Problem 30 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ |x+4| \leq 0 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Absolute Value Properties

When dealing with absolute inequalities, it's important to understand the property of absolute values. An absolute value yields non-negative results. Hence, the minimum value an absolute expression like \(|x+4|\) can attain is zero.

Step 2: Setting the Absolute Expression Equal to Zero

Given the inequality \(|x+4| \leq 0\), we find the value of \(|x+4|\) where it equals zero because absolute values are either positive or zero. Solve the equation \(x+4 = 0\).

Step 3: Solving the Equation

To solve \(x+4 = 0\), isolate \(x\) by subtracting 4 from both sides: \(x = -4\).

Step 4: Interpreting the Solution

Since \(|x+4|\) can only be equal to zero when \(x = -4\), and given the inequality \(|x+4| \leq 0\), only \(x = -4\) satisfies this condition.

Step 5: Expressing the Solution in Interval Notation

In interval notation, a single value \(x = -4\) is represented as the interval \([-4, -4]\).

Key Concepts

Absolute Value

The concept of absolute value refers to the distance of a number from zero on a number line, without considering its direction. It's always a non-negative value. For any number \(x\), the absolute value is expressed as \(|x|\). This means:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

Absolute values are crucial in handling inequalities as they guarantee non-negative results. In the inequality \(|x+4| \leq 0\), the expression \(|x+4|\) must be zero, as it can’t be less than zero. The only point where an absolute value of a number equals zero is when the inside expression is itself zero, indicating a boundary case of an absolute value function. This helps us find solutions by transforming the inequality into an equation like \(x+4=0\). By understanding this core principle, solving absolute value inequalities becomes more intuitive.

Interval Notation

Interval notation is a compact way to describe a range of numbers in mathematics. It uses parentheses \(()\) or brackets \([]\) to indicate the endpoints of an interval:

• Parentheses \(()\) indicate that the endpoint is not included, called an open interval.
• Brackets \([]\) indicate that the endpoint is included, called a closed interval.

For example, \((-3, 2)\) describes all numbers greater than \(-3\) and less than \(2\). \([-4, -4]\), as seen in our exercise, represents a situation where there is only one endpoint, meaning \(x\) can only be \(-4\). In this way, a single point is represented as a closed interval with the same start and endpoint. This notation simplifies the expression of solutions considerably, especially in mathematical contexts involving inequalities or specific solution sets.

Solving Equations

Solving an equation involves isolating the variable to find its value. Consider the simple approach applied in this exercise: \(x+4=0\). The goal is to determine \(x\) such that the equation holds true. To find the solution:

• Subtract \(4\) from both sides: \(x+4-4=0-4\)
• This simplifies to \(x = -4\)

This process emphasizes moving terms across the equation to isolate the variable. By performing the same operation on both sides of the equation, you maintain equality. This method is also applicable in more complex equations, where multiple terms or coefficients might be involved. It becomes fundamental when solving absolute value inequalities or any other algebraic equations. Understanding and practicing this process is essential for solving mathematical problems efficiently.