Solve each equation. |2x + 3/ 3x - 4 | = 1

Verified step by step guidance

1. Step 1: Recognize that the equation involves an absolute value, which means we need to consider two separate cases: one where the expression inside the absolute value is equal to 1, and another where it is equal to -1.
2. Step 2: Set up the first equation by removing the absolute value and setting the expression equal to 1: \( \frac{2x + 3}{3x - 4} = 1 \).
3. Step 3: Solve the first equation by cross-multiplying to eliminate the fraction: \( 2x + 3 = 3x - 4 \).
4. Step 4: Set up the second equation by removing the absolute value and setting the expression equal to -1: \( \frac{2x + 3}{3x - 4} = -1 \).
5. Step 5: Solve the second equation by cross-multiplying to eliminate the fraction: \( 2x + 3 = -(3x - 4) \).

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 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. In the context of equations, the absolute value can lead to two separate cases that must be solved individually.

Equations with Absolute Values

When solving equations involving absolute values, such as |A| = B, where B is a positive number, we create two separate equations: A = B and A = -B. This approach allows us to find all possible solutions that satisfy the original equation. It is crucial to check each solution in the context of the original equation to ensure they are valid.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In the equation given, |(2x + 3)/(3x - 4)| = 1, understanding how to manipulate and simplify rational expressions is essential. This includes finding common denominators, factoring, and ensuring that the denominator does not equal zero, as this would make the expression undefined.