Question

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

Accepted Answer

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 $$-A if$$ $$A \u003c 0. $$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$$ \u003c 0$$, so $$|$$x^2 - 9$$| = $$-(x^2 - 9$$) = $$-x^2 + 9$. The equation becomes:

-x^2$ + 9 = x + 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$. Solve each quadratic equation using factoring, completing the square, or the quadratic formula to find the possible values of x$ for each case. Check each solution against the original case condition (x^2$ - 9 \geq 0$$ for Case 1 and $$x^2 - 9$$ \u003c 0$$ for Case 2) to ensure the solution is valid. Discard any solutions that do not satisfy the respective condition.

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

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Key Concepts

Absolute Value Properties

Quadratic Expressions and Factoring

Solving Equations Involving Absolute Values and Quadratics