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Chapter 2: Problem 8

Decide whether each of the following is an expression or an equation. $$ 4(x+3)-2(x+1)+10 $$

Short Answer

Expert verified
Expression

Step by step solution

01

Understand the Difference Between Expression and Equation

An expression is a combination of numbers, variables, and operations without an equality sign. An equation includes an equality sign and shows that two expressions are equal.
02

Identify Components

Look at the given sequence: \[4(x+3)-2(x+1)+10\] Observe if there is an '=' sign that separates two expressions.
03

Determine If It鈥檚 an Expression or Equation

Since \[4(x+3)-2(x+1)+10\] doesn't have an equality sign ('='), it does not equate two expressions. This makes it an expression.

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are fundamental in algebra. They are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. What makes algebraic expressions unique is that they do not contain an equality sign '='. For example, in the expression \(4(x+3)-2(x+1)+10\), we see numbers, variables, and operations mixed together.
You can think of an algebraic expression as a mathematical phrase. It can be simplified but not solved in the traditional sense because it doesn't set one part equal to another. Simplifying expressions involves combining like terms and applying arithmetic operations. Here鈥檚 why it鈥檚 important:
  • They represent real-life quantities.
  • Can be part of larger equations or formulas.
  • Useful in problem-solving and mathematical modeling.
Exploring Algebraic Equations
Unlike expressions, algebraic equations include an equality sign (鈥�=鈥�), indicating that two expressions are equal. This means equations can be solved to find the value of the variables. An equation expresses a balance; something like \(4(x+3)-2(x+1)+10 = 20\) means both sides are equal.
Equations are vital because they allow us to solve for unknowns. In real life, equations can model relationships and changes in different variables. Understanding equations involves:
  • Identifying the parts of the equation.
  • Using various methods to isolate the variable.
  • Checking the solutions to ensure they are correct.
Identifying Algebra Components
Knowing the components of algebraic expressions and equations is crucial for understanding and solving them. Let鈥檚 break down the key components:
  • Variables: Symbols (often letters) representing unknown quantities. In the expression \(4(x+3)-2(x+1)+10\), 鈥榵鈥� is the variable.
  • Coefficients: Numbers multiplying the variables, such as 鈥�4鈥� and 鈥�2鈥� in the expression \(4(x+3)-2(x+1)+10\).
  • Terms: Individual parts separated by '+' or '-' signs. 鈥�4(x+3)鈥� and 鈥�-2(x+1)鈥� are terms in the given expression.
  • Operators: Signs indicating mathematical operations like addition (+), subtraction (-), multiplication (*), and division (/).
  • Constants: Numbers without variables, like 鈥�10鈥� in the expression \(4(x+3)-2(x+1)+10\).
Identifying these components helps in simplifying expressions and solving equations effectively.

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Most popular questions from this chapter

For each compound inequality, decide whether intersection or union should be used. Then give the solution set in both interval and graph form. See Examples \(2-4\) and \(6-8 .\) $$ x<4 \text { or } x<-2 $$

Express each set in the simplest interval form. $$ [3,6] \cup(4,9) $$

Dan Abbey has invested \(\$ 12,000\) in bonds paying \(6 \% .\) How much additional money should he invest in a certificate of deposit paying \(3 \%\) simple interest so that the total return on the two investments will be \(4 \% ?\)

Give the correct solution set. Dr. Paul Donohue writes a syndicated column in which readers question him on a variety of health topics. Reader C. J. wrote, 鈥淢any people say they can weigh more because they have a large frame. How is frame size determined?鈥� Here is Dr. Donohue鈥檚 response: "For a man, a urist circumference between 6.75 and 7.25 in. / inclusive/ indicates \(a\) medium frame. Anything above is a large frame and anything below, a small frame." Using \(x\) to represent wrist circumference in inches, write an inequality or a three-part inequality that represents wrist circumference for a male with the following. (a) a small frame (b) a medium frame (c) a large frame

Express each set in the simplest interval form. $$ (-\infty,-1] \cap[-4, \infty) $$
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