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. |3x - 1| = |x + 5|

Verified step by step guidance

1. Recognize that the equation |3x - 1| = |x + 5| can be rewritten using the property that if |u| = |v|, then u = v or u = -v. Here, let u = 3x - 1 and v = x + 5.
2. Set up the two separate equations based on the property: 3x - 1 = x + 5 and 3x - 1 = -(x + 5).
3. Solve the first equation 3x - 1 = x + 5 by isolating x: subtract x from both sides and add 1 to both sides to get 2x = 6, then solve for x.
4. Solve the second equation 3x - 1 = -x - 5 by distributing the negative sign, then combine like terms: 3x - 1 = -x - 5, add x to both sides and add 1 to both sides to get 4x = -4, then solve for x.
5. Check both solutions by substituting them back into the original equation to verify that the absolute values are equal.

Key Concepts

Absolute Value Definition

The absolute value of a number or expression represents its distance from zero on the number line, always yielding a non-negative result. For any expression u, |u| equals u if u is non-negative, and -u if u is negative. This concept is fundamental when solving equations involving absolute values.

Equating Absolute Values

When two absolute values are equal, |u| = |v|, it implies that either u = v or u = -v. This property allows us to rewrite the equation without absolute value bars by considering both positive and negative scenarios, which is essential for finding all possible solutions.

Solving Linear Equations

After rewriting the absolute value equation as two separate linear equations, solving each involves isolating the variable using algebraic operations like addition, subtraction, multiplication, or division. Understanding how to solve linear equations is crucial to find the values of x that satisfy the original equation.