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 - 14x | = 5

Verified step by step guidance

1. Identify the absolute value equation: \(|3x^2 - 14x| = 5\).
2. Recognize that the absolute value equation \(|A| = B\) implies two separate equations: \(A = B\) and \(A = -B\).
3. Set up the first equation: \(3x^2 - 14x = 5\).
4. Set up the second equation: \(3x^2 - 14x = -5\).
5. Solve each quadratic equation separately to find the 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 factoring, completing the square, or 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 for its roots to find the values of x.

Properties of Inequalities

When solving inequalities, certain properties must be considered, such as the fact that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Additionally, when dealing with absolute values, the resulting equations can lead to multiple cases that must be analyzed separately. Understanding these properties is essential for correctly interpreting and solving the given equation.