Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 4/x = 5/2x + 3

Verified step by step guidance

1. Identify the denominators in the equation: $x$ and $2x$.
2. Determine the values of the variable that make each denominator zero: $x=0$ and $2x=0$.
3. These values are restrictions: $x\ne 0$.
4. To solve the equation, find a common denominator for all terms, which is $2x$.
5. Multiply each term by the common denominator $2x$ to eliminate the fractions and solve the resulting equation.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. To solve these equations, one must find a common denominator to eliminate the fractions, allowing for easier manipulation and solution of the equation. Understanding how to work with rational expressions is crucial for solving these types of equations.

Restrictions on Variables

Restrictions on variables in rational equations arise when the denominator equals zero, as division by zero is undefined. Identifying these restrictions is essential because they determine the values that the variable cannot take. For example, in the equation 4/x = 5/2x + 3, the variable x cannot be zero, as it would make the denominators undefined.

Solving for Variables

Solving for variables in rational equations involves isolating the variable on one side of the equation after addressing any restrictions. This often requires cross-multiplication or finding a common denominator to simplify the equation. Once the variable is isolated, it can be solved, ensuring that the solution adheres to the previously identified restrictions.