Algebra Chapter 6 Flashcards

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Questions and Answers

What is the result of multiplying rational expressions A/B and C/D?

  • AD/BC
  • A-B/D
  • AC/BD (correct)
  • A+B/D

What is the result of dividing rational expressions A/B by C/D?

  • A/B * D/C (correct)
  • C/D / A/B
  • AD/BC (correct)
  • A/B / C/D

What is the definition of adding rational expressions?

A/D + B/D = (A+B)/D

What is the definition of subtracting rational expressions?

<p>A/D - B/D = (A-B)/D</p> Signup and view all the answers

What does LCD stand for?

<p>Least common denominator</p> Signup and view all the answers

What are the steps for adding and subtracting rational expressions?

<p>Find the LCD, write each expression with the LCD, add/subtract numerators, simplify.</p> Signup and view all the answers

What does it mean if two terms are opposites?

<p>The terms are the same but differ in sign.</p> Signup and view all the answers

What is a complex rational expression?

<p>A rational expression whose numerator and/or denominator contain rational expressions.</p> Signup and view all the answers

What defines a complex fraction?

<p>A fraction whose numerator or denominator or both contain a fraction.</p> Signup and view all the answers

How do you simplify complex fractions using division?

<p>Write numerator/denominator as single rational expressions, multiply by reciprocal, simplify.</p> Signup and view all the answers

How do you simplify complex fractions using the LCD?

<p>Find the LCD, multiply by 1 (LCD/LCD), perform operations, simplify.</p> Signup and view all the answers

What is a formula?

<p>An equation that states a known relationship between two or more variables.</p> Signup and view all the answers

What is an extraneous solution?

<p>A possible solution that does not satisfy the original equation.</p> Signup and view all the answers

What is a rational equation?

<p>An equation that contains one or more rational expressions.</p> Signup and view all the answers

What are the steps for solving rational equations?

<p>Determine invalid solutions, multiply by LCD, solve, check solutions.</p> Signup and view all the answers

What is the rate of work?

<p>1/x of the job is completed per hour.</p> Signup and view all the answers

What is the formula for distance?

<p>Distance = rate * time (r * t).</p> Signup and view all the answers

What is a ratio?

<p>The quotient of two numbers or quantities with the same units.</p> Signup and view all the answers

What is the rate?

<p>The quotient of two quantities with different units.</p> Signup and view all the answers

What is a proportion?

<p>A mathematical statement that two ratios or two rates are equal.</p> Signup and view all the answers

What is the fundamental property of proportions?

<p>If a/b = c/d, then ad = bc (and vice versa).</p> Signup and view all the answers

What is a term in a proportion?

<p>What each of the four numbers in a proportion are called.</p> Signup and view all the answers

What are extremes in a proportion?

<p>The first and fourth terms.</p> Signup and view all the answers

What are means in a proportion?

<p>The second and third terms.</p> Signup and view all the answers

What characterizes similar triangles?

<p>Triangles with corresponding sides in proportion.</p> Signup and view all the answers

What is the cross product in proportions?

<p>ab and cd when diagonally multiplying proportions.</p> Signup and view all the answers

What is the formula for interest?

<p>I = prt</p> Signup and view all the answers

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Study Notes

Multiplying and Dividing Rational Expressions

  • Multiplying: For rational expressions A/B and C/D, the product is AC/BD.
  • Dividing: For A/B divided by C/D, use A/B * D/C to get AD/BC.

Adding and Subtracting Rational Expressions

  • Addition: To sum A/D and B/D, combine the numerators: (A+B)/D.
  • Subtraction: For A/D minus B/D, the result is (A-B)/D.

Least Common Denominator (LCD)

  • The least common denominator is the smallest multiple shared by the denominators of the rational expressions.

Steps for Adding and Subtracting Rational Expressions

  • Find the LCD.
  • Rewrite each expression with the LCD as the new denominator.
  • Combine numerators and place over the LCD.
  • Simplify the final expression if possible.

Opposite Terms

  • Opposite terms share the same value but differ in sign.

Complex Rational Expressions

  • These involve rational expressions in either the numerator, denominator, or both.

Complex Fractions

  • Defined as fractions that include a fraction in their numerator or denominator.

Simplifying Complex Fractions via Division

  • Formulate the numerator and denominator as single rational expressions.
  • Perform multiplication by taking the reciprocal of the denominator.
  • Simplify if applicable.

Simplifying Complex Fractions Using LCD

  • Identify the LCD of all components in the complex fraction.
  • Multiply the fraction by 1, expressed as LCD/LCD.
  • Execute operations in the numerator and denominator ensuring no fractions remain.
  • Simplify the resultant expression.

Formulas

  • Equations that describe established relationships among multiple variables.

Extraneous Solutions

  • Solutions that arise from the process of solving but do not satisfy the original equation.

Rational Equations

  • Equations containing one or more rational expressions.

Solving Rational Equations

  • Identify values that cannot be solutions.
  • Multiply by the LCD to eliminate fractions.
  • Solve the simplified equation.
  • Confirm all solutions by substituting back into the original equation.

Rates and Ratios

  • Rate of Work: Complete a job in x hours implies a completion rate of 1/x per hour.
  • Distance: Computed by the formula distance = rate × time.
  • Ratio: The quotient of two same-unit quantities.
  • Rate: The quotient of two quantities with different units.

Proportions

  • A statement that two ratios or rates are equivalent.
  • Fundamental Property: If a/b = c/d, then ad = bc.
  • Terms in a Proportion: Four numbers; the first and fourth are termed extremes, the second and third are means.

Similar Triangles

  • Defined by corresponding sides that are proportional.

Cross Product

  • In a proportion, the cross product involves multiplying the extremes and means: ab and cd.

Interest Calculation

  • Interest calculated using the formula I = prt (Interest = principal × rate × time).

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