Problem 15 - College Algebra with Corequisite Support

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: Identify the Inequality

The inequality given is \(|x + 9| \geq -6\). Notice that the absolute value expression \(|x + 9|\) is always non-negative (zero or positive) because absolute values represent distances. Hence, for any value of \(x\), \(|x + 9|\) will always be greater than or equal to \(-6\).

Step 2: Analyze the Inequality Condition

Since any absolute value is always greater than or equal to zero, and zero is greater than \(-6\), the inequality \(|x + 9| \geq -6\) holds true for all real numbers. This means there is no restriction on \(x\), so every real number is a solution.

Step 3: Writing the Solution in Interval Notation

Since any real number satisfies the inequality, the solution in interval notation is \((-\infty, \infty)\), which represents the entire set of real numbers.

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to describe the set of solutions to an inequality. It represents the range of values that satisfy a given condition. For inequalities, such as those involving variables, it's a compact way to express which values of a variable make the inequality true.

In interval notation:

• Parentheses, "(", ")", are used to denote that an endpoint is not included in the interval. For example, "(a, b)" means everything between but not including "a" and "b".
• Square brackets, "[", "]", are used to denote that an endpoint is included. For example, "[a, b]" means everything from "a" to "b", inclusive.
• "(−∞, ∞)" represents all real numbers, meaning no boundaries. In this example, it indicates that every real number is a solution to the inequality."

Real Numbers

Real numbers encompass all numbers that can be found on the number line. This includes both rational numbers (like 1/2, 0.75, etc.) and irrational numbers (like the square root of 2, or π). They are a broad category that includes most of the numbers used in everyday life.

Key properties of real numbers:

• They can be positive, negative, or zero.
• They can be decimals or whole numbers.
• They fill the entire number line, with no gaps.

Real numbers form the basis for real-world measurements and are essential when solving inequalities, as demonstrated in our given inequality involving the absolute value, where the solution encompasses every real number.

Inequality Solution

Part of understanding inequality solutions is grasping what inequalities seek to express. Inequalities are mathematical statements that indicate one quantity is greater than, less than, or equal to another. For example, "<" indicates less than, while ">=" denotes greater than or equal to.

For absolute value inequalities, such as \(|x + 9| \geq -6\), the goal is to determine the range of values for x that make the statement true. Absolute value fundamentally measures distance, so it's always non-negative, meaning it can't be less than a negative number. Therefore, \(|x + 9| \geq -6\) holds for every real number, as no real number squared results in a negative number.

The solution, therefore, doesn't restrict x at all, leading to our end result being \( (-\infty, \infty) \), as discussed in the final step of the solution process.