Problem 79 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\left(\frac{x+1}{x+7}\right)^{2} \div\left(\frac{x+1}{x+7}\right)^{4}=0$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -1 \).

Step 1: Simplify the Equation

First, subtract the exponents of the similar term. \[ \left( \frac{x+1}{x+7} \right)^{2-4} = 0 \] This simplifies to \[ \left( \frac{x+1}{x+7} \right)^{-2} = 0 \]

Step 2: Reorganize the Equation

Next, when you have a negative exponent, simply swap the position of the numerator and denominator to lose the negative sign. Thus, \[ \left( \frac{x+7}{x+1} \right)^{2} = 0 \]

Step 3: Take the Square Root of Both Sides

Taking the square root of both sides, we have \[ \frac{x+7}{x+1} = 0 \] After cross-multiplying, this becomes \( x+1 = 0 \).

Step 4: Solve for x

Solving this for x yields \( x = -1 \).

Key Concepts

Negative Exponents

Negative exponents can be a little tricky at first, but they are quite simple once you understand the basic rule. When you have an expression like \( a^{-n} \), it means that \( a^n \) is in the denominator. This is a shorthand way of writing \( \frac{1}{a^n} \). So, a negative exponent tells you to take the reciprocal of the base raised to the positive exponent.

For example, if you have \( \left( \frac{x+1}{x+7} \right)^{-2} \), you should first understand that it is the same as \( \frac{1}{\left( \frac{x+1}{x+7} \right)^{2}} \). In simpler terms, the negative exponent has flipped the fraction \( \frac{x+1}{x+7} \) to \( \frac{x+7}{x+1} \), making the exponent positive. This act of flipping the fraction is key when working with negative exponents.

Remember, working with negative exponents is all about reciprocals and making the exponent positive by changing the position of the numerator and denominator.

Cross-Multiplication

Cross-multiplication is a powerful tool when solving equations involving fractions. It involves multiplying the numerator of one fraction by the denominator of the other fraction and vice-versa. This technique helps to get rid of fractions, simplifying the equation.

In the problem \( \frac{x+7}{x+1} = 0 \), cross-multiplication becomes quite straightforward. Since one side of the equation equals zero, cross-multiplying involves multiplying \( (x+1) \) by 0, resulting in 0. Thus, we have:

• The equation \( (x+7) \times 1 = 0 \times (x+1) \)

After simplifying, this leaves us with:

• \( x + 7 = 0 \)

Cross-multiplication is particularly useful when simplifying expressions to solve equations as it allows fractions to be eliminated in a straightforward manner.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation. Once you have an equation like \( x+7 = 0 \), solving it is about isolating \( x \) on one side to figure out its value. This is generally achieved through operations such as addition, subtraction, multiplication, or division.

In our example, you have:

• \( x + 7 = 0 \)

To solve for \( x \), you need to isolate it:

• Subtract 7 from both sides: \( x = -1 \)

Finding the right value of \( x \) means that when you substitute it back into the original equation, both sides equal each other, confirming your solution. Make sure to double-check your work by plugging your solution back into the equation.

Simplifying Expressions

Simplifying an expression is the process of transforming it into a simpler or more comprehensible form without changing its value. This can involve several steps like combining like terms, canceling terms, or using rules of exponents.

In solving rational equations, simplifying the terms is crucial. Initially, the given rational equation:

• \( \left(\frac{x+1}{x+7}\right)^2 \div \left(\frac{x+1}{x+7}\right)^4 = 0 \)

requires simplifying the terms using the rules of exponents. By subtracting the exponents, you have:

• \( \left(\frac{x+1}{x+7}\right)^{2-4} \)

Which simplifies to:

• \( \left(\frac{x+1}{x+7}\right)^{-2} \)

This step is vital because it reduces the complexity of the expression, making it easier to work with. Always remember to look for opportunities to simplify expressions early in the process, as this can make solving the equation much more straightforward.