Add the following and simplify the sum if possible. x 2 − 4 x 2 − x − 6 + x + 2 x 2 − x − 6 \frac{x^2-4}{x^2-x-6}+\frac{x+2}{x^2-x-6}

x − 1 x − 3 \frac{x-1}{x-3}

How do you add rational expressions with common denominators?

Adding rational expressions with common denominators is similar to adding fractions. First, ensure the denominators are the same. Then, combine the numerators by adding them together while keeping the denominator unchanged. For example, if you have 5 18x + 1 18x , you add the numerators: 5 + 1 = 6, so the expression becomes 6 18x . The final step is to simplify the expression by factoring and canceling common factors. Here, 6 and 18 share factors 3 and 2, so after canceling, the simplified form is 1 3 x . This process ensures the expression is in its simplest form.

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Common Denominators

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Common Denominators: Videos & Practice Problems

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Topic summary

Adding or subtracting rational expressions mirrors combining rational numbers by using a common denominator and combining numerators. Simplification involves factoring polynomials, such as trinomials, and canceling common factors. For example, subtracting x2x-1 - -x+2x-1 requires distributing the negative and factoring the numerator to x-1⋅x+2x-1, then canceling x-1. This process emphasizes understanding polynomial degree, terms, and coefficients for effective simplification.

concept

Adding and Subtracting Rational Expressions with Common Denominators

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Adding and Subtracting Rational Expressions with Common Denominators Video Summary

Adding and subtracting rational expressions follows the same fundamental principles as working with rational numbers, with the added step of managing variables and simplifying expressions. When two rational expressions share the same denominator, you combine their numerators directly while keeping the denominator unchanged. For example, adding $\frac{5}{18}$ and $\frac{1}{18}$ results in $\frac{5 + 1}{18} = \frac{6}{18}$. Simplifying this fraction involves factoring both numerator and denominator into their prime factors: $6 = 3 \times 2$ and $18 = 3 \times 3 \times 2$. Canceling common factors leaves $\frac{1}{3}$ as the simplified result.

This approach extends naturally to rational expressions with variables. For instance, adding $\frac{5}{18x}$ and $\frac{1}{18x}$ yields $\frac{5 + 1}{18x} = \frac{6}{18x}$. Simplification again involves factoring the numeric part and canceling common factors, while keeping the variable terms intact. Factoring \$6$ and $18$ as before and canceling common factors leaves \(\frac{1}{3x}$ as the simplified expression.

When subtracting rational expressions with a common denominator, it is crucial to distribute the negative sign across all terms in the numerator of the second fraction before combining. For example, subtracting $\frac{-x + 2}{x - 1}$ from $\frac{x^2}{x - 1}$ involves rewriting the expression as:

\[\frac{x^2}{x - 1} - \frac{-x + 2}{x - 1} = \frac{x^2 - (-x + 2)}{x - 1} = \frac{x^2 + x - 2}{x - 1}\]

Next, factor the numerator, which is a quadratic expression. To factor \)x^2 + x - 2$, find two numbers that multiply to $-2$ and add to $1$. These numbers are $2$ and $-1\(, so the factorization is:

\[x^2 + x - 2 = (x - 1)(x + 2)\]

Substituting back, the expression becomes:

\[\frac{(x - 1)(x + 2)}{x - 1}\]

Since \)x - 1$ appears in both numerator and denominator, it cancels out, leaving the simplified result:

\[x + 2\]

This process highlights the importance of factoring and simplifying after combining rational expressions. Whether adding or subtracting, always identify the common denominator, combine the numerators carefully—distributing negatives when necessary—and then factor and reduce the resulting expression to its simplest form. Mastery of these steps ensures efficient handling of rational expressions in algebra.

Problem

Add the following and simplify the sum if possible.
$\frac{2}{5} + \frac{3}{5}$

$2$

$\frac{1}{2}$

$\frac{3}{5}$

$1$

Problem

Add the following and simplify the sum if possible.
$\frac{x}{7} + \frac{4 x}{7}$

$4 x over 7$

$5 x over 7$

$x over 7$

$5 x$

Problem

Subtract the following and simplify the difference if possible.
$\frac{5}{8} - \frac{3}{8}$

$\frac{1}{8}$

$1$

$2$

$\frac{1}{4}$

Problem

Subtract the following and simplify the difference if possible.
$\frac{4 y}{3} - \frac{- 5 y}{3}$

$3$

$3 y$

$9$

$9 y$

Problem

Add the following and simplify the sum if possible.
$\frac{x + 1}{3 x} + \frac{2 x - 1}{3 x}$

$1$

$negative 1$

$x$

$- x$

Problem

Add the following and simplify the sum if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} + \frac{x + 2}{x^{2} - x - 6}$

$\frac{x + 2}{x - 3}$

$\frac{x + 2}{x + 3}$

$\frac{x - 1}{x - 3}$

$\frac{x - 1}{x + 2}$

Problem

Subtract the following and simplify the difference if possible.
$\frac{x + 5}{4 x} - \frac{2 x - 1}{4 x}$

$\frac{x - 6}{4 x}$

$\frac{6 - x}{4 x}$

$\frac{5}{4} x$

$6 over 4 x$

Problem

Subtract the following and simplify the difference if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} - \frac{x + 2}{x^{2} - x - 6}$

$\frac{1}{x - 3}$

$x$

$1$

$\frac{x - 3}{x + 2}$

example

Adding and Subtracting Rational Expressions with Common Denominators Example 1

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Adding and Subtracting Rational Expressions with Common Denominators Example 1 Video Summary

When simplifying rational expressions with a common denominator, the first step is to combine the numerators while keeping the denominator unchanged. For example, given the expression:

$\frac{2y^2 + 7y + 2}{2y - 5} - \frac{10y - 7}{2y - 5} + \frac{4y^2}{2y - 5}$

Since all terms share the denominator \$2y - 5\(, you can combine the numerators directly:

$\frac{(2y^2 + 7y + 2) - (10y - 7) + 4y^2}{2y - 5}$

Distributing the minus sign across the second numerator changes the signs inside the parentheses:

$\frac{2y^2 + 7y + 2 - 10y + 7 + 4y^2}{2y - 5}$

Next, combine like terms in the numerator. The \)y^2$ terms add up to $2y^2 + 4y^2 = 6y^2$. The $y$ terms combine as $7y - 10y = -3y$. The constants sum to $2 + 7 = 9\(. This simplifies the expression to:

$\frac{6y^2 - 3y + 9}{2y - 5}$

To simplify further, factor out the greatest common factor (GCF) from the numerator. Here, 3 is the GCF:

$\frac{3(2y^2 - y + 3)}{2y - 5}$

At this point, check if the quadratic \)2y^2 - y + 3$ can be factored. To factor a quadratic $ay^2 + by + c$, look for two numbers that multiply to \(a \times c$ and add to \)b$. For $2y^2 - y + 3$, multiply \(2 \times 3 = 6$ and find two numbers that add to \)-1$. Possible pairs like $-2$ and $3$ or $2$ and $-3\( do not satisfy both conditions simultaneously. Since no suitable pair exists, the quadratic does not factor over the integers.

Because the quadratic cannot be factored further and does not share any common factors with the denominator \)2y - 5$, the expression is already in its simplest form:

$\frac{3(2y^2 - y + 3)}{2y - 5}$

This process highlights the importance of combining like terms, factoring out common factors, and testing for further factorization to simplify rational expressions effectively. When no further factoring or cancellation is possible, the expression should be left in its simplest form to ensure accuracy and clarity.

Here’s what students ask on this topic:

Adding rational expressions with common denominators is similar to adding fractions. First, ensure the denominators are the same. Then, combine the numerators by adding them together while keeping the denominator unchanged. For example, if you have $\frac{5}{18x} + \frac{1}{18x}$ , you add the numerators: 5 + 1 = 6, so the expression becomes $\frac{6}{18x}$ . The final step is to simplify the expression by factoring and canceling common factors. Here, 6 and 18 share factors 3 and 2, so after canceling, the simplified form is $\frac{1}{3}$ . This process ensures the expression is in its simplest form.

When subtracting rational expressions with the same denominator, start by distributing the negative sign to all terms in the numerator of the second expression. For example, consider $x^{2} - - x + 2$ over the common denominator $x - 1$ . Distribute the negative to get $\frac{x^{2} + x - 2}{x}$ . Next, factor the numerator if possible. Here, $x^{2} + x - 2$ factors into $(x - 1)(x + 2)$ . Cancel the common factor $x - 1$ from numerator and denominator, leaving $x + 2$ as the simplified result.

Simplifying rational expressions after addition or subtraction is crucial because it reduces the expression to its simplest form, making it easier to understand and work with. Simplification often involves factoring polynomials in the numerator and denominator and canceling common factors. This process helps avoid errors in further calculations and clarifies the expression's behavior. For example, after adding $\frac{5}{18x}$ and $\frac{1}{18x}$ , simplifying $\frac{6}{18x}$ to $\frac{1}{3}$ makes the expression more manageable. Simplification is a key skill in algebra that ensures clarity and accuracy in problem-solving.

When subtracting rational expressions, after distributing the negative sign, you often need to factor the numerator to simplify the expression. For example, if the numerator is a quadratic like $x^{2} + x - 2$ , find two numbers that multiply to the constant term (-2) and add to the coefficient of the middle term (1). Here, those numbers are 2 and -1. So, factor the quadratic as $(x + 2)(x - 1)$ . This factoring allows you to cancel common factors with the denominator if they exist, simplifying the expression. Factoring is essential for reducing rational expressions to their simplest form.

The key steps to add or subtract rational expressions with common denominators are: (1) Confirm the denominators are the same. (2) For addition, combine the numerators by adding them; for subtraction, distribute the negative sign to the second numerator and then combine. (3) Keep the common denominator unchanged. (4) Simplify the resulting expression by factoring numerators and denominators and canceling common factors. For example, adding $\frac{5}{18x}$ and $\frac{1}{18x}$ results in $\frac{6}{18x}$ , which simplifies to $\frac{1}{3}$ . These steps ensure accurate and simplified results.

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Adding and Subtracting Rational Expressions with Common Denominators: Videos & Practice Problems

Adding and Subtracting Rational Expressions with Common Denominators

Adding and Subtracting Rational Expressions with Common Denominators Video Summary

Add the following and simplify the sum if possible.
$\frac{2}{5} + \frac{3}{5}$

Add the following and simplify the sum if possible.
$\frac{x}{7} + \frac{4 x}{7}$

Subtract the following and simplify the difference if possible.
$\frac{5}{8} - \frac{3}{8}$

Subtract the following and simplify the difference if possible.
$\frac{4 y}{3} - \frac{- 5 y}{3}$

Add the following and simplify the sum if possible.
$\frac{x + 1}{3 x} + \frac{2 x - 1}{3 x}$

Add the following and simplify the sum if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} + \frac{x + 2}{x^{2} - x - 6}$

Subtract the following and simplify the difference if possible.
$\frac{x + 5}{4 x} - \frac{2 x - 1}{4 x}$

Subtract the following and simplify the difference if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} - \frac{x + 2}{x^{2} - x - 6}$

Adding and Subtracting Rational Expressions with Common Denominators Example 1

Adding and Subtracting Rational Expressions with Common Denominators Example 1 Video Summary

Here’s what students ask on this topic:

How do you add rational expressions with common denominators?

What is the process for subtracting rational expressions with the same denominator?

Why is simplifying rational expressions important after adding or subtracting?

How do you factor the numerator when subtracting rational expressions?

What are the key steps to add or subtract rational expressions with common denominators?

Your Beginning Algebra tutors

Adding and Subtracting Rational Expressions with Common Denominators: Videos & Practice Problems

Adding and Subtracting Rational Expressions with Common Denominators

Adding and Subtracting Rational Expressions with Common Denominators Video Summary

Add the following and simplify the sum if possible.25+35\(\frac\)25+\(\frac\)35

Add the following and simplify the sum if possible.x7+4x7\(\frac{x}{7}\)+\(\frac{4x}{7}\)

Subtract the following and simplify the difference if possible.58−38\(\frac\)58-\(\frac\)38

Subtract the following and simplify the difference if possible.4y3−−5y3\(\frac{4y}{3}\)-\(\frac{-5y}{3}\)

Add the following and simplify the sum if possible.x+13x+2x−13x\(\frac{x+1}{3x}\)+\(\frac{2x-1}{3x}\)

Add the following and simplify the sum if possible.x2−4x2−x−6+x+2x2−x−6\(\frac{x^2-4}{x^2-x-6}\)+\(\frac{x+2}{x^2-x-6}\)

Subtract the following and simplify the difference if possible.x+54x−2x−14x\(\frac{x+5}{4x}\)-\(\frac{2x-1}{4x}\)

Subtract the following and simplify the difference if possible.x2−4x2−x−6−x+2x2−x−6\(\frac{x^2-4}{x^2-x-6}\)-\(\frac{x+2}{x^2-x-6}\)

Adding and Subtracting Rational Expressions with Common Denominators Example 1

Adding and Subtracting Rational Expressions with Common Denominators Example 1 Video Summary

Here’s what students ask on this topic:

How do you add rational expressions with common denominators?

How do you add rational expressions with common denominators?

What is the process for subtracting rational expressions with the same denominator?

What is the process for subtracting rational expressions with the same denominator?

Why is simplifying rational expressions important after adding or subtracting?

Why is simplifying rational expressions important after adding or subtracting?

How do you factor the numerator when subtracting rational expressions?

How do you factor the numerator when subtracting rational expressions?

What are the key steps to add or subtract rational expressions with common denominators?

What are the key steps to add or subtract rational expressions with common denominators?

Your Beginning Algebra tutors

Add the following and simplify the sum if possible.
$\frac{2}{5} + \frac{3}{5}$

Add the following and simplify the sum if possible.
$\frac{x}{7} + \frac{4 x}{7}$

Subtract the following and simplify the difference if possible.
$\frac{5}{8} - \frac{3}{8}$

Subtract the following and simplify the difference if possible.
$\frac{4 y}{3} - \frac{- 5 y}{3}$

Add the following and simplify the sum if possible.
$\frac{x + 1}{3 x} + \frac{2 x - 1}{3 x}$

Add the following and simplify the sum if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} + \frac{x + 2}{x^{2} - x - 6}$

Subtract the following and simplify the difference if possible.
$\frac{x + 5}{4 x} - \frac{2 x - 1}{4 x}$

Subtract the following and simplify the difference if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} - \frac{x + 2}{x^{2} - x - 6}$