In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|4 - (5/2)x| + 6 = 18

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|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations that involve expressions within these bars, as it leads to two possible cases based on the definition.

Linear Equations

Linear equations are mathematical statements that establish equality between two expressions, typically in the form of ax + b = c, where a, b, and c are constants. These equations represent straight lines when graphed. Solving linear equations often involves isolating the variable, which is essential when working with absolute value equations that can be transformed into linear forms.

Case Analysis

Case analysis is a method used to solve equations involving absolute values by considering different scenarios. For an equation like |x| = a, where a is non-negative, we create two cases: x = a and x = -a. This approach is necessary for absolute value equations, as it allows us to find all possible solutions by addressing both the positive and negative scenarios of the expression within the absolute value.