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Chapter 6: Problem 100

Write four equivalent forms for each rational expression. $$ -\frac{2 x-9}{7 x-1} $$

Short Answer

Expert verified
Equivalent forms include \( \frac{2x-9}{-(7x-1)}\), \(\frac{-(2x-9)}{7x-1}\), \( -\frac{(2x-9+k) - k}{7x-1} \), and \(-\frac{2(2x-9)}{2(7x-1)}\).

Step by step solution

01

Identify the given rational expression

The given rational expression is \(-\frac{2x-9}{7x-1}\).
02

Multiply numerator and denominator by -1

To create an equivalent form, multiply both the numerator and the denominator by -1: \[ -\frac{2x-9}{7x-1} \times \frac{-1}{-1} = \frac{-(2x-9)\times(-1)}{(7x-1)\times(-1)} = \frac{2x-9}{-(7x-1)} \]
03

Factor out common integers

To simplify the expression, one can factor out common integers from the numerator: \[ -\frac{2x-9}{7x-1} = -\frac{1}{1} \times \frac{2x-9}{7x-1} = \frac{-(2x-9)}{7x-1} \]
04

Add and subtract the same term in the numerator

By adding and subtracting the same term in the numerator, we can form an equivalent expression: \[ -\frac{2x-9 + k - k}{7x-1} = -\frac{(2x-9+k) - k}{7x-1} \]
05

Combine constants in the denominator

Another form can be achieved by distributing elements in the denominator: \[ -\frac{2x-9}{7x-1} \times \frac{2}{2} = -\frac{2(2x-9)}{2(7x-1)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equivalent Forms
When working with rational expressions, it's essential to understand that multiple forms can express the same value. These are known as equivalent forms. An equivalent form of a rational expression is another expression that represents the same value for all permissible values of the variable. For example, different algebraic manipulations such as multiplying or dividing by the same non-zero quantity or adding and subtracting the same values can yield equivalent forms.
Factoring
Factoring is the process of breaking down an expression into simpler multiplicative components. For rational expressions, finding common factors in the numerator and the denominator can often help simplify the expression. For instance, if both parts of the expression share a common factor, you can divide that factor out. However, it's crucial to remember that any factoring performed must maintain the equality of the original expression.
Simplification
Simplifying rational expressions makes it easier to work with them. Simplification involves reducing the expression to its lowest terms by canceling out common factors in the numerator and the denominator. For example, in the given exercise, one way to simplify is to multiply both parts by -1, which simplifies the expression without changing its value. Simplifying can also involve factoring out integers or combining like terms to refine the expression further.
Adding and Subtracting Terms
Adding and subtracting terms in rational expressions can provide alternate equivalent forms. This is done by adding and then immediately subtracting the same term within the numerator or denominator. This technique doesn't change the overall value of the expression but can make it easier to understand or work with. For example, if we have a rational expression where the numerator is \(2x-9\), adding and subtracting a term k will still keep the expression equivalent.

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Most popular questions from this chapter

Which complex fraction is equivalent to \(\frac{2-\frac{1}{4}}{3-\frac{1}{2}} ?\) Answer this question without showing any work, and explain your reasoning. A. \(\frac{2+\frac{1}{4}}{3+\frac{1}{2}}\) B. \(\frac{2-\frac{1}{4}}{-3+\frac{1}{2}}\) C. \(\frac{-2-\frac{1}{4}}{-3-\frac{1}{2}}\) D. \(\frac{-2+\frac{1}{4}}{-3+\frac{1}{2}}\)

Divide. Write each answer in lowest terms. $$ \frac{2 x}{x-1} \div \frac{x^{2}}{x+2} $$

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{1+p^{3}}{1+p} $$

Only one of these choices is equal to \(\frac{\frac{1}{3}+\frac{1}{12}}{\frac{1}{2}+\frac{1}{4}} .\) Which one is it? Answer this question without showing any work, and explain your reasoning. A. \(\frac{5}{9}\) B. \(-\frac{5}{9}\) C. \(-\frac{9}{5}\) D. \(-\frac{1}{12}\)

Divide. Write each answer in lowest terms. $$ \frac{7 t+7}{-6} \div \frac{4 t+4}{15} $$
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