Solve the Rational Equation for x (x+5)/5=6/(x-2)
The given problem is an algebraic equation involving rational expressions, which are fractions that have polynomials in their numerators and/or denominators. You are asked to find the value(s) of the variable x that make the equation true, essentially finding the common value(s) that satisfy both sides of the equation once simplified. This process will likely involve finding a common denominator, multiplying both sides to clear the fractions, and then solving for x like a typical algebraic equation.
Cross-multiply the two fractions to eliminate the denominators. This involves multiplying the numerator of the first fraction by the denominator of the second and vice versa. Write the equation as
Begin solving for
First, expand the product
Use the FOIL Method to expand the binomials.
Distribute each term in the first binomial across the second binomial:
Distribute
Continue distribution:
Combine like terms in the equation.
Simplify each term in the equation.
Square
Combine the
Multiply
Add the
Calculate the product of
Subtract
Factor the quadratic equation
Use the AC method to factor the quadratic.
Find two numbers that multiply to
Write the equation in factored form:
Apply the zero-product property, which states that if a product equals zero, then at least one of the factors must be zero.
Solve for
Set the first factor equal to zero:
Solve for
Solve for
Set the second factor equal to zero:
Solve for
Combine the solutions to find all values of
To solve a rational equation where the variables are in the numerator and the denominator, we often use cross-multiplication to eliminate the fractions. This is possible when we have an equation of the form
The FOIL Method stands for First, Outer, Inner, Last, and is used to expand the product of two binomials. The distributive property is applied to multiply each term in the first binomial by each term in the second.
Combining like terms is a process of adding or subtracting terms that have the same variable raised to the same power.
The zero-product property is used to solve quadratic equations that have been factored into the form
Factoring is a method used to solve quadratic equations by expressing them as a product of two binomials. The AC method is one of the techniques used for factoring trinomials of the form
Finally, the solutions to the equation are the values of