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 - 9 | = x + 3

Verified step by step guidance

1. Start by recognizing that the equation involves an absolute value: \(|x^2 - 9| = x + 3\). Recall that the absolute value \(|A|\) equals \(A\) if \(A \geq 0\), and equals \(-A\) if \(A < 0\).
2. Set up two separate cases based on the definition of absolute value: Case 1: \(x^2 - 9 \geq 0\), so \(|x^2 - 9| = x^2 - 9\). The equation becomes: \(x^2 - 9 = x + 3\). Case 2: \(x^2 - 9 < 0\), so \(|x^2 - 9| = -(x^2 - 9) = -x^2 + 9\). The equation becomes: \(-x^2 + 9 = x + 3\).
3. Solve each case separately: For Case 1, rearrange the equation to standard quadratic form: \(x^2 - 9 = x + 3 \implies x^2 - x - 12 = 0\). For Case 2, rearrange similarly: \(-x^2 + 9 = x + 3 \implies -x^2 - x + 6 = 0\) or equivalently \(x^2 + x - 6 = 0\) after multiplying both sides by \(-1\).
4. Solve each quadratic equation using factoring, completing the square, or the quadratic formula to find the possible values of \(x\) for each case.
5. Check each solution against the original case condition (\(x^2 - 9 \geq 0\) for Case 1 and \(x^2 - 9 < 0\) for Case 2) to ensure the solution is valid. Discard any solutions that do not satisfy the respective condition.

Key Concepts

Absolute Value Properties

Absolute value represents the distance of a number from zero on the number line, always non-negative. For an equation |A| = B, if B ≥ 0, then A = B or A = -B. Understanding how to split absolute value equations into two cases is essential for solving equations like |x² - 9| = x + 3.

Quadratic Expressions and Factoring

The expression inside the absolute value, x² - 9, is a quadratic that can be factored as (x - 3)(x + 3). Factoring helps identify critical points and simplifies solving the equation by breaking it into manageable parts, especially when combined with absolute value properties.

Solving Equations Involving Absolute Values and Quadratics

To solve |x² - 9| = x + 3, consider the domain where the right side is non-negative, then split into cases: x² - 9 = x + 3 and -(x² - 9) = x + 3. Solve each quadratic equation separately and check for extraneous solutions, ensuring the solutions satisfy the original absolute value equation.