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.) | x^2 + 5x + 5 | = 1

Verified step by step guidance

1. Step 1: Recognize that the equation involves an absolute value, which means we need to consider two separate cases: one where the expression inside the absolute value is equal to 1, and another where it is equal to -1.
2. Step 2: Set up the first equation by removing the absolute value and setting the expression equal to 1: \( x^2 + 5x + 5 = 1 \).
3. Step 3: Solve the first equation \( x^2 + 5x + 5 = 1 \) by first subtracting 1 from both sides to get \( x^2 + 5x + 4 = 0 \). Then, factor the quadratic equation or use the quadratic formula to find the values of \( x \).
4. Step 4: Set up the second equation by removing the absolute value and setting the expression equal to -1: \( x^2 + 5x + 5 = -1 \).
5. Step 5: Solve the second equation \( x^2 + 5x + 5 = -1 \) by first adding 1 to both sides to get \( x^2 + 5x + 6 = 0 \). Then, factor the quadratic equation or use the quadratic formula to find the values of \( x \).

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, disregarding its sign. 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 property allows us to set up two separate cases to solve for the variable, as the expression inside the absolute value can equal either the positive or negative of the other side.

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. In the context of the given problem, the expression x^2 + 5x + 5 is a quadratic that may need to be solved after considering the absolute value cases.

Inspection Method

The inspection method involves solving equations or inequalities by analyzing them intuitively rather than through formal algebraic manipulation. This approach can be particularly useful for simpler equations or when the solutions are evident. In the context of the given problem, the hint suggests that some solutions can be identified quickly without extensive calculations, making it a valuable strategy for efficiency.