Problem 33 - College Algebra

Question

Graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-1|>2 $$

Step-by-Step Solution

Answer

The solution set is \((- fty, -1) \cup (3, fty)\).

Step 1: Understanding the Inequality

The inequality given is \(|x-1|>2\). This means we need to find all \(x\) values for which the distance between \(x\) and 1 is greater than 2. We will split this into two separate inequalities.

Step 2: Breaking Down the Absolute Value Inequality

We can split the inequality \(|x-1|>2\) into two separate inequalities: \(x-1 > 2\) and \(x-1 < -2\). We will solve these inequalities separately.

Step 3: Solve First Inequality

For \(x-1 > 2\), adding 1 to both sides gives us \(x > 3\). This means \(x\) must be greater than 3.

Step 4: Solve Second Inequality

For \(x-1 < -2\), adding 1 to both sides gives us \(x < -1\). This means \(x\) must be less than -1.

Step 5: Combine the Solutions

The solution for the inequality \(|x-1|>2\) is the union of the solutions from the two inequalities: \(x < -1\) or \(x > 3\). We can express this in interval notation as \((-fty, -1) \cup (3, fty)\).

Step 6: Graph the Solution

On a number line, we shade to the left of -1 and to the right of 3, indicating the regions where the inequality holds true. The regions \((-fty, -1)\) and \((3, fty)\) do not include -1 and 3 themselves, as indicated by open circles at -1 and 3.

Step 7: Final Representation and Interval Notation

The solution set on the number line corresponds to values of \(x\) outside the interval \([-1, 3]\). The solution to the inequality \(|x-1|>2\) in interval notation is \((-fty, -1) \cup (3, fty)\).

Key Concepts

Graphing Inequalities

When it comes to graphing inequalities on a number line, the main goal is to visually represent the solution set of the inequality. For an inequality like \(|x-1|>2\), we need to show where the inequality holds true. This involves identifying on the number line:

• The regions that satisfy the inequality.
• Whether certain points are included or excluded.

To graph \(|x-1|>2\), we've already determined the two separate inequalities: \(x < -1\) and \(x > 3\). On the number line, we:

• Shade all numbers less than -1.
• Leave -1 itself unshaded, indicating it's not part of the solution (open circle).
• Also shade numbers greater than 3.
• Leave 3 itself unshaded (open circle as well).

This shading visually represents all the \(x\) values that make \(|x-1|>2\) true, making it easier to understand the solution set.

Interval Notation

Interval notation is a concise way to describe sets of numbers on a number line. It's especially handy for inequalities because it clearly shows which numbers are included or excluded from the solution set. For instance, let's take the solution to \(|x-1|>2\): \(x < -1\) or \(x > 3\).

• This translates to two separate intervals: \((-\infty, -1)\) and \((3, \infty)\).
• Parentheses \(()\) are used because -1 and 3 are not included in the solution, which matches the open circles on the graph.
• The union symbol \(\cup\) is used to combine the two intervals, indicating the complete solution set.

Presenting the solution set in this way not only highlights where the inequality holds but also visually matches the graph drawn.

Solution Sets

A solution set is simply the collection of numbers that satisfy a given inequality. When dealing with inequalities involving absolute values, the solution set often comprises multiple intervals. For our example \(|x-1|>2\), the solution set includes:

• All \(x\) values less than -1.
• All \(x\) values greater than 3.

These intervals reflect the regions on the number line that make the inequality true. Each part of the absolute value inequality contributes a separate interval to the solution set. Together, they form the complete collection of \(x\) values that satisfy \(|x-1|>2\). Recognizing and combining these intervals into a solution set is crucial for understanding how the graph and interval notation connect.

Inequality Solving Steps

Solving absolute value inequalities involves a structured approach to ensure all solutions are found. Here's how it's done:

• Step 1: Understand the Inequality: Absolute value inequalities like \(|x-1|>2\) measure distance from a point on a number line. We're looking for points more than 2 units away from 1.
• Step 2: Split the Inequality: The inequality is divided into two: \(x - 1 > 2\) and \(x - 1 < -2\). This comes from the nature of absolute values needing to consider both the positive and negative distances.
• Step 3: Solve Each Part Separately: Solve \(x - 1 > 2\) by adding 1 to find \(x > 3\). Solve \(x - 1 < -2\) similarly to find \(x < -1\).
• Step 4: Combine the Solutions: Merge the separate solutions using union, resulting in \(x < -1\) or \(x > 3\), and express in interval notation: \((-\infty, -1) \cup (3, \infty)\).

Each step builds on the last, maintaining clarity and thoroughness in finding all possible solutions to the inequality. This methodical breakdown ensures nothing is overlooked and each part of the solution is clearly identified.