Simplify The Expression: 5 X X − 8 + 3 X − 2 \frac{5x}{x-8} + \frac{3}{x-2} X − 8 5 X ​ + X − 2 3 ​

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Introduction

In mathematics, simplifying expressions is a crucial step in solving equations and inequalities. It involves combining like terms, canceling out common factors, and rearranging the expression to make it more manageable. In this article, we will focus on simplifying the expression $\frac{5x}{x-8} + \frac{3}{x-2}$ using various techniques.

Understanding the Expression

The given expression is a sum of two fractions, each with a different denominator. The first fraction has a variable $x$ in the numerator and a constant $8$ in the denominator, while the second fraction has a constant $3$ in the numerator and a variable $x-2$ in the denominator. To simplify this expression, we need to find a common denominator and combine the two fractions.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the two denominators, $x-8$ and $x-2$. The LCM of these two expressions is $(x-8)(x-2)$, which can be expanded as $x^2-10x+16$.

Simplifying the Expression

Now that we have found the common denominator, we can rewrite each fraction with the common denominator. The first fraction becomes $\frac{5x(x-2)}{(x-8)(x-2)}$, and the second fraction becomes $\frac{3(x-8)}{(x-8)(x-2)}$. We can then combine the two fractions by adding their numerators.

Combining the Fractions

To combine the fractions, we need to add the numerators and keep the common denominator. The expression becomes $\frac{5x(x-2)+3(x-8)}{(x-8)(x-2)}$. We can simplify the numerator by expanding and combining like terms.

Expanding and Combining Like Terms

Expanding the numerator, we get $5x^2-10x+3x-24$. Combining like terms, we get $5x^2-7x-24$. The expression now becomes $\frac{5x^2-7x-24}{(x-8)(x-2)}$.

Factoring the Numerator

To simplify the expression further, we can try to factor the numerator. Factoring the numerator, we get $(5x+3)(x-8)$. The expression now becomes $\frac{(5x+3)(x-8)}{(x-8)(x-2)}$.

Canceling Out Common Factors

Now that we have factored the numerator, we can cancel out common factors between the numerator and the denominator. The common factor is $(x-8)$, which can be canceled out.

Simplifying the Expression

After canceling out the common factor, the expression becomes $\frac{5x+3}{x-2}$. This is the simplified form of the original expression.

Conclusion

In this article, we have simplified the expression $\frac{5x}{x-8} + \frac{3}{x-2}$ using various techniques. We found a common denominator, combined the fractions, expanded and combined like terms, factored the numerator, and canceled out common factors. The simplified expression is $\frac{5x+3}{x-2}$. This expression can be used as a building block for more complex mathematical problems.

Final Answer

The final answer is $\boxed{\frac{5x+3}{x-2}}$.

Related Topics

• Simplifying expressions
• Combining like terms
• Factoring numerators
• Canceling out common factors
• Least common multiple (LCM)

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

In our previous article, we simplified the expression $\frac{5x}{x-8} + \frac{3}{x-2}$ using various techniques. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional examples.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to identify the type of expression and determine the best approach to simplify it. This may involve finding a common denominator, combining like terms, or factoring the numerator.

Q: How do I find a common denominator?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the two denominators. The LCM can be found by listing the multiples of each denominator and finding the smallest multiple that appears in both lists.

Q: What is the difference between a common denominator and a least common multiple (LCM)?

A: A common denominator is a denominator that is common to both fractions, while a least common multiple (LCM) is the smallest multiple that appears in both lists of multiples.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. However, you need to be careful when canceling out common factors, as this may result in an expression with a variable in the denominator.

Q: How do I know when to factor the numerator?

A: You should factor the numerator when it can be expressed as a product of two or more factors. Factoring the numerator can help you simplify the expression and cancel out common factors.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent. To do this, you need to rewrite the expression with a positive exponent and then simplify it.

Q: How do I know when to cancel out common factors?

A: You should cancel out common factors when they appear in both the numerator and the denominator. Canceling out common factors can help you simplify the expression and make it easier to work with.

Examples

Example 1: Simplify the expression $\frac{2x}{x+3} + \frac{4}{x+3}$

To simplify this expression, we need to find a common denominator, which is $(x+3)$. We can then combine the two fractions by adding their numerators.

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

We can simplify the numerator by combining like terms.

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

Example 2: Simplify the expression $\frac{3x}{x-2} + \frac{2}{x-2}$

To simplify this expression, we need to find a common denominator, which is $(x-2)$. We can then combine the two fractions by adding their numerators.

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

We can simplify the numerator by combining like terms.

$\frac{3x+2}{x-2} = \frac{3(x+2/3)}{x-2}$

Conclusion

In this article, we have answered some frequently asked questions related to simplifying expressions and provided additional examples. We have also discussed the importance of finding a common denominator, combining like terms, and factoring the numerator. By following these steps, you can simplify expressions and make them easier to work with.

Final Answer

The final answer is $\boxed{\frac{5x+3}{x-2}}$.

Related Topics

• Simplifying expressions
• Combining like terms
• Factoring numerators
• Canceling out common factors
• Least common multiple (LCM)

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.