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Chapter 13: Problem 3

Find each difference. $$(9 x+5)-(4 x+3)$$

Short Answer

Expert verified
The difference is \(5x + 2\).

Step by step solution

01

Distribute the Negative Sign

We start with the expression \((9x + 5) - (4x + 3)\). To remove the parentheses, first distribute the negative sign into the second set of parentheses: this gives us \((9x + 5) - 4x - 3\).
02

Combine Like Terms

Now, group the terms with \(x\) together and the constant terms together: \(9x - 4x + 5 - 3\).
03

Simplify the Expression

Simplify the expression by performing the operations: \(9x - 4x = 5x\) and \(5 - 3 = 2\). This results in the simplified expression \(5x + 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a useful algebraic rule. It helps when you need to remove parentheses, especially when there is a negative sign involved. For example, in the expression \((9x + 5) - (4x + 3)\), we encounter a subtraction of the entire expression \((4x + 3)\). To simplify, the distributive property tells us to multiply the negative sign through the term inside the parentheses. That means you have:
  • -1 times \(4x\)
  • -1 times \(3\)
This changes \((4x + 3)\) into \(-4x - 3\). Hence, the original expression becomes \(9x + 5 - 4x - 3\). Once you apply the distributive property correctly, parentheses are removed, making it easier to combine like terms.
Combining Like Terms
Once you've removed parentheses using the distributive property, it's time to combine like terms. This step focuses on terms that share the same variable or are constants. For the expression \(9x + 5 - 4x - 3\), you need to:
  • Identify and group terms with \(x\) together (\(9x\) and \(-4x\))
  • Identify and group constant terms (\(+5\) and \(-3\))
This regrouping process helps simplify your task as you're focusing on similar types of terms. Adding or subtracting like terms simplifies the expression further. In this case, \(9x - 4x = 5x\), and \(5 - 3 = 2\). So, combining like terms gives us the expression \(5x + 2\).
Simplifying Expressions
Simplifying expressions is the final and satisfying step of many algebra problems. It involves reducing an expression to its simplest form by dealing with all terms within it.After combining like terms, our task is often nearing completion. For example, from the problem above, we simplify \(9x - 4x + 5 - 3\) into \(5x + 2\) by first removing parentheses, then grouping and simplifying terms.The key parts of simplification include:
  • Ensure all like terms are combined
  • Ensure constants are combined
  • Ensure all operations are performed accurately
Once simplified, expressions are often less cluttered, making future calculations cleaner and faster.

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Most popular questions from this chapter

Seth is making a rectangular dog pen. He needs the length of the pen to be 5 feet less than 4 times the width. Write an expression for the area \(A\) of the dog pen. A \(10 x-10 \mathrm{ft}^{2}\) B \(4 x^{2}-5 x \mathrm{ft}^{2}\) C \(4 x-5 \mathrm{ft}^{2}\) D \(x^{2}-5 x \mathrm{ft}^{2}\)

Determine whether each equation or table represents a linear or nonlinear function. Explain. $$\begin{array}{|r|r|} \hline x & y \\ \hline-4 & 12 \\ \hline-2 & 0 \\ \hline 0 & 4 \\ \hline 2 & 0 \\ \hline \end{array}$$

Find each sum. $$(11 x+2 y)+(x-5 y)$$

The table shows the years in which the first six million patents were issued. Is the number of patents issued a linear function of time? Explain. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Number of Patents Issued} \\ \hline 1911 & 1 \text { million } \\ \hline 1936 & 2 \text { million } \\ \hline 1961 & 3 \text { million } \\ \hline 1976 & 4 \text { million } \\ \hline 1991 & 5 \text { million } \\ \hline 1999 & 6 \text { million } \\ \hline \end{array}$$

Express each percent as a decimal. $$8 \%$$
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