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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 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 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.