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Chapter 7: Problem 102

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (2 x+3)+(x-5) $$

Short Answer

Expert verified
The simplified expression is \( 3x - 2 \).

Step by step solution

01

Identify Like Terms

In the expression \( (2x+3)+(x-5) \), identify the like terms. The like terms here are those that contain the variable \( x \) and those that are constants. The \( x \) terms are \( 2x \) and \( x \), and the constants are \( 3 \) and \( -5 \).
02

Combine Like Terms with Variables

Add the coefficients of the \( x \) terms: \( 2x + x \). This gives you \( (2+1)x = 3x \).
03

Combine Constant Terms

Add the constant terms: \( 3 + (-5) \). This simplifies to \( -2 \).
04

Form the Simplified Expression

Combine the results of Steps 2 and 3 to form the simplified expression: \( 3x - 2 \).

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Key Concepts

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

Combine Like Terms
In algebra, combining like terms is a fundamental step in simplifying expressions. It involves grouping terms that have identical variable parts. For instance, in the expression \((2x+3)+(x-5)\), the like terms are those containing the variable \(x\), such as \(2x\) and \(x\), as well as the constant numbers, \(3\) and \(-5\).
  • Always start by identifying terms: like terms are those that share the same variables raised to the same powers.
  • Group the terms: For \(x\) terms, you have \(2x\) and \(x\). For constants, you have \(3\) and \(-5\).
  • Add the coefficients of like terms together. This means for \(x\) terms, you calculate \(2 + 1 = 3\), resulting in \(3x\).

This process is crucial as it helps to reorganize expressions into more manageable forms. It sets the stage for further simplification or solving equations.
Polynomials
Polynomials are mathematical expressions consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication. They do not include division by variable terms. Consider the example:
\[(2x+3)+(x-5)\]This expression showcases basic polynomial components. Each term in the polynomial is a product of a coefficient and a power of a variable.
  • The terms \(2x\) and \(x\) are linear terms (degree 1) due to the variable \(x\).
  • The constants \(3\) and \(-5\) are zero-degree terms.

These expressions are foundational in algebra because they can describe a wide array of problems through equations and functions. Understanding polynomials is essential for navigating more complex algebraic topics, such as factoring, expanding, and solving polynomial equations.
Simplifying Expressions
Simplifying expressions is the process of transforming expressions into a simpler or more usable form. It's an essential skill in solving equations efficiently. Using our case of \((2x+3)+(x-5)\), simplification involves several steps:

  • Identify and Combine Like Terms: Find the like terms and combine them, as discussed previously.
  • Perform Arithmetic Operations: Add or subtract constant terms after combining variable terms. Here, we simplify \(3 + (-5)\) to get \(-2\).
  • Re-write the Expression: After combining, write the simplified expression in its most reduced form. In this instance, it is \(3x - 2\).

By breaking expressions down, simplifying helps in making further calculations easier and in solving equations. Simplification often precedes solving as it lays out a clearer path to obtaining solutions.

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Most popular questions from this chapter

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{\frac{x^{4}}{16}} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{z^{21}}{27 x^{3}}} $$

Use a calculator to approximate each square root to 3 decimal places. $$ \sqrt{200} $$

Identify the domain and then graph each function. $$ f(x)=\sqrt[3]{x}+1 $$

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical(s) simplify to \(-3 ?\) a. \(\sqrt{9}\) b. \(\sqrt{-9}\) c. \(\sqrt[3]{27}\) d. \(\sqrt[3]{-27}\)
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