Problem 76 - Intermediate Algebra

Question

Let \(f(x)=x-2 .\) Find all values of \(x\) for which \(f(x)>5\) or \(f(x)<-1 .\)

Step-by-Step Solution

Answer

The solution is \(x < 1\) or \(x > 7\).

Step 1: Understand the Inequality

We need to find values of \(x\) for which \(f(x) > 5\) or \(f(x) < -1\). Given \(f(x) = x - 2\), this becomes the inequality \(x - 2 > 5\) or \(x - 2 < -1\). We will solve each part separately.

Step 2: Solve the First Inequality

To solve \(x - 2 > 5\), add 2 to both sides to isolate \(x\):\[x - 2 + 2 > 5 + 2\]which simplifies to \(x > 7\).

Step 3: Solve the Second Inequality

To solve \(x - 2 < -1\), add 2 to both sides to isolate \(x\):\[x - 2 + 2 < -1 + 2\]which simplifies to \(x < 1\).

Step 4: Combine the Solutions

The solution is the union of the two solutions: \(x > 7\) or \(x < 1\). Therefore, \(x\) must be either greater than 7 or less than 1. In interval notation, the solution is \((-8, 1) \cup (7, +8)\).

Key Concepts

Solving Inequalities

Understanding and solving inequalities is fundamental in algebra. Here, we deal with finding values for a variable that make an inequality true. For the given exercise, we need to solve two separate inequalities:

• For the first inequality, \(x - 2 > 5\), add 2 to both sides to isolate \(x\). This results in \(x > 7\).
• The second inequality, \(x - 2 < -1\), is tackled similarly by adding 2 to both sides, simplifying to \(x < 1\).

The key process here is to perform the same mathematical operation on both sides of the inequality to isolate the variable, making solving inequalities systematic and predictable. Remember, if you multiply or divide an inequality by a negative number, you need to flip the inequality sign, though it's not needed in this example.

Interval Notation

Interval notation provides a concise way to express the range of solutions for inequalities. It uses parentheses and brackets to show the start and end of a solution set, indicating open and closed intervals.

• Parentheses \((\) or \()\) indicate that a boundary is not included (an open interval) like in \((-\infty, 1)\) and \((7, \infty)\).
• Brackets \([\) or \(]\) show that a boundary is included (a closed interval). For example, \([1, 7]\) would include 1 and 7, but we don’t use them in this exercise as the solution only involves open intervals.

In our case, the solution \((-\infty, 1) \cup (7, \infty)\) represents all numbers less than 1 and greater than 7, using infinity signs to show that the intervals continue indefinitely in those directions.

Union of Intervals

In mathematics, the union of intervals is used to combine multiple solution sets into one comprehensive set. This is symbolized by the union symbol \(\cup\).

• The exercise solution \(x < 1\) or \(x > 7\) forms two separate sets of numbers.
• By using the union, we write this as \((-\infty, 1) \cup (7, \infty)\), which simply joins the two intervals, signifying that values can belong to either interval to satisfy the inequality.

Unions are particularly useful in expressing solutions where multiple ranges of numbers meet the criteria, as it gives a complete picture of all possible solutions that make the original inequality statement valid.