Ranger's Friend, Foster, Wanted To Challenge Him By Creating This Expression Involving Two Sets Of Parentheses With Fractions: \frac{1}{3}(9x - 15) + 4\left(x + \frac{1}{2}\right ]Ranger Showed His First Two Steps Below:1. Distribute

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Introduction

In mathematics, simplifying algebraic expressions is a crucial skill that helps us solve equations and inequalities. It involves combining like terms, removing parentheses, and performing other operations to make the expression more manageable. In this article, we will explore how to simplify a complex algebraic expression involving fractions and parentheses.

The Expression

The expression given by Foster is:

13(9x−15)+4(x+12)\frac{1}{3}(9x - 15) + 4\left(x + \frac{1}{2}\right)

This expression involves two sets of parentheses, fractions, and variables. Our goal is to simplify it by following a series of steps.

Step 1: Distribute

The first step in simplifying the expression is to distribute the terms inside the parentheses. This involves multiplying each term inside the parentheses by the coefficient outside the parentheses.

\frac{1}{3}(9x - 15) = \frac{1}{3} \cdot 9x - \frac{1}{3} \cdot 15
= 3x - 5

Similarly, we distribute the term inside the second set of parentheses:

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

Step 2: Combine Like Terms

Now that we have distributed the terms, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

3x - 5 + 4x + 2 = (3x + 4x) + (-5 + 2)
= 7x - 3

Step 3: Simplify the Expression

The final step is to simplify the expression by removing any unnecessary parentheses or brackets.

7x - 3

Conclusion

In this article, we have simplified a complex algebraic expression involving fractions and parentheses. We followed a series of steps, including distributing terms, combining like terms, and simplifying the expression. By following these steps, we can simplify any algebraic expression and make it more manageable.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

  • Distribute terms carefully: When distributing terms, make sure to multiply each term inside the parentheses by the coefficient outside the parentheses.
  • Combine like terms: Like terms are terms that have the same variable raised to the same power. Combine these terms to simplify the expression.
  • Simplify the expression: Once you have combined like terms, simplify the expression by removing any unnecessary parentheses or brackets.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

  • Simplify the expression: 12(x+3)+2(x−1)\frac{1}{2}(x + 3) + 2(x - 1)
  • Simplify the expression: 3(x−2)+2(x+1)3(x - 2) + 2(x + 1)
  • Simplify the expression: 14(x−2)+3(x+1)\frac{1}{4}(x - 2) + 3(x + 1)

Answer Key

Here are the answers to the practice problems:

  • 12x+32+2x−2=72x−12\frac{1}{2}x + \frac{3}{2} + 2x - 2 = \frac{7}{2}x - \frac{1}{2}
  • 3x−6+2x+2=5x−43x - 6 + 2x + 2 = 5x - 4
  • 14x−12+3x+3=134x+52\frac{1}{4}x - \frac{1}{2} + 3x + 3 = \frac{13}{4}x + \frac{5}{2}
    Simplifying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored how to simplify algebraic expressions involving fractions and parentheses. We followed a series of steps, including distributing terms, combining like terms, and simplifying the expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

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

A: The first step in simplifying an algebraic expression is to distribute the terms inside the parentheses. This involves multiplying each term inside the parentheses by the coefficient outside the parentheses.

Q: How do I distribute terms in an algebraic expression?

A: To distribute terms, you multiply each term inside the parentheses by the coefficient outside the parentheses. For example, if you have the expression 13(9x−15)\frac{1}{3}(9x - 15), you would multiply 13\frac{1}{3} by each term inside the parentheses: 13⋅9x−13⋅15\frac{1}{3} \cdot 9x - \frac{1}{3} \cdot 15.

Q: What is the difference between distributing and combining like terms?

A: Distributing involves multiplying each term inside the parentheses by the coefficient outside the parentheses, while combining like terms involves adding or subtracting terms that have the same variable raised to the same power.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms, you add or subtract terms that have the same variable raised to the same power. For example, if you have the expression 3x−5+4x+23x - 5 + 4x + 2, you would combine the like terms 3x3x and 4x4x to get 7x7x, and then combine the constants −5-5 and 22 to get −3-3.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to simplify the expression by removing any unnecessary parentheses or brackets.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you follow these steps:

  1. Distribute the terms inside the parentheses.
  2. Combine like terms.
  3. Simplify the expression by removing any unnecessary parentheses or brackets.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not distributing terms correctly: Make sure to multiply each term inside the parentheses by the coefficient outside the parentheses.
  • Not combining like terms correctly: Make sure to add or subtract terms that have the same variable raised to the same power.
  • Not simplifying the expression correctly: Make sure to remove any unnecessary parentheses or brackets.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems, such as the ones provided in our previous article. You can also try simplifying expressions on your own and checking your work with a calculator or a friend.

Conclusion

In this article, we have answered some frequently asked questions about simplifying algebraic expressions. We have covered topics such as distributing terms, combining like terms, and simplifying the expression. By following these steps and avoiding common mistakes, you can simplify algebraic expressions with confidence.

Tips and Tricks

Here are some additional tips and tricks to help you simplify algebraic expressions:

  • Use a calculator or a friend to check your work: This can help you catch any mistakes and ensure that your answer is correct.
  • Practice, practice, practice: The more you practice simplifying algebraic expressions, the more comfortable you will become with the process.
  • Take your time: Don't rush through the process of simplifying an algebraic expression. Take your time and make sure to follow each step carefully.