The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. $\mid x^2-6\mid =\mid 5x\mid$

Verified step by step guidance

1. Recognize that the equation |x^2 - 6| = |5x| can be rewritten using the property that if |u| = |v|, then u = v or u = -v. Here, let u = x^2 - 6 and v = 5x.
2. Set up the two separate equations based on the property: 1) x^2 - 6 = 5x and 2) x^2 - 6 = -5x.
3. Solve the first equation x^2 - 6 = 5x by rearranging all terms to one side to form a quadratic equation: x^2 - 5x - 6 = 0.
4. Solve the second equation x^2 - 6 = -5x by rearranging all terms to one side to form another quadratic equation: x^2 + 5x - 6 = 0.
5. Use the quadratic formula or factoring method to find the roots of both quadratic equations, which will give the solutions to the original absolute value equation.

Key Concepts

Absolute Value Definition and Properties

Absolute value represents the distance of a number from zero on the number line, always yielding a non-negative result. For any expressions u and v, the equation |u| = |v| means u = v or u = -v, reflecting that two values can have the same magnitude but opposite signs.

Solving Equations Involving Absolute Values

To solve equations with absolute values, rewrite the equation without absolute value bars by setting the inside expressions equal to each other and to their negatives. This creates two separate equations to solve, which together provide all possible solutions.

Quadratic Equations and Factoring

When solving equations like |x^2 - 6| = |5x|, the resulting equations often involve quadratics. Understanding how to rearrange, factor, or use the quadratic formula is essential to find all real solutions for x.