Algebraic Fractions Practice Problems & Solutions

Master adding and subtracting algebraic fractions with step-by-step practice problems. Learn factorization techniques and find common denominators easily.

๐Ÿ“šWhat You'll Master in This Practice Session
  • Factorize denominators using difference of squares and perfect square trinomials
  • Find common denominators for complex algebraic fractions efficiently
  • Add and subtract algebraic fractions with unlike denominators step-by-step
  • Simplify resulting expressions by identifying common factors
  • Apply the 6-step method for solving algebraic fraction problems
  • Solve real algebraic fraction problems similar to xยฒ-9 and xยฒ-6x+9 examples

Understanding Addition and Subtraction of Algebraic Fractions

Complete explanation with examples

The key to adding or subtracting algebraic fractions is to make all denominators equal, that is, to find the common denominator.
To do this, we will need to factorize according to the different methods we have learned.

Action steps:

  1. We will factorize all the denominators we have.
  2. We will note the common denominator and, in this way, we will know how to meticulously carry out the third step.
  3. We will multiply each of the numerators by the same number that we need to multiply its denominator in order to reach the common denominator.
  4. We will write the exercise with a single denominator, the common denominator, and among the numerators, we will keep the same mathematical operations that were in the original exercise.
  5. After opening the parentheses, it may happen that we encounter another expression that needs to be factorized. We will factorize it and see if we can simplify it.
  6. We will obtain a common fraction and solve it.
Detailed explanation

Practice Addition and Subtraction of Algebraic Fractions

Test your knowledge with 19 quizzes

Identify the field of application of the following fraction:

\( \frac{7}{13+x} \)

Examples with solutions for Addition and Subtraction of Algebraic Fractions

Step-by-step solutions included
Exercise #1

Determine if the simplification below is correct:

3โ‹…77โ‹…3=0 \frac{3\cdot7}{7\cdot3}=0

Step-by-Step Solution

We will divide the fraction exercise into two different multiplication exercises.

As this is a multiplication exercise, you can use the substitution property:

77ร—33=1ร—1=1 \frac{7}{7}\times\frac{3}{3}=1\times1=1

Therefore, the simplification described is false.

Answer:

Incorrect

Video Solution
Exercise #2

Determine if the simplification below is correct:

4โ‹…84=18 \frac{4\cdot8}{4}=\frac{1}{8}

Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

44ร—81= \frac{4}{4}\times\frac{8}{1}=

We simplify:

1ร—81=8 1\times\frac{8}{1}=8

Therefore, the described simplification is false.

Answer:

Incorrect

Video Solution
Exercise #3

Determine if the simplification shown below is correct:

77โ‹…8=8 \frac{7}{7\cdot8}=8

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

77ร—18 \frac{7}{7}\times\frac{1}{8}

We simplify:

1ร—18=18 1\times\frac{1}{8}=\frac{1}{8}

Therefore, the described simplification is false.

Answer:

Incorrect

Video Solution
Exercise #4

Determine if the simplification below is correct:

5โ‹…88โ‹…3=53 \frac{5\cdot8}{8\cdot3}=\frac{5}{3}

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

88ร—53 \frac{8}{8}\times\frac{5}{3}

We simplify:

1ร—53=53 1\times\frac{5}{3}=\frac{5}{3}

Answer:

Correct

Video Solution
Exercise #5

Determine if the simplification below is correct:

6โ‹…36โ‹…3=1 \frac{6\cdot3}{6\cdot3}=1

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality:

6ฬธโ‹…3ฬธ6ฬธโ‹…3ฬธ=?1โ†“1=!1 \frac{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }1\\ \downarrow\\ 1\stackrel{!}{= }1 therefore, the described simplification is correct.

Therefore, the correct answer is A.

Answer:

Correct

Video Solution

Frequently Asked Questions

Everything you need to know about Addition and Subtraction of Algebraic Fractions

What is the first step when adding algebraic fractions with different denominators?

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The first step is to factorize all denominators completely using methods like difference of squares, perfect square trinomials, or common factoring. This reveals the structure needed to find the least common denominator.

How do you find the common denominator for algebraic fractions?

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After factorizing all denominators, identify all unique factors and take the highest power of each factor that appears. For example, if you have (x-3)(x+3) and (x-3)ยฒ, the common denominator is (x+3)(x-3)ยฒ.

What are the 6 steps for adding and subtracting algebraic fractions?

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1) Factorize all denominators, 2) Find the common denominator, 3) Multiply each numerator by the factor needed to reach the common denominator, 4) Write with single denominator keeping original operations, 5) Simplify and factorize if needed, 6) Obtain the final simplified fraction.

Why do we need to factorize denominators in algebraic fractions?

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Factorizing denominators reveals common factors between fractions, making it easier to find the least common denominator. Without factorization, you might miss opportunities to simplify and could create unnecessarily complex denominators.

How do you multiply numerators when finding common denominators?

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Multiply each numerator by exactly the same factors you multiply its denominator by to reach the common denominator. If the denominator needs to be multiplied by (x+3), then multiply the numerator by (x+3) as well.

What common mistakes should I avoid with algebraic fractions?

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Common mistakes include: not factorizing denominators completely, forgetting to multiply numerators by the same factor as denominators, making sign errors when distributing, and not simplifying the final answer by canceling common factors.

When can you simplify algebraic fractions after adding or subtracting?

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You can simplify when the numerator and denominator share common factors after combining terms. Always check if the final numerator can be factored and if any factors match those in the denominator for cancellation.

How do you handle negative signs in algebraic fraction operations?

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Be extra careful with negative signs when multiplying numerators and when combining terms in the final numerator. Use parentheses to group terms clearly and distribute negative signs properly to avoid sign errors.

More Addition and Subtraction of Algebraic Fractions Questions

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Factorization and Algebraic Fractions

Solve Complex Rational Equation: ((2x-1)ยฒ)/(x-2) + ((x-2)ยฒ)/(2x-1) = 4.5x Calculate A and B in the Equation: Solving \((\sqrt{x}-\sqrt{x+1})/(x+1)=1\) Solve the Equation: 1/(x-2)ยฒ + 1/(x-2) = 1 Step-by-Step Solve the Rational Equation: (xยณ+1)/(x-1)ยฒ = x+4 Identify the Application Context of the Fraction x/16: Mathematical Analysis

Continue Your Math Journey

Master Addition and Subtraction of Algebraic Fractions first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Factoring using contracted multiplicationFactorizationExtracting the common factor in parenthesesFactorization: Common factor extractionFactoring Trinomials

Topics Learned in Later Sections

Algebraic FractionsSimplifying Algebraic FractionsFactoring Algebraic FractionsMultiplication and Division of Algebraic FractionsSolving Equations by FactoringUses of Factorization