Add the following expressions and simplify if possible: 3 x + 2 x + 2 \frac{3}{x}+\frac{2}{x+2}

5 x + 6 x ( x + 2 ) \frac{5x+6}{x\left(x+2\right)}

5 x + 4 x ( x + 2 ) \frac{5x+4}{x\left(x+2\right)}

Add the following expressions and simplify if possible: x x + 2 + 2 ( x + 2 ) ( x + 3 ) \frac{x}{x+2}+\frac{2}{\left(x+2\right)\left(x+3\right)}

1 ( x + 3 ) \frac{1}{\left(x+3\right)}

( x + 1 ) ( x + 4 ) ( x + 2 ) ( x + 3 ) \frac{\left(x+1\right)\left(x+4\right)}{\left(x+2\right)\left(x+3\right)}

1 ( x + 2 ) ( x + 3 ) \frac{1}{\left(x+2\right)\left(x+3\right)}

Add the following expressions and simplify if possible: 2 x 2 − 1 + 3 x + 1 \frac{2}{x^2-1}+\frac{3}{x+1}

3 x − 1 ( x + 1 ) ( x − 1 ) \frac{3x-1}{\left(x+1\right)\left(x-1\right)}

3 x + 1 ( x + 1 ) ( x − 1 ) \frac{3x+1}{\left(x+1\right)\left(x-1\right)}

Add the following expressions and simplify if possible: x 2 x 2 − 4 + 2 x 4 − x 2 \frac{x^2}{x^2-4}+\frac{2x}{4-x^2}

x 2 ( x − 2 ) ( x + 2 ) \frac{x^2}{\left(x-2\right)\left(x+2\right)}

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Different Denominators

8. Rational Expressions and Equations

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

Video Lessons

Topic summary

Adding and subtracting rational expressions requires finding the least common denominator (LCD), similar to working with rational numbers. The LCD is found by factoring denominators, often involving binomials like $x + 4$ and $x - 8$ . Equivalent rational expressions are created by multiplying numerator and denominator by missing factors, enabling combination of terms. Simplifying involves distributing coefficients and combining like terms, resulting in a single rational expression. Understanding terms, coefficients, and polynomial degrees is essential for mastering these operations and simplifying expressions effectively.

concept

Adding and Subtracting Rational Expressions with Unlike Denominators

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Adding and Subtracting Rational Expressions with Unlike Denominators Video Summary

Adding and subtracting rational expressions with unlike denominators involves a process similar to working with rational numbers. The key step is to find the least common denominator (LCD), which allows you to rewrite each rational expression as an equivalent expression with a common denominator. For example, when adding fractions like $\frac{1}{30}$ and $\frac{1}{20}$, the LCD is 60. Multiplying the numerator and denominator of each fraction by the missing factor (2 for $\frac{1}{30}$ and 3 for $\frac{1}{20}$) converts them to $\frac{2}{60}$ and $\frac{3}{60}$, respectively. Adding these gives $\frac{5}{60}$, which simplifies to $\frac{1}{12}$.

This same principle applies to rational expressions. Suppose you have expressions with denominators \$30x$ and $20x^2$. The LCD here is $60x^2$. To rewrite each expression with this common denominator, multiply the numerator and denominator of $\frac{1}{30x}$ by $2x$ to get $\frac{2x}{60x^2}$, and multiply $\frac{1}{20x^2}$ by $\frac{3}{3}$ to get $\frac{3}{60x^2}$. Adding these numerators results in $\frac{2x + 3}{60x^2}\(, which is the simplified sum of the rational expressions.

When working with algebraic expressions such as \)\frac{2}{x+4}$ and $\frac{7}{x-8}$, the LCD is the product of the distinct factors in the denominators, here $(x+4)(x-8)$. To combine these, multiply the numerator and denominator of each fraction by the missing factor: multiply $\frac{2}{x+4}$ by $\frac{x-8}{x-8}$ and $\frac{7}{x-8}$ by $\frac{x+4}{x+4}$. This yields equivalent expressions with the common denominator $(x+4)(x-8)$. Subtracting the numerators gives:

\[\frac{2(x-8) - 7(x+4)}{(x+4)(x-8)}.\]

Distributing the terms in the numerator results in:

\[\frac{2x - 16 - 7x - 28}{(x+4)(x-8)} = \frac{-5x - 44}{(x+4)(x-8)}.\]

This expression is fully simplified and represents the difference of the original rational expressions as a single rational expression.

Understanding how to find the least common denominator, rewrite rational expressions as equivalent expressions, and then combine them by adding or subtracting numerators is essential for manipulating rational expressions effectively. This process not only simplifies complex expressions but also lays the foundation for solving equations involving rational expressions and further algebraic operations.

Problem

Add the following expressions and simplify if possible:
$\frac{3}{x} + \frac{2}{x + 2}$

$\frac{5 x + 6}{x (x + 2)}$

$\frac{11}{x + 2}$

$\frac{5 x + 4}{x (x + 2)}$

$\frac{9}{x + 2}$

Problem

Add the following expressions and simplify if possible:
$\frac{x}{x + 2} + \frac{2}{(x + 2) (x + 3)}$

$\frac{1}{(x + 3)}$

$\frac{(x + 1) (x + 4)}{(x + 2) (x + 3)}$

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

$\frac{1}{(x + 2) (x + 3)}$

Problem

Add the following expressions and simplify if possible:
$\frac{4}{x - 5} + \frac{3}{5 - x}$

$\frac{1}{x + 5}$

$\frac{x}{x - 5}$

$\frac{1}{5 - x}$

$\frac{1}{x - 5}$

Problem

Add the following expressions and simplify if possible:
$\frac{2}{x^{2} - 1} + \frac{3}{x + 1}$

$\frac{3 x + 1}{(x + 1) (x - 1)}$

$\frac{3 x - 1}{(x + 1) (x - 1)}$

$\frac{3 x - 1}{x^{2} - 1}$

$\frac{3 x - 1}{x^{2} + 1}$

Problem

Add the following expressions and simplify if possible:
$\frac{x}{x^{2} + 3 x} + \frac{3}{x + 3}$

$\frac{4}{x^{2} + 3 x}$

$\frac{4 x}{x + 3}$

$\frac{4}{x + 3}$

$4 over x$

Problem

Add the following expressions and simplify if possible:
$\frac{x^{2}}{x^{2} - 4} + \frac{2 x}{4 - x^{2}}$

$\frac{x^{2}}{(x - 2) (x + 2)}$

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

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

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

Problem

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{3}{x} - \frac{2}{x + 2}$

$\frac{3 x + 6}{x^{2} + 2 x}$

$\frac{x + 6}{x + 2}$

$\frac{x + 6}{x^{2} + 2 x}$

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

Problem

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x + 2}{x^{2} - 4} - \frac{1}{x - 2}$

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

$\frac{x + 2}{(x - 2) (x + 2)}$

$\frac{x - 2}{(x - 2) (x + 2)}$

$0$

Problem

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x^{2}}{x^{2} - 25} - \frac{5 x}{25 - x^{2}}$

$\frac{x}{x + 5}$

$\frac{x}{x - 5}$

$\frac{x}{(x - 5) (x + 5)}$

$\frac{x + 5}{x - 5}$

example

Adding and Subtracting Rational Expressions with Unlike Denominators Example 1

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

Rational equations are equations that include a variable in the denominator of a fraction. To solve these equations, the goal remains the same: find a value for $ x $ that satisfies the equation. However, it is crucial to recognize that the solution must not make any denominator equal to zero, as this would render the equation undefined. For instance, in the equation $ \frac{1}{x - 1} = 12 $, the denominator $ x - 1 $ cannot equal zero, leading to the restriction that $ x \neq 1 $.

To solve a rational equation, follow these steps:

1. **Determine Restrictions**: Identify values that would make any denominator zero. For example, if the denominator is $ x - 1 $, setting it to zero gives $ x = 1 $, indicating that $ x $ cannot be 1.

2. **Multiply by the Least Common Denominator (LCD)**: This step eliminates the fractions and transforms the rational equation into a linear equation. For example, in the equation $ \frac{x}{x - 1} = 76 $, the denominators are $ x - 1 $ and 6. The LCD is $ 6(x - 1) $. Multiply each term by the LCD to simplify the equation.

3. **Solve the Linear Equation**: After eliminating the fractions, you will have a linear equation. For instance, after multiplying, you might arrive at $ 6x = 7(x - 1) $. Distributing and rearranging terms leads to a solvable linear equation. In this case, you would distribute the 7 to get $ 6x = 7x - 7 $, then isolate $ x $ by moving terms around.

4. **Check the Solution Against Restrictions**: Finally, verify that the solution does not violate any restrictions identified in the first step. If the solution is $ x = 7 $, and since 7 does not equal 1, it is valid.

In summary, solving rational equations involves identifying restrictions, eliminating fractions through multiplication by the LCD, solving the resulting linear equation, and checking the solution against any restrictions to ensure it is valid.

Here’s what students ask on this topic:

To find the least common denominator (LCD) when adding or subtracting rational expressions, first factor each denominator completely. For example, if the denominators are binomials like $x + 4$ and $x - 8$ , the LCD is their product: $(x + 4) (x - 8)$ . The LCD must include all factors from each denominator, ensuring it is the smallest expression that both denominators divide into evenly. This step is crucial because it allows you to rewrite each rational expression with a common denominator, enabling addition or subtraction of the numerators.

Once the LCD is found, rewrite each rational expression as an equivalent expression with the LCD as the denominator. Identify the missing factor for each denominator by dividing the LCD by the original denominator. Multiply both the numerator and denominator of each rational expression by this missing factor. For example, if the original denominator is $x + 4$ and the LCD is $(x + 4) (x - 8)$ , the missing factor is $x - 8$ . Multiply numerator and denominator by this to get an equivalent expression with the LCD denominator. This step ensures all rational expressions have the same denominator, allowing you to add or subtract their numerators directly.

After rewriting rational expressions with a common denominator, add or subtract the numerators as indicated. Then, simplify the resulting numerator by distributing any coefficients and combining like terms. For example, if the numerator is $2 (x - 8) - 7 (x + 4)$ , distribute to get $2x - 16 - 7x - 28$ . Combine like terms: $(2x - 7x) + (- 16 - 28) = - 5x - 44$ . This simplified numerator is then placed over the common denominator to form the final simplified rational expression.

Multiplying the numerator and denominator by the missing factor is essential to create equivalent rational expressions with a common denominator. This process does not change the value of the expression because multiplying by a form of one (the missing factor over itself) preserves equivalence. It allows you to rewrite each rational expression so that all denominators match, enabling you to add or subtract the numerators directly. Without this step, the denominators would remain different, and you could not combine the expressions properly.

To add rational expressions with denominators $x + 4$ and $x - 8$ , first find the LCD, which is $(x + 4) (x - 8)$ . Then, multiply the numerator and denominator of the first expression by $x - 8$ and the second expression by $x + 4$ to get equivalent expressions with the LCD denominator. Add the numerators, distribute any coefficients, and combine like terms. Finally, write the sum over the common denominator. This method ensures the expressions are combined correctly and simplified.

Your Beginning Algebra tutors

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

Adding and Subtracting Rational Expressions with Unlike Denominators

Adding and Subtracting Rational Expressions with Unlike Denominators Video Summary

Add the following expressions and simplify if possible:
$\frac{3}{x} + \frac{2}{x + 2}$

Add the following expressions and simplify if possible:
$\frac{x}{x + 2} + \frac{2}{(x + 2) (x + 3)}$

Add the following expressions and simplify if possible:
$\frac{4}{x - 5} + \frac{3}{5 - x}$

Add the following expressions and simplify if possible:
$\frac{2}{x^{2} - 1} + \frac{3}{x + 1}$

Add the following expressions and simplify if possible:
$\frac{x}{x^{2} + 3 x} + \frac{3}{x + 3}$

Add the following expressions and simplify if possible:
$\frac{x^{2}}{x^{2} - 4} + \frac{2 x}{4 - x^{2}}$

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{3}{x} - \frac{2}{x + 2}$

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x + 2}{x^{2} - 4} - \frac{1}{x - 2}$

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x^{2}}{x^{2} - 25} - \frac{5 x}{25 - x^{2}}$

Adding and Subtracting Rational Expressions with Unlike Denominators Example 1

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

Here’s what students ask on this topic:

How do you find the least common denominator (LCD) when adding or subtracting rational expressions with different denominators?

What is the process for rewriting rational expressions as equivalent expressions with a common denominator?

How do you simplify the numerator after adding or subtracting rational expressions?

Why is it important to multiply numerator and denominator by the missing factor when adding rational expressions?

Can you explain how to add rational expressions with denominators like $x + 4$ and $x - 8$ ?

Your Beginning Algebra tutors

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

Adding and Subtracting Rational Expressions with Unlike Denominators

Adding and Subtracting Rational Expressions with Unlike Denominators Video Summary

Add the following expressions and simplify if possible:3x+2x+2\frac{3}{x}+\frac{2}{x+2}

Add the following expressions and simplify if possible:xx+2+2(x+2)(x+3)\frac{x}{x+2}+\frac{2}{\left(x+2\right)\left(x+3\right)}

Add the following expressions and simplify if possible:4x−5+35−x\frac{4}{x-5}+\frac{3}{5-x}

Add the following expressions and simplify if possible:2x2−1+3x+1\frac{2}{x^2-1}+\frac{3}{x+1}

Add the following expressions and simplify if possible:xx2+3x+3x+3\frac{x}{x^2+3x}+\frac{3}{x+3}

Add the following expressions and simplify if possible:x2x2−4+2x4−x2\frac{x^2}{x^2-4}+\frac{2x}{4-x^2}

Subtract the following rational expressions and write the difference in simplest form if possible.3x−2x+2\frac{3}{x}-\frac{2}{x+2}

Subtract the following rational expressions and write the difference in simplest form if possible.x+2x2−4−1x−2\frac{x+2}{x^2-4}-\frac{1}{x-2}

Subtract the following rational expressions and write the difference in simplest form if possible. x2x2−25−5x25−x2\frac{x^2}{x^2-25}-\frac{5x}{25-x^2}

Adding and Subtracting Rational Expressions with Unlike Denominators Example 1

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

Here’s what students ask on this topic:

How do you find the least common denominator (LCD) when adding or subtracting rational expressions with different denominators?

How do you find the least common denominator (LCD) when adding or subtracting rational expressions with different denominators?

What is the process for rewriting rational expressions as equivalent expressions with a common denominator?

What is the process for rewriting rational expressions as equivalent expressions with a common denominator?

How do you simplify the numerator after adding or subtracting rational expressions?

How do you simplify the numerator after adding or subtracting rational expressions?

Why is it important to multiply numerator and denominator by the missing factor when adding rational expressions?

Why is it important to multiply numerator and denominator by the missing factor when adding rational expressions?

Can you explain how to add rational expressions with denominators like x+4 and x-8?

Can you explain how to add rational expressions with denominators like x+4 and x-8?

Your Beginning Algebra tutors

Add the following expressions and simplify if possible:
$\frac{3}{x} + \frac{2}{x + 2}$

Add the following expressions and simplify if possible:
$\frac{x}{x + 2} + \frac{2}{(x + 2) (x + 3)}$

Add the following expressions and simplify if possible:
$\frac{4}{x - 5} + \frac{3}{5 - x}$

Add the following expressions and simplify if possible:
$\frac{2}{x^{2} - 1} + \frac{3}{x + 1}$

Add the following expressions and simplify if possible:
$\frac{x}{x^{2} + 3 x} + \frac{3}{x + 3}$

Add the following expressions and simplify if possible:
$\frac{x^{2}}{x^{2} - 4} + \frac{2 x}{4 - x^{2}}$

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{3}{x} - \frac{2}{x + 2}$

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x + 2}{x^{2} - 4} - \frac{1}{x - 2}$

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x^{2}}{x^{2} - 25} - \frac{5 x}{25 - x^{2}}$

Can you explain how to add rational expressions with denominators like $x + 4$ and $x - 8$ ?