Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{4(x+2)^{-1}-3}{3(x+2)^{-1}-1} $$

Short answer

\( \frac{4(x+2)^{-1}-3}{3(x+2)^{-1}-1} \)

Step-by-step solution

Substitute the terms

Let’s substitute the term \(y = (x+2)^{-1}\). The given expression now becomes \frac{4y - 3}{3y - 1}\.

Factor numerator and denominator

Examine the numerator and the denominator to see if they can be factored. In this case, they cannot be factored further.

Replace y with the original term

Now, replace \(y\) back with \((x+2)^{-1}\). So the expression \frac{4y - 3}{3y - 1}\ becomes \frac{4(x+2)^{-1} - 3}{3(x+2)^{-1} - 1}\.

Simplify the expression

Observe that no further simplification can be done while keeping the expression in factored form. The result is already simplified as required.

Key concepts

Factored Form

Factored form is a way of expressing a mathematical expression as a product of its factors. This form often makes it easier to solve equations or simplify expressions, especially in algebra where simplification is crucial. To express something in factored form, you break it down into simpler parts that multiply to give the original expression.
In our exercise, we are already given the expression that doesn't need further factoring because both the numerator and the denominator are in their simplest forms when we substitute back the variable. The aim was to perform operations and ensure the expression stays in factored form.
To recognize factored form in any expression:

• Look for common factors.
• Use techniques like factoring by grouping or using formulas like the difference of squares.

Maintaining expressions in factored form can make subsequent mathematical operations more manageable.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. They are a significant part of algebra and calculus because they model relationships between variables. Rational expressions follow the same rules as numerical fractions.
Here are some key points:

• The denominator should never be zero since division by zero is undefined.
• Simplify the expressions by canceling out common factors in the numerator and denominator.

In the given exercise, the rational expression is \frac{4(x+2)^{-1}-3}{3(x+2)^{-1}-1}. By substituting \(y = (x+2)^{-1}\), the expression becomes more approachable:
\frac{4y-3}{3y-1}
This substitution step makes it easier to handle the terms and see if further simplification is possible.
Finally, reversing the substitution retains the algebraic form but in a simplified manner.

Algebraic Substitution

Algebraic substitution is a powerful technique to simplify complex expressions or solve equations. By replacing a complex term with a simpler variable, you make the equation easier to work with.
In our provided exercise, we started with a relatively complicated expression:
\frac{4(x+2)^{-1}-3}{3(x+2)^{-1}-1}
Algebraic substitution was used to set \(y = (x+2)^{-1}\), transforming the expression into:
\frac{4y-3}{3y-1}
This substitution simplifies performing the operations and checking for possible factorizations. After realizing that no further simplification or factorization is possible, we revert the substitution to restore the original variables.
Steps for successful algebraic substitution:

• Identify a complex term in the equation that can be replaced.
• Replace the term with a simpler variable.
• Perform the necessary operations and simplify.

Finally, replace the substitution variable back with the original term to get the final simplified expression.