Problem 26 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x+4| \leq 0 $$

Step-by-Step Solution

Answer

The solution to the inequality is \([-4, -4]\).

Step 1: Understand the Absolute Value Inequality

An absolute value inequality like \(|x+4| \leq 0\) means that the distance between \(x+4\) and 0 on the number line is less than or equal to zero. Since absolute value represents distance, it can only be zero or a positive number.

Step 2: Set the Absolute Value to Zero

For the absolute value \(|x+4|\) to be equal to zero, the expression inside the absolute value must be equal to zero itself. Therefore, set \(x + 4 = 0\).

Step 3: Solve the Equation

Solve the equation \(x + 4 = 0\) for \(x\). Subtract 4 from both sides to isolate \(x\): \[ x = -4 \]

Step 4: Analyze the Solution

Since the inequality \(|x+4| \leq 0\) is satisfied only when the expression is exactly zero, the only solution is \(x = -4\). There are no other values of \(x\) that would satisfy the inequality because absolute value cannot be negative.

Step 5: Express the Solution in Interval Notation

Since \(x = -4\) is the only solution, express this as a single point in interval notation: \[ [-4, -4] \]

Key Concepts

Interval Notation for Absolute Value Inequalities

Understanding interval notation is key when expressing the solutions to inequalities, especially those involving absolute values. Interval notation uses brackets and parentheses to describe a set of numbers, showing which numbers are included in a solution.

• Square brackets \[ a, b \] indicate that both values are included. This is known as a closed interval.
• Round brackets (a, b) mean that the endpoints are not included, which is called an open interval.

In the inequality \(|x+4| \leq 0\), we found that the solution is \(x = -4\). Since the solution consists of a single number, it's expressed as \([-4, -4]\) in interval notation, showing that only \(x = -4\) satisfies the condition.

Solving Equations Involving Absolute Values

To tackle equations that include absolute values, it’s essential to comprehend how absolute values work. An absolute value \(|a|\) represents the distance from zero to \(a\) on a number line, which can only be zero or positive.Whenever you have an equation like \(|x+4| \leq 0\), your first step involves understanding the conditions where this inequality holds true. An absolute value equals zero only when the expression inside is exactly zero. This gives us a simple equation: - Set the expression inside the absolute value equal to zero: \(x + 4 = 0\).- Solve this equation by isolating \(x\). Subtract 4 from each side to get \(x = -4\).There are no other solutions for the inequality \(|x+4| \leq 0\), because absolute values can't yield negative results.

Properties of Absolute Value in Inequalities

Absolute value properties play a crucial role in determining the solution set of inequalities. Specifically, they help us understand the nature of the solutions we are dealing with. Here are a few key points to remember:

• The absolute value \(|a|\) of any number \(a\\) gives its distance from zero, meaning it is always non-negative.
• When you encounter an inequality like \(|x+4| \leq 0\), it implies that the distance is zero, meaning \(x+4 = 0\\)
• No values other than those which make the expression inside the absolute value zero can satisfy such an inequality because absolute values can't be negative.

Understanding these properties ensures that when solving absolute value inequalities, you can quickly identify when an equality, such as \(|x+4| = 0\), leads to a point solution rather than a range of values.