Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |3 x-9|>0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 3) \cup (3, \infty)\).

Step 1: Understanding the Absolute Value Inequality

The given inequality is \(|3x - 9| > 0\). This means that the expression inside the absolute value, \(3x - 9\), can be any value except zero. We need to find the values of \(x\) for which this expression is either positive or negative.

Step 2: Identify Critical Point

The expression \(3x - 9\) equals zero when \(x = 3\). We find this by setting \(3x - 9 = 0\) and solving for \(x\). Thus, the critical point divides the number line into regions where the inequality may change.

Step 3: Test Intervals

We have two intervals to consider based on the critical point \(x = 3\):1. \(x < 3\)2. \(x > 3\)For both intervals, the inequality holds because in either interval \(3x - 9\) doesn’t equal zero, meaning the inequality \(|3x - 9| > 0\) is true.

Step 4: Write Solution in Interval Notation

The inequality \(|3x - 9| > 0\) is satisfied for all \(x\) except \(x = 3\). Therefore, the solution in interval notation is \((-\infty, 3) \cup (3, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers between specified limits. It's like a shorthand way to express a range of values.

When we deal with inequalities, especially after solving them, we often end up with a range or a set of possible numbers that satisfy the condition. Instead of listing all those numbers, interval notation is quick and efficient.

• The symbol “(” or “)” is used to denote that a boundary is not included.
• The symbol “{” or “}” would denote inclusion, but with intervals, we'll stick to round brackets for open intervals like this case.
• For example, in \((-\infty, 3) \cup (3, \infty)\), it means all numbers less than 3 and more than 3 are included, but not 3 itself.

These symbols create a clean and efficient way to communicate solutions to inequalities, especially those involving absolute values.

Inequality Solving

Inequality solving is different from solving equations. Instead of finding a specific number, we often find a range of numbers that satisfy the inequality condition.

To solve an inequality, there are several important steps:

• Remove the absolute value: Recognize that absolute values involve two expressions, one positive, and the other negative.
• Critical Points Identification: Find the values where the inequality can change. Absolute values create these critical points which divide the number line into sections where the inequality behaves differently.
• Test Intervals: Check regions separated by critical points to decide where the inequality holds true.

The solution to the inequality provides insight into which interval or range of numbers fit the conditions given in our inequality.

Critical Points

Understanding critical points in the context of inequalities is essential to find the intervals where a condition is true.

Critical points are specific points on the number line where the value inside an absolute value could potentially change. These occur when the expression inside the absolute value is equal to zero.

For the inequality \(|3x - 9| > 0\), when solving \((3x - 9) = 0\), we find this point at \(x = 3\). This divides the number line into segments which are to be checked separately.

• It’s important because this helps us determine where on the number line our inequality holds or breaks.
• After finding these points, test the values around these points to establish which satisfy the inequality.

Recognizing this concept allows you to break down complex inequalities into manageable parts that are easier to solve.