Find b such that (4x - b)/(x - 5) = 3 has a solution set given by {Ø}.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding how to manipulate and simplify these expressions is crucial for solving equations involving them. In this case, the expression (4x - b)/(x - 5) must be analyzed to determine conditions under which it can equal 3.

Solution Set

The solution set of an equation is the set of all values that satisfy the equation. In this context, the notation {Ø} indicates that there are no solutions. This implies that the equation must be structured in such a way that it leads to a contradiction or an undefined expression, which is essential for determining the value of b.

Undefined Expressions

An expression is undefined when it involves division by zero. In the equation (4x - b)/(x - 5) = 3, the denominator (x - 5) becomes zero when x equals 5. To ensure the solution set is empty, we need to find a value of b that makes the numerator also zero at this point, leading to an undefined expression.