality notation to show that the value of \( 4-x \) must be more than -5 but less than 5. Through manipulation, like subtraction and multiplication, we aim to solve for 'x'. Importantly, when we multiply or divide both sides of an inequality by a negative value, the direction of the inequality symbols must be reversed to preserve the inequalities’ true meaning. This rule is essential to correctly solve absolute value inequalities.

After solving the inequalities, we represent the solution using interval notation, which is a compact way of describing sets of numbers. This notation is particularly useful when expressing the solution set of an inequality.

For example, the solution to the inequality \( -1