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.) | 4x² - 23x - 6 | = 0

Verified step by step guidance

1. Recognize that the equation involves an absolute value expression set equal to zero: \(|4x^2 - 23x - 6| = 0\).
2. Recall the property of absolute value: \(|A| = 0\) if and only if \(A = 0\). This means we can set the inside of the absolute value equal to zero: \(4x^2 - 23x - 6 = 0\).
3. Solve the quadratic equation \(4x^2 - 23x - 6 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -23\), and \(c = -6\).
4. Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots.
5. Use the values of \(a\), \(b\), and \(c\) in the quadratic formula to find the solutions for \(x\), which will be the solutions to the original absolute value equation.

Key Concepts

Absolute Value and Its Properties

The absolute value of a number represents its distance from zero on the number line, always non-negative. For an expression |A| = 0, the only solution is when A = 0, since absolute value cannot be negative. Understanding this property helps simplify equations involving absolute values.

Solving Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax² + bx + c = 0. Solutions can be found by factoring, completing the square, or using the quadratic formula. Identifying and solving the quadratic inside the absolute value is essential here.

Equation Solving Strategy for Absolute Value Equations

When solving |f(x)| = 0, set the inside function f(x) equal to zero and solve. For inequalities or other absolute value equations, consider splitting into cases based on the definition of absolute value. This method ensures all possible solutions are found.