Problem 69 - College Algebra

Question

Solve absolute value inequality. \(|x|>3\)

Step-by-Step Solution

Answer

The solution for the inequality \(|x|>3\) is \(x>3\) or \(x<-3\), or expressed in interval notation: (-∞,-3) U (3,∞).

Step 1: Understand the inequality

First, understand that \(|x|>3\) means x is either less than -3 or greater than 3. This inequality can be expressed as \(x>3\) or \(x<-3\). This is because the absolute value signifies the distance from zero on a number line, hence values greater than 3 would be both numbers greater than 3 and less than -3.

Step 2: Solve for x

Since our inequality is broken into two parts, no further steps are necessary to isolate x. The solution is already stated in step 1, that is: \(x>3\) or \(x<-3\). It could also be expressed as an interval notation (-∞,-3) U (3,∞).

Step 3: Plot the solution on a number line

The last step is to illustrate the solution on a number line. Points -3 and 3 will not be included in the solution, therefore a parenthesis or an open dot will be used to signify this. The line extending to the left of -3 and to the right of 3 will mark the solution set.

Key Concepts

Solving Absolute Value Inequalities

When dealing with absolute value inequalities, it is crucial to understand what the absolute value represents. The absolute value of a number, denoted as \(|x|\), describes the distance of that number from zero on the number line. In the inequality \(|x|>3\), we are looking for all the values of \(x\) whose distance from zero is greater than three. This translates mathematically into two separate inequalities: \(x>3\) and \(x<-3\). These inequalities denote that \(x\) can be any number greater than 3 or any number less than -3.

To summarize: \(|x|>3\) is solved by separating it into \(x>3\) and \(x<-3\). Always remember, the key to solving absolute value inequalities is to split them into two simpler inequalities that can easily be solved separately.

Understanding Inequality Solutions

The solutions to inequalities, such as the one mentioned, are sets of numbers rather than a single value. For the inequality \(|x|>3\), the solutions are all numbers greater than 3 and less than -3. In essence, the solution covers both ends of the number scale where the values lie outside the range of \(-3, 3\).

Remember, inequalities are not strict equations. They do not tie \(x\) to a fixed number, but rather to a range or a set of permissible values, which in this case are split into two regions on the number line.

• Numbers greater than 3: \(x>3\)
• Numbers less than -3: \(x<-3\)

Finding solutions to inequalities requires understanding and interpreting these ranges correctly.

Interval Notation and Its Use

Interval notation is a compact way of expressing a range of values that satisfy an inequality. For the inequality \(|x|>3\), the solution \(x>3\) or \(x<-3\) can be written in interval notation as \((-\infty,-3) \cup (3,\infty)\).

The symbols involved in interval notation have specific meanings:

• '(' or ')' indicate that the endpoints are not included in the interval, representing ''open''. This is crucial here, as 3 and -3 are not part of the solutions.
• '\cup' is used to denote a union between two sets, meaning any number in either set is a solution.

Using interval notation helps to clearly and succinctly present solution sets, especially useful when writing mathematics or communicating solutions.

Number Line Graphing

Graphing the solutions of inequalities on a number line provides a visual representation that makes understanding the solution easier. For the inequality \(|x|>3\), plotting involves marking the open intervals \([-\infty, -3)\) and \( (3, \infty)\). In this graph:

• Open circles (or parentheses) are placed on the points -3 and 3 to indicate they are not included in the solution.
• A line or arrow extends to the left of -3 and another to the right of 3, showing the regions where the inequality holds true.

This graphing method provides a straightforward and effective way to comprehend and observe where solutions to absolute value inequalities exist on the number line.