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. 5/2x - 8/9 = 1/18 - 1/3x

Verified step by step guidance

1. Identify the denominators in the equation: \(2x\), \(9\), \(18\), and \(3x\).
2. Determine the values of \(x\) that make any denominator zero. For \(2x\) and \(3x\), set each equal to zero: \(2x = 0\) and \(3x = 0\), which gives \(x = 0\). Thus, \(x = 0\) is a restriction.
3. To solve the equation, find a common denominator for all terms. The least common multiple of \(2x\), \(9\), \(18\), and \(3x\) is \(18x\).
4. Multiply every term in the equation by \(18x\) to eliminate the denominators: \(18x \cdot \frac{5}{2x} - 18x \cdot \frac{8}{9} = 18x \cdot \frac{1}{18} - 18x \cdot \frac{1}{3x}\).
5. Simplify each term: \(45 - 16x = x - 6\). Now, solve for \(x\) while keeping in mind the restriction \(x \neq 0\).

Key Concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. To solve these equations, it is essential to find a common denominator and eliminate the fractions, which simplifies the equation. Understanding how to manipulate these fractions is crucial for finding solutions.

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 critical because they determine the values that the variable cannot take, ensuring that the solutions to the equation are valid and do not lead to undefined expressions.

Solving for Variables

Solving for variables in rational equations involves isolating the variable on one side of the equation. This process often includes combining like terms, applying inverse operations, and checking for extraneous solutions that may arise from the manipulation of the equation. A thorough understanding of algebraic principles is necessary to effectively solve these equations.