Solve each equation or inequality. | 8x + 5| = 0

Verified step by step guidance

1. Identify the absolute value equation: |8x + 5| = 0.
2. Recall that the absolute value of a number is zero only if the number itself is zero.
3. Set the expression inside the absolute value equal to zero: 8x + 5 = 0.
4. Solve the equation 8x + 5 = 0 for x by isolating x.
5. Subtract 5 from both sides to get 8x = -5, then divide both sides by 8 to solve for x.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For any real number x, |x| is always non-negative, meaning it can never be less than zero. This property is crucial for solving equations involving absolute values, as it dictates the possible values that satisfy the equation.

Solving Absolute Value Equations

To solve an equation involving absolute values, such as |A| = B, we consider two cases: A = B and A = -B. This is because the absolute value of a number can be equal to a positive number in two scenarios: the number itself or its negative counterpart. In the case of |A| = 0, the only solution is A = 0, as absolute values cannot be negative.

Zero as a Solution

In the context of equations, zero is a unique solution that indicates the absence of a quantity. When solving equations like |8x + 5| = 0, we find that the expression inside the absolute value must equal zero for the equation to hold true. This leads to a straightforward solution process, as it simplifies to solving a linear equation.