Chapter 7

A _________ is the quotient of two numbers or the quotient of two quantities with the same units. A ___________ is a quotient of two quantities that have different units.

Fill in the blanks. Equations that contain one or more rational expressions, such as \(\frac{x}{x+2}=4+\frac{10}{x+2},\) are called _____________ equations.

Fill in the blanks. The expression \(\frac{\frac{2}{3}-\frac{1}{x}}{\frac{x-3}{4}}\) is called a ___________ rational expression or a ________________ fraction.

Fill in the blanks. \(\frac{x}{x-7}\) and \(\frac{1}{x-7}\) have like denominators. \(\frac{x+5}{x-7}\) and \(\frac{4 x}{x+7}\) have ____ denominators.

Fill in the blanks. The rational expressions \(\frac{7}{6 n}\) and \(\frac{n+1}{6 n}\) have the common _______ \(6 n\).

Fill in the blanks. The ______ of \(\frac{x^{2}+6 x+1}{10 x}\) is \(\frac{10 x}{x^{2}+6 x+1}\)

Fill in the blanks. A quotient of two polynomials, such as \(\frac{x^{2}+x}{x^{2}-3 x},\) is called a ____ expression.

A __________ is a mathematical statement that two ratios or two rates are equal.

Fill in the blanks. To __________ a rational equation we find all the values of the variable that make the equation true.

Fill in the blanks. In a complex fraction, the numerator is above the __________ fraction bar and the _________ is below it.

Fill in the blanks. The polynomials \(x-3\) and \(3-x\) are ____.

Fill in the blanks. The ________ common denominator of \(\frac{x-8}{x+6}\) and \(\frac{6-5 x}{x}\) is \(x(x+6)\).

Fill in the blanks. To simplify a rational expression, we remove common ____ of the numerator and denominator.

Choose the equation that can be used to solve the following problem: If the same number is added to the numerator and the denominator of the fraction \(\frac{5}{8},\) the result is \(\frac{2}{3} .\) Find the number. (i) \(\frac{5}{8}+x=\frac{2}{3}\) (ii) \(\frac{5+x}{8}=\frac{2}{3}\) (iii) \(\frac{5+x}{8+x}=\frac{2}{3}\) (iv) \(\frac{5}{8}=\frac{2+x}{3+x}\)

Fill in the blanks. To ___________ a rational equation of fractions, multiply both sides by the LCD of all rational expressions in the equation.

Fill in the blanks. Method 1: To simplify a complex fraction, write its numerator and denominator as _________________ rational expressions. Then perform the indicated ____________ by multiplying the numerator of the complex fraction by the _____________ of the denominator.

Write each denominator in factored form. a. \(\frac{x+1}{20 x^{2}}\) b. \(\frac{3 x^{2}-4}{x^{2}+4 x-12}\)

Fill in the blanks. To _________ a rational expression, we multiply it by a form of \(1 .\) For example: \(\frac{2}{n^{2}} \cdot \frac{8 n}{8 n}=\frac{16 n}{8 n^{3}}\).

Fill in the blanks. a. To multiply rational expressions, multiply their ______ and multiply their ______. To divide two rational expressions, multiply the first by ______ of the second. In symbols, b. \(\frac{A}{B} \cdot \frac{C}{D}=\) and \(\quad \frac{A}{B} \div \frac{C}{D}=\frac{A}{B} \cdot\)

Fill in the blanks. Because of the division by \(0,\) the expression \(\frac{8}{0}\) is ____.

The _______ products for the proportion \(\frac{5}{2}=\frac{6}{x}\) are \(5 x\) and 12

Fill in the blank: If a job can be completed in \(t\) hours, then the rate of work can be expressed as \(\frac{1}{\underline{\phantom{xx}}}\) of the job is completed per hour.

Fill in the blanks. When solving a rational equation, if we obtain a number that does not satisfy the original equation, the number is called an ______________ solution.

Fill in the blanks. Method 2: To simplify a complex fraction, find the LCD of ______________ the rational expressions within the complex fraction. Multiply the complex fraction by 1 in the form

The factorizations of the denominators of two rational expressions are given. Find the LCD. a. \(12 a=2 \cdot 2 \cdot 3 \cdot a\) \(18 a^{2}=2 \cdot 3 \cdot 3 \cdot a \cdot a\) b. \(x^{2}-36=(x+6)(x-6)\) \(3 x-18=3(x-6)\)

Fill in the blanks. \(\frac{16 n}{8 n^{3}}\) and \(\frac{2}{n^{2}}\) are _________ expressions. They have the same value for all values of \(n,\) except for \(n=0\)

Simplify each expression. \(\frac{(x+7) \cdot 2 \cdot 5}{5(x+1)(x+7)(x-9)}\)

Fill in the blanks. The binomials \(x-15\) and \(15-x\) are called _____ because their terms are the same, except that they are opposite in sign.

a. It takes a night security officer 45 minutes to check each of the doors in an office building to make sure they are locked. What is the officer's rate of work? b. It takes an elementary school teacher 4 hours to make out the semester report cards. What part of the job does she complete in \(x\) hours?

Is 5 a solution of the given rational equation? $$ \text { a. } \frac{1}{x-1}=1-\frac{3}{x-1} $$ $$ \text { b. } \frac{x}{x-5}=3+\frac{5}{x-5} $$

Fill in the blanks. $$ \text { Consider: } \frac{\frac{x-3}{4}}{\frac{1}{12}-\frac{x}{6}} $$ a. What is the numerator of the complex fraction? Is it a single rational expression? b. What is the denominator of the complex fraction? Is it a single rational expression?

What is the LCD for \(\frac{x-1}{x+6}\) and \(\frac{1}{x+3} ?\)

Fill in the blanks. To add or subtract rational expressions that have the same denominator, add or subtract the _______, and write the sum or difference over the common ________ In symbols, \(\frac{A}{D}+\frac{B}{D}=\) and \(\frac{A}{D}-\frac{B}{D}=\)

Simplify each expression. \(\frac{y \cdot y \cdot y(15-y)}{y(y-15)(y+1)}\)

When we simplify \(\frac{x^{2}+5 x}{4 x+20},\) the result is \(\frac{x}{4} .\) These equivalent expressions have the same value for all real numbers, except \(x=-5 .\) Show that they have the same value for \(x=1\)

Two triangles with the same shape, but not necessarily the same size, are called _______ triangles.

Hospitals. An experienced employee can sterilize an operating room in 3 hours. It takes a new employee 5 hours to sterilize the same room. Select the best estimate below of the time it will take them to sterilize the room if they work together. Less than 3 hours Between 3 and 5 hours More than 5 hours

A student was asked to solve a rational equation. The first step of his solution is as follows: $$ 12 x\left(\frac{5}{x}+\frac{2}{3}\right)=12 x\left(\frac{7}{4 x}\right) $$ a. What equation was he asked to solve? b. What LCD is used to clear the equation of fractions?

Fill in the blanks. $$ \text { Consider the complex fraction: } \frac{\frac{1}{y}-\frac{1}{3}}{\frac{5}{6}+\frac{1}{y}} $$ a. What is the LCD of all the rational expressions in the complex fraction? b. To simplify the complex fraction using method \(2,\) it should be multiplied by what form of \(1 ?\)

The LCD for \(\frac{1}{9 n^{2}}\) and \(\frac{37}{15 n^{3}}\) is \(3 \cdot 3 \cdot 5 \cdot n \cdot n \cdot n=45 n^{3}\) If we want to add these rational expressions, what form of 1 should be used a. to build \(\frac{1}{9 n^{2}} ?\) b. to build \(\frac{37}{15 n^{3}} ?\)

Fill in the blanks. When adding or subtracting rational expressions, always write the result in _______ form by removing any factors common to the numerator and denominator.

a. Write \(3 x+5\) in fractional form. b. What is the reciprocal of \(18 x ?\)

Determine whether each pair of polynomials are opposites. Write yes or no. a. \(y+7\) and \(y-7\) b. \(b-20\) and \(20-b\) c. \(x^{2}+2 x-1\) and \(-x^{2}-2 x-1\)

Fill in the blanks. $$ \frac{\frac{12}{y^{2}}}{\frac{4}{y^{3}}} \text { means } \frac{12}{y^{2}} \quad \frac{4}{y^{3}} $$

a. Solve \(d=r t\) for \(t\) b. Solve \(I=P r t\) for \(P\)

Consider the rational equation \(\frac{x}{x-3}=\frac{1}{x}+\frac{2}{x-3}\) a. What values of \(x\) make a denominator \(0 ?\) b. What values of \(x\) make a rational expression undefined? c. What numbers can't be solutions of the equation?

Fill in the blanks. To build \(\frac{x}{x+2}\) so that it has a denominator of \(5(x+2),\) we multiply it by 1 in the form of _____.

The sum of two rational expressions is \(\frac{4 x+4}{5(x+1)} .\) Factor the numerator and then simplify the result.

Find the product of the rational expression and its reciprocal. \(\frac{3}{x+2} \cdot \frac{x+2}{3}\)

Simplify each expression, if possible. a. \(\frac{x-8}{x-8}\) b. \(\frac{x-8}{8-x}\) c. \(\frac{x+8}{8+x}\) d. \(\frac{x+8}{x}\)