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.3/(x + 4) - 7 = - 4/(x + 4)

Verified step by step guidance

1. Identify the denominators in the equation: \(x + 4\).
2. Determine the value that makes the denominator zero: Set \(x + 4 = 0\) and solve for \(x\).
3. The restriction is \(x \neq -4\) because it would make the denominator zero.
4. Combine the fractions on the left side of the equation: \(\frac{3}{x+4} - \frac{-4}{x+4}\).
5. Solve the resulting equation by isolating \(x\) and keeping the restriction in mind.

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 identify any restrictions on the variable, particularly values that would make the denominator zero, as these values are not permissible in the solution set.

Restrictions on Variables

Restrictions on variables in rational equations arise when the denominator equals zero, leading to undefined expressions. For example, in the equation 3/(x + 4), the restriction is x ≠ -4, since substituting -4 would result in division by zero. Identifying these restrictions is crucial before attempting to solve the equation.

Solving Rational Equations

To solve rational equations, one typically eliminates the denominators by multiplying both sides of the equation by the least common denominator (LCD). After simplifying, the resulting equation can be solved for the variable, ensuring that any solutions found do not violate the previously identified restrictions.