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 - 9 | = x + 3

Verified step by step guidance

1. Start by considering the definition of absolute value: $|a|=b$ implies $a=b$ or $a=-b$.
2. Apply this definition to the equation $|x^2-9|=x+3$, resulting in two separate equations: $x^2-9=x+3$ and $x^2-9=-(x+3)$.
3. Solve the first equation $x^2-9=x+3$ by rearranging it to form a quadratic equation: $x^2-x-12=0$.
4. Solve the second equation $x^2-9=-(x+3)$ by simplifying it to $x^2+x-6=0$.
5. Solve both quadratic equations using factoring, the quadratic formula, or completing the square to find the possible values of $x$.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. This concept is crucial for solving equations involving absolute values, as it leads to the creation of two separate cases to consider.

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 the quadratic formula. In the context of the given problem, recognizing that |x^2 - 9| can lead to a quadratic equation is essential for finding the values of x.

Inequalities and Their Properties

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like <, >, ≤, or ≥. When solving inequalities involving absolute values, it is important to consider the implications of the absolute value definition, which can lead to multiple cases. Understanding how to manipulate and solve inequalities is key to finding the solution set for the given equation.