Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 4x^2 - 23x - 6 | = 0

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. In equations, this means that |A| = B implies A = B or A = -B, which is crucial for solving equations involving absolute values.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding how to manipulate and solve these equations is essential for addressing the given problem.

Inspection Method

The inspection method involves solving equations or inequalities by identifying solutions through observation rather than formal algebraic manipulation. This approach is particularly useful for simpler equations or when the solutions are evident. In the context of the given problem, recognizing that the absolute value equation can yield straightforward solutions by considering the nature of the quadratic expression is key.