Problem 19 - College Algebra

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |x-2|+4 \geq 10 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((- fty, -4] \cup [8, fty)\).

Step 1: Understand the Absolute Value Inequality

Given the inequality \(|x-2|+4 \geq 10\). To solve, first isolate the absolute value by subtracting 4 from both sides. This gives \(|x-2| \geq 6\).

Step 2: Break Down the Absolute Value

The inequality \(|x-2| \geq 6\) implies two cases. The first case is when the expression inside the absolute value is greater than or equal to 6. The second case is when the expression is less than or equal to -6. This gives two separate inequalities: \(x - 2 \geq 6\) and \(x - 2 \leq -6\).

Step 3: Solve the First Inequality

For \(x - 2 \geq 6\), add 2 to both sides to solve for \(x\). This gives \(x \geq 8\).

Step 4: Solve the Second Inequality

For \(x - 2 \leq -6\), add 2 to both sides to solve for \(x\). This gives \(x \leq -4\).

Step 5: Combine Solutions Using Interval Notation

The solution to the inequality includes all \(x\) such that either \(x \geq 8\) or \(x \leq -4\). This is represented in interval notation as \((-fty, -4] \cup [8, fty)\).

Key Concepts

Interval Notation

Interval notation is a method of writing down subsets of the real numbers. It is particularly useful for expressing solutions to inequalities because it provides a clear way to show all the numbers between two endpoints, as well as include multiple intervals. Understanding interval notation makes it easier to interpret results from inequality problems.

• Parentheses \( (\) \, \( ) \) indicate that an endpoint is not included in the interval. This is known as an 'open' interval.
• Brackets \([\) \, \()]\) indicate that an endpoint is included or closed in the interval.
• For example, the interval \((-3, 5]\) means all numbers greater than -3 and up to and including 5.
• An infinite interval like \((-\infty, -4]\) includes all values less than or equal to -4.
• "Union" symbol \((\cup)\) is used to connect separate intervals showing that numbers can belong to one set or another. \((-\infty, -4] \cup [8, \infty)\) represents a union of two parts: numbers less than -4 or greater than or equal to 8.

Understanding how to express answers with interval notation is key when writing solutions for absolute value inequalities.

Inequality Solving Steps

Solving inequalities can be approached through a series of thoughtful steps. These steps guide you towards finding the range of values that satisfy an inequality. Whether dealing with a simple or an absolute value inequality, they help structure the solution process clearly.
Here’s a general outline for solving absolute value inequalities:

• **Isolate the Absolute Value Expression:** Start by isolating the absolute value expression on one side of the inequality. This might require adding or subtracting from both sides.
• **Break the Absolute Value Expression into Two Cases:** For an inequality such as \( |x-a| \geq b\), it can be divided into two separate inequalities: \( x-a \geq b\) and \( x-a \leq -b\). This is because the expression inside the absolute value can either be positive or negative while still having a magnitude greater than or equal to the number on the other side.
• **Solve Each Inequality:** Go through the steps to solve each inequality separately. Find the values of the variable that satisfy each inequality.
• **Combine Solutions:** Use `or` to combine the solutions from both inequalities. This typically forms a compound inequality, which can then be expressed using interval notation.

Each of these steps helps consolidate understanding and ensures that no part of the solution is overlooked, leading to a more robust mastery of solving absolute value inequalities.

Compound Inequalities

Compound inequalities involve two individual inequalities combined into one expression by the words "and" or "or". These terms are key because they affect how the solution is constructed and interpreted.

• **"And" Compound Inequalities:** This term indicates that both conditions must be satisfied simultaneously. The solution to such inequalities is the intersection of the solutions to the two parts. It is usually represented by the overlapping region of two intervals.
• **"Or" Compound Inequalities:** In this scenario, only one of the conditions needs to be satisfied. This results in a union of the intervals from each part of the inequality solution. As in the original problem, \( x \leq -4 \) or \( x \geq 8\), is an example of an "or" compound inequality.

With each approach, understanding whether you are dealing with "and" or "or" affects how results are combined in interval notation.
Compound inequalities are quite practical as they allow for representing more complex solutions that might arise in real-world scenarios and mathematical problem-solving alike.