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. |4x - 3| = |4x - 5|

Verified step by step guidance

1. Recognize that the equation |4x - 3| = |4x - 5| can be rewritten using the property that if |u| = |v|, then u = v or u = -v. Here, let u = 4x - 3 and v = 4x - 5.
2. Set up the two separate equations based on the property: 4x - 3 = 4x - 5 and 4x - 3 = -(4x - 5).
3. Solve the first equation: 4x - 3 = 4x - 5. Subtract 4x from both sides to isolate constants and check for solutions.
4. Solve the second equation: 4x - 3 = -4x + 5. Combine like terms by adding 4x to both sides and adding 3 to both sides to isolate x.
5. Check the solutions obtained from both equations to ensure they satisfy the original absolute value equation.

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 dealing with equations involving absolute values.

Equations Involving Two Absolute Values

When an equation has two absolute value expressions set equal, such as |u| = |v|, it implies that either u = v or u = -v. This property allows us to rewrite the equation without absolute value bars and solve two separate linear equations, covering all possible solutions.

Solving Linear Equations

After rewriting the absolute value equation as two linear equations, solving each involves isolating the variable using algebraic operations like addition, subtraction, multiplication, or division. Understanding how to manipulate and solve linear equations is essential to find all valid solutions.