In Exercises 67–70, find all values of x such that y = 0. y = 1/(5x + 5) - 3/(x + 1) + 7/5

Verified step by step guidance

1. Set the equation equal to zero: $\frac{1}{5x+5}-\frac{3}{x+1}+\frac{7}{5}=0$.
2. Find a common denominator for the fractions. The common denominator is $(5x+5)(x+1)$.
3. Rewrite each term with the common denominator: $\frac{x+1}{(5x+5)(x+1)}-\frac{3(5x+5)}{(5x+5)(x+1)}+\frac{7(x+1)}{5(5x+5)(x+1)}=0$.
4. Combine the fractions into a single fraction: $\frac{(x+1)-3(5x+5)+7(x+1)}{(5x+5)(x+1)}=0$.
5. Set the numerator equal to zero and solve for $x$: $(x+1)-3(5x+5)+7(x+1)=0$.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function involves terms like 1/(5x + 5) and -3/(x + 1), which are rational expressions. Understanding how to manipulate and simplify these expressions is crucial for solving equations involving them.

Finding Roots of an Equation

Finding the values of x such that y = 0 involves solving the equation for its roots. This means determining the x-values where the function intersects the x-axis. This process often requires setting the rational function equal to zero and solving for x, which may involve finding a common denominator.

Domain Restrictions

When dealing with rational functions, it is important to consider the domain, which includes all possible x-values for which the function is defined. In this case, values that make the denominators zero (5x + 5 = 0 and x + 1 = 0) must be excluded from the solution set, as they lead to undefined expressions.