Problem 15 - College Algebra

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |x+9| \geq-6 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Absolute Values

The absolute value of any number or expression is always non-negative. This means that for any expression \(|x + 9|\), the result is always greater than or equal to 0. Therefore, if the inequality states \(|x + 9| \geq -6\), we can safely assume it's always true because the absolute value cannot be negative and is always greater than -6.

Step 2: Inequality Analysis

Since \(|x + 9|\) is always non-negative, it satisfies \(|x + 9| \geq -6\) for any real number \(x\). This means the inequality holds true for all possible values of \(x\).

Step 3: Solution in Interval Notation

Given that the inequality is valid for any real number, we express the solution in interval notation as \((-\infty, \infty)\).

Key Concepts

Inequality Solutions

In mathematics, solving inequalities means finding all possible values for variables that satisfy the given inequality. Inequalities often involve expressions that are not equal. Inequalities can range from very simple, like linear inequalities, to complex ones involving absolute values or multiple variables.
An absolute value inequality, such as \(|x+9| \geq -6\), demonstrates a unique aspect of inequalities. Importantly, the absolute value of any expression ensures that the result is always non-negative. This implies that, in cases where the inequality includes negative numbers as in this example, the inequality holds true for all possible values within the domain. Understanding this fundamental characteristic of absolute value helps simplify these types of problems incredibly.

• Absolute value inequalities can describe the range of solutions.
• They often include expressions that must be greater than or less than zero.
• It is critical to understand the non-negative nature of absolute values.

When you deal with an absolute value inequality where the inequality symbol is combined with a negative number, it will usually result in a solution that covers the entire number set.

Interval Notation

Interval notation is a concise way of expressing sets, mainly when discussing solutions to inequalities. Instead of listing every number that satisfies the inequality, we use interval notation to describe the entire range of possible solutions.
The primary components include brackets and parentheses:

• Use parentheses \((-\infty, \, \infty)\) to indicate that endpoints are not included ("open interval").
• Brackets, like in \([a, b]\), indicate that endpoints are included ("closed interval").

In the case of solving absolute value inequalities with solutions covering all numbers, such as \(|x+9| \geq -6\), the interval notation \((-\infty, \infty)\) is used to succinctly state that any real number satisfies the condition. It's important to master interval notation, as it provides a clear, standardized way to communicate which numbers or ranges satisfy an equation or inequality.

Real Numbers

Real numbers form the backbone of the number system widely used in mathematics. They encompass all the numbers on the number line, from negative to positive infinity, including all rational and irrational numbers.
Understanding real numbers is crucial, especially when examining absolute value inequalities, as these inequalities typically concern values that must fit within the realm of real numbers.

• Real numbers include integers, fractions, and decimals.
• An inequality holding for all real numbers implies every possible value on the number line satisfies the inequality.
• The symbols \((-\infty, \infty)\) represent all real numbers in interval notation.

When we solve an inequality like \(|x+9| \geq -6\), it inherently suggests that all real numbers will fit the solution set. This understanding helps know that inequalities dealing with absolute values often translate to mean that the domain of solutions is unrestricted along the real number line.