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.) | x² + 5x + 5 | = 1

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|x^2 + 5x + 5| = 1\). The absolute value of an expression equals 1 means the expression inside the absolute value can be either 1 or -1.
2. Set up two separate equations based on the definition of absolute value: 1) \(x^2 + 5x + 5 = 1\) 2) \(x^2 + 5x + 5 = -1\)
3. Solve the first quadratic equation \(x^2 + 5x + 5 = 1\) by moving all terms to one side to set it equal to zero: \(x^2 + 5x + 5 - 1 = 0\) which simplifies to \(x^2 + 5x + 4 = 0\)
4. Solve the second quadratic equation \(x^2 + 5x + 5 = -1\) similarly by moving all terms to one side: \(x^2 + 5x + 5 + 1 = 0\) which simplifies to \(x^2 + 5x + 6 = 0\)
5. Use factoring, completing the square, or the quadratic formula to find the roots of each quadratic equation. The solutions to these equations will be the solutions to the original absolute value equation.

Key Concepts

Absolute Value Equations

An absolute value equation involves the expression |A| = B, where B ≥ 0. It means that the quantity inside the absolute value, A, can be either B or -B. To solve, split the equation into two cases: A = B and A = -B, then solve each separately.

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. Understanding how to solve quadratics is essential when the absolute value expression contains a quadratic.

Properties of Absolute Value Inequalities and Equations

Properties of absolute value help simplify and solve equations or inequalities involving absolute values. For example, |A| = B implies A = B or A = -B, while |A| ≤ B implies -B ≤ A ≤ B. Recognizing these properties allows for correct splitting and solving of the given equation.