Advanced Rational Equations: No Solution Scenario

This page explores a more complex rational equation example:

2x + 5 ------- = 4 x - 1

The solution process begins similarly to the previous example, with domain determination. Here, the domain is defined by x - 1 ≠ 0, or x ≠ 1.

Vocabulary: An equation with no solution is called an inconsistent equation.

After cross-multiplication and simplification, we arrive at:

2x + 5 = 4x - 4

Further simplification leads to:

0 = -9

This result indicates that the equation has no solution, as 0 ≠ -9 is always true.

Highlight: When solving rational equations, it's possible to encounter scenarios where no solution exists. This occurs when the simplification process leads to a false statement.

Example: In this case, the equation 2x + 5 = 4(x - 1) simplifies to 0 = -9, which is never true for any value of x.

This example underscores the importance of careful algebraic manipulation and interpretation of results when working with rational equations. It also demonstrates that not all equations have solutions, a crucial concept in advanced algebra and mathematical problem-solving.