Use the method described in Exercises 83–86, if applicable, and properties ofabsolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 canbe solved by inspection.) | 3x^2 + x | = 14

Verified step by step guidance

1. Identify the absolute value equation: \(|3x^2 + x| = 14\).
2. Recognize that the absolute value equation \(|A| = B\) implies two separate equations: \(A = B\) and \(A = -B\).
3. Set up the first equation: \(3x^2 + x = 14\).
4. Set up the second equation: \(3x^2 + x = -14\).
5. Solve each quadratic equation separately 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 denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. In equations, this means that |A| = B implies A = B or A = -B, which is crucial for solving equations involving absolute values.

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 inside the absolute value is a quadratic, which will need to be solved after removing the absolute value.

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 easily identifiable. In the context of the problem, the hint suggests that some solutions can be found quickly by recognizing patterns or values that satisfy the equation without extensive calculations.