91Ó°ÊÓ

Solving inequalities is a crucial skill in mathematics, similar to solving equations. However, inequalities differ because they express a range of possible solutions. When solving inequalities, we aim to find all values of the variable that make the inequality true.
For the exercise given, we are dealing with two inequalities:

• The first is \( f(x) > 5 \), which translates to \( x - 2 > 5 \). Solving this involves isolating the variable \( x \) by adding 2 to both sides, yielding \( x > 7 \).
• The second inequality is \( f(x) < -1 \), which becomes \( x - 2 < -1 \). Similarly, adding 2 gives us \( x < 1 \).

To solve these inequalities, we isolate the variable \( x \) through basic algebraic manipulations, resulting in a range of potential solutions. Eventually, we combine these into a compound statement: either \( x < 1 \) or \( x > 7 \). This means any \( x \) that satisfies either part is a solution to the compound inequality.

When working with inequalities, especially those involving functions, analyzing the function itself helps us understand the range of values it can take. The function from the given exercise is \( f(x) = x - 2 \).
This is a linear function, which graphically represents a straight line with a slope of 1 and a y-intercept of -2.

• Understanding the function's behavior provides insight into how we approach problems like inequality solving.
• Specifically, altering \( x \) affects both the direction of the inequality and its corresponding solutions significantly.

For example, the statement \( f(x) > 5 \) implies we are looking for \( x \) values that place the function output above 5 on this line. Similarly, \( f(x) < -1 \) seeks where the function’s value is below -1. Knowing this helps immensely during interpretation and graphically representing solutions, because it reveals where our critical points of interest lie (at \( x = 1 \) and \( x = 7 \)).

Once we solve an inequality, we frequently express our results in interval notation. This form of notation is concise and a preferred method for representing ranges of solutions.
For the exercise, we arrived at the combination

• \( x < 1 \)
• \( x > 7 \)

In interval notation, this is written as \( (-\infty, 1) \cup (7, \infty) \). Here’s how it breaks down:

• The use of parentheses \( \) indicates that 1 and 7 are not included within the intervals, as indicated by open circles on a graph.
• The symbols \( -\infty \) and \( \infty \) suggest that the ranges extend indefinitely in their respective directions.

The union symbol \( \cup \) signifies the solution set is a combination of separate intervals, further simplifying how we communicate compound inequality solutions neatly. This notation complements the graph by providing clarity in numerical terms, ensuring the solution is understandable even without a visual aid.