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.) | 3x² + x | = 14

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Recognize that the equation involves an absolute value: $|3x^2 + x| = 14$. The absolute value of an expression equals 14 means the expression inside can be either 14 or -14.

Set up two separate equations to remove the absolute value: $3x^2 + x = 14$ and $3x^2 + x = -14$.

Rewrite each equation in standard quadratic form by moving all terms to one side: For the first, $3x^2 + x - 14 = 0$; for the second, $3x^2 + x + 14 = 0$.

Solve each quadratic equation using an appropriate method such as factoring, completing the square, or the quadratic formula. Remember to identify coefficients $a$, $b$, and $c$ for each equation.

Check the solutions obtained by substituting them back into the original absolute value equation to ensure they satisfy $|3x^2 + x| = 14$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Equations

An absolute value equation involves expressions within absolute value bars, which represent the distance from zero on the number line. To solve |A| = B, where B ≥ 0, we set A = B and A = -B, creating two separate equations to solve. Understanding this principle is essential for breaking down the given equation.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them involves factoring, completing the square, or using the quadratic formula. Since the expression inside the absolute value is quadratic, solving the resulting equations requires knowledge of these methods.

Properties of Absolute Value

Properties of absolute value include that |x| ≥ 0 for all x, and |x| = 0 if and only if x = 0. Also, |xy| = |x||y| and |x/y| = |x|/|y| (y ≠ 0). These properties help simplify and manipulate expressions involving absolute values, aiding in solving equations or inequalities like the one given.

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