Problem 64 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ |x-0.3|>1.5 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1.2) \cup (1.8, \infty)\).

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|x - 0.3| > 1.5\). An absolute value inequality of the form \(|A| > B\) means that either \(A > B\) or \(A < -B\).

Step 2: Break Down into Two Cases

Using the rule from Step 1, split the inequality into two separate inequalities: 1. \(x - 0.3 > 1.5\)2. \(x - 0.3 < -1.5\).

Step 3: Solve the First Case

To solve \(x - 0.3 > 1.5\), add \(0.3\) to both sides:\[x > 1.8\]

Step 4: Solve the Second Case

For \(x - 0.3 < -1.5\), add \(0.3\) to both sides:\[x < -1.2\]

Step 5: Combine the Results

The solutions to the two individual inequalities can be combined using the word "or":\[x > 1.8 \text{ or } x < -1.2\]

Step 6: Write the Solution Set

Express the solution set using interval notation, which is:\((-\infty, -1.2) \cup (1.8, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers between two endpoints. It is especially useful for expressing solutions to inequalities. It provides a compact and visually clear way of presenting information. Instead of wordy descriptions, we use symbols to show the range of values that satisfy the given inequality of the problem.

When we solved the inequality \(|x - 0.3| > 1.5\), we found that \( x > 1.8 \) or \( x < -1.2 \). To express this solution using interval notation, we use:

• \((1.8, \, \infty)\) for all numbers greater than 1.8.
• \((-\infty, \,-1.2)\) for all numbers less than -1.2.

The union symbol \(\cup\) is used to combine the two distinct solution intervals, showing that any number in either interval satisfies the inequality. Thus, the complete solution set is \((-\infty, \,-1.2) \cup (1.8, \, \infty)\). This notation is efficient and clear, making it easy to convey complex solutions.

Solving Inequalities

Solving inequalities involves finding all values of the variable that make the inequality true. When working with inequalities like \(|x - 0.3| > 1.5\), we need to understand what the absolute value represents. In this case, it is looking at the distance of \(x\) from 0.3 on the number line.

For an inequality of the form \(|A| > B\), there are generally two cases:

• \(A > B\)
• \(A < -B\)

Breaking it into these cases removes the absolute value, and each case results in a different inequality to solve. This step is crucial in transforming the problem into a more familiar algebraic form.

After identifying the cases, solve each resulting inequality separately, as we did with:

• \(x - 0.3 > 1.5\)
• \(x - 0.3 < -1.5\)

By adding 0.3 to both sides, we simplify these to \(x > 1.8\) and \(x < -1.2\). Solving each inequality results in finding the ranges of values for \(x\) that satisfy the original inequality.

Mathematical Reasoning

Mathematical reasoning is the process that allows us to logically break down, analyze, and solve problems. It's essential when dealing with inequalities, as it helps us understand what the inequality represents and how to manipulate it correctly.

When given \(|x - 0.3| > 1.5\), mathematical reasoning tells us to decode the absolute value by determining the distance from 0.3 to be more than 1.5. This translates the problem into real-world terms, making the solution more intuitive.

Here’s how we apply reasoning to each step:

• Understand that absolute values represent a distance, guiding us to split the inequality into two conditions.
• Use logic to separate the expression into \( x - 0.3 > 1.5 \) and \( x - 0.3 < -1.5 \) based on the definition of absolute value.
• Seamlessly transition into solving for \(x\) by adding or subtracting, using arithmetic to isolate \(x\).

Each step requires reasoning to ensure we are accurately simplifying and combining the solutions. Through thorough reasoning, we achieve a complete understanding of the solution, expressed in interval notation, and grasp all the nuances of the process.