Problem 36 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x \geq 1 \quad \text { or } \quad x \geq 8 $$

Step-by-Step Solution

Answer

The solution set is \(x \geq 1\) or \([1, \infty)\).

Step 1: Identify the Compound Inequalities

There are two inequalities given: 1. \(x \geq 1\) 2. \(x \geq 8\)

Step 2: Analyze Each Inequality Separately

First, consider the inequality \(x \geq 1\). Any value of \(x\) that is 1 or greater satisfies this inequality.Next, consider the inequality \(x \geq 8\). Any value of \(x\) that is 8 or greater satisfies this inequality.

Step 3: Combine the Inequalities

Since the compound inequality uses 'or', combine the solutions of both inequalities. This means any value that satisfies either inequality will be included in the solution set.Since \(x \geq 1\) already includes all values where \(x \geq 8\), the combined solution is simply \(x \geq 1\).

Step 4: Write the Solution in Interval Notation

The solution in interval notation can be written as \[[1, \infty)\]

Step 5: Graph the Solution Set

On a number line, start at 1 and draw a line extending to the right towards infinity. Use a closed dot at 1 to indicate that 1 is included in the solution set.

Key Concepts

Inequality Solving

When solving inequalities, the goal is to find the range of values that make the given statement true. Inequality symbols include \(>\), \(<\), \(\geq\) (greater than or equal to), and \(\leq\) (less than or equal to). Unlike equations, solutions to inequalities are often ranges of values instead of specific numbers. In the given problem, we are dealing with compound inequalities, which involve two or more inequalities combined together using the words 'and' or 'or'. For example, in the exercise, we have \(x \geq 1\) or \(x \geq 8\). When solving compound inequalities, treat each component separately before combining the results based on whether you use 'and' or 'or'.

Interval Notation

In interval notation, we write solutions to inequalities in a way that concisely describes the range of values that are included. Interval notation uses brackets \([]\) and parentheses \(( )\) to show which endpoints are included or excluded.

• A closed bracket \([\) or \(]\) means the endpoint is included.
• An open parenthesis \( ( \) or \([\) means the endpoint is excluded.

In the exercise, we solved the inequality \(x \geq 1\) or \(x \geq 8\). Since all values from 1 and upwards satisfy this compound inequality, we express it using interval notation as \[ [1, \infty) \]. Here, 1 is included (closed bracket), and infinity is not included (open parenthesis) because infinity denotes that values go on indefinitely. Interval notation is a clear and precise way to represent the solution set of an inequality.

Graphing Inequalities

Graphing inequalities helps visualize the solution set on a number line. For example, in the exercise, the inequality \(x \geq 1\) means any value of \(x\) greater than or equal to 1 is part of the solution. To graph this:

• Draw a number line.
• Locate the point 1 and draw a closed dot at this point, indicating that 1 is included in the solution.
• Draw a line extending to the right from 1 towards infinity to show that all values greater than or equal to 1 are part of the solution set.

The visual representation on the number line provides a clear understanding of the range of values that satisfy the inequality. By understanding graphing, you can easily compare and combine solution sets for more complex inequalities.