Problem 38 - Algebra for College Students

Question

Solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. \(x-4<-2\) or \(x-4>2\)

Step-by-Step Solution

Answer

The solution is \((-\infty, 2) \cup (6, \infty)\).

Step 1: Solve the First Inequality

Begin by solving the inequality \(x-4<-2\). Add 4 to both sides of the inequality to isolate \(x\). This gives us: \(x<-2+4\). Simplify the right side: \(x<2\).

Step 2: Solve the Second Inequality

Now solve the second inequality \(x-4>2\). Add 4 to both sides to isolate \(x\): \(x>2+4\). Simplify the right side: \(x>6\).

Step 3: Analyze the Compound Inequality

Since the compound inequality is connected by 'or', we consider the union of the two individual solution sets. This means \(x<2\) or \(x>6\).

Step 4: Express the Solution in Interval Notation

The solution in interval notation is \((-\infty, 2) \cup (6, \infty)\).

Step 5: Graph the Solution Sets

To graph \((-\infty, 2) \cup (6, \infty)\), draw a number line. Shade the section of the number line extending to the left from 2 with an open circle at 2, and extend to the right from 6 with an open circle at 6.

Key Concepts

Inequality Solutions

Understanding how to solve inequalities is a fundamental skill in algebra. Inequalities express a relationship where two expressions are not necessarily equal. For the problem given, we are solving the compound inequalities:

• \(x-4<-2\)
• \(x-4>2\)

To find the solution for each part:1. Start by isolating the variable \(x\) on one side of the inequality. 2. For \(x-4<-2\), add 4 to both sides, resulting in \(x<2\).3. For \(x-4>2\), also add 4 on both sides, giving \(x>6\).

The solutions, \(x<2\) and \(x>6\), show where the inequalities are true on a number line.
This involves looking at all possible values of \(x\) that satisfy these conditions. If we say \(x<2\), it means any number less than 2 works. Similarly, \(x>6\) means any number greater than 6 is a solution.
Essentially, solving inequalities involves reversing operations while respecting the inequality direction.

Interval Notation

Interval notation is a concise way to express the solution set of inequalities. It's particularly useful when dealing with continuous ranges of numbers, like in our solution.
In interval notation, we use parentheses \(()\) to indicate open intervals, meaning the values at the endpoints are not included, and brackets \([]\) for closed intervals where endpoints are included. This exercise uses open intervals as the inequality signs \(<\) and \(>\) exclude values at 2 and 6, respectively.
The solution sets from our inequalities can be expressed in interval notation as:

• For \(x<2\), the interval notation is \((-\infty, 2)\).
• For \(x>6\), it is \((6, \infty)\).

Infinity (\(\infty\)) is always expressed with parentheses because they suggest a boundless range beyond any specific number.
Interval notation provides a clear and standardized method to present solutions, making it easier to communicate complex sets of numbers precisely.

Union of Sets

The term "union" is a fundamental concept from set theory. In mathematics, the union of two sets is a set containing all elements that are in either set. For our compound inequality with an 'or' statement, we need the union of the two inequalities' solutions.

In our example, the union of the solutions \(x<2\) and \(x>6\) is represented as:

• \((-\infty, 2) \cup (6, \infty)\)

The symbol \(\cup\) denotes the union operation. Union here effectively combines both solution sets, meaning any number less than 2 or greater than 6 satisfies the compound inequality.

By understanding the union of sets, we see how combining solutions from individual inequalities constructs a comprehensive solution for the entire system. This is particularly useful when dealing with compound inequalities connected by 'or' because it captures all possible solutions in a unified form.