Simplifying (x-2)/(x+9) - 4: Step-by-Step Explanation

Let's examine a complex rational expression and understand its components.Here's our expression: the fraction x minus 2 over x plus 9, minus 4.First, let's identify the main components. We have a rational expression - a fraction - followed by subtraction of a whole number.The rational part consists of a numerator x minus 2, and a denominator x plus 9.This is followed by a subtraction operation and the whole number 4.An important consideration when working with rational expressions is the domain restriction.Since we can't divide by zero, x cannot equal negative 9, as this would make the denominator zero.To summarize the structure of our expression: we have a rational expression in fraction form, a subtraction operation, and a whole number term.Now that we understand the components of our expression, we're ready to begin converting it to a single fraction.To convert this mixed expression into a single fraction, we need to focus on the whole number 4.When subtracting a whole number from a fraction, we first need to convert the whole number into a fraction with the same denominator.We can do this by multiplying 4 by a special form of 1, which is x plus 9 over x plus 9.Let's convert 4 into an equivalent fraction. First, we multiply 4 by our fraction.Then we can distribute 4 to both terms inside the parentheses.Now we have our original fraction minus the new equivalent fraction, both with the same denominator of x plus 9.Remember that our domain restriction of x not equal to negative 9 still applies, as we can't divide by zero.Now that we have a common denominator, we're ready to subtract these fractions.Now that we have both terms with the same denominator, we can subtract the numerators.We'll place the subtraction inside brackets to keep our work organized.Next, we need to distribute 4 to each term inside the parentheses x plus 9.When we multiply 4 times x, we get 4x. And when we multiply 4 times 9, we get 36.Now we can write our expression with all terms in the numerator, keeping our denominator the same.This sets us up for our next step where we'll combine like terms in the numerator.Now that we have our expression with a common denominator, let's simplify the numerator by combining like terms.First, let's identify the terms with x. We have x and negative four x.When we combine these terms, x plus negative four x equals negative three x.Next, let's look at our constant terms. We have negative two and negative thirty-six.Adding these negative numbers gives us negative thirty-eight.Putting these terms together in our fraction, negative three x minus thirty-eight over x plus nine is our simplified numerator.This expression cannot be simplified further in the numerator, as we have completely combined all like terms.Our final simplified expression is negative three x minus thirty-eight, all over x plus nine.Remember our critical domain restriction: x cannot equal negative nine.Let's verify our answer by testing with x equals one in both the original and simplified expressions.In the original expression, when we substitute x equals one...Simplifying the numerator and denominator...Converting to decimals and subtracting four...Now let's check our simplified expression with x equals one...Simplifying the numerator...And reducing to a decimal...Notice how both expressions give us the same result: negative four point one.The domain restriction x cannot equal negative nine is crucial because it prevents division by zero in both the original and simplified expressions.This verification confirms that our simplification is correct, and both expressions are equivalent for all values in the domain.