# 2: Introduction to the Language of Algebra

- Page ID
- 21666

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)You may not realize it, but you already use algebra every day. Perhaps you figure out how much to tip a server in a restaurant. Maybe you calculate the amount of change you should get when you pay for something. It could even be when you compare batting averages of your favorite players. You can describe the algebra you use in specific words, and follow an orderly process. In this chapter, you will explore the words used to describe algebra and start on your path to solving algebraic problems easily, both in class and in your everyday life.

- 2.1: Use the Language of Algebra (Part 1)
- An expression is a number, a variable, or a combination of numbers and variables and operation symbols. An equation is made up of two expressions connected by an equal sign. An inequality is used in algebra to compare two quantities that have different values. Exponential notation is used in algebra to represent a quantity multiplied by itself several times.

- 2.2: Use the Language of Algebra (Part 2)
- When simplifying mathematical expressions perform the operations in the following order: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first. Simplify all expressions with exponents. Perform all multiplication and division in order from left to right. Perform all addition and subtraction in order from left to right. Multiplication and division, and addition and subtraction have equal priority.

- 2.3: Evaluate, Simplify, and Translate Expressions (Part 1)
- To evaluate an algebraic expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations. We can also simplify an expression by combining the like terms. A term is a constant or the product of a constant and one or more variables. Terms that are either constants or have the same variables with the same exponents are like terms.

- 2.4: Evaluate, Simplify, and Translate Expressions (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.

- 2.5: Solving Equations Using the Subtraction and Addition Properties of Equality (Part 1)
- To determine whether a number is a solution to an equation, first substitute the number for the variable in the equation. Then simplify the expressions on both sides of the equation and determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. The Subtraction and Addition Properties of Equality help in solving for the variable in an equation.

- 2.6: Solving Equations Using the Subtraction and Addition Properties of Equality (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.

- 2.7: Find Multiples and Factors (Part 1)
- A number is a multiple of n if it is the product of a counting number and n. If a number m is a multiple of n, then we say that m is divisible by n. If a • b = m, then a and b are factors of m, and m is the product of a and b. To find all the factors of a counting number, divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor. Then, list all the factor pairs and write all the factors in order from smallest to largest.

- 2.8: Find Multiples and Factors (Part 2)
- A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

- 2.9: Prime Factorization and the Least Common Multiple (Part 1)
- The prime factorization of a number is the product of prime numbers that equals the number. This can be found using either the tree method or the ladder method. The tree method involves writing the factors below the number and connecting them to the number with small line segments. The ladder method involves dividing the given number by its smallest prime factor. The composite number is the product of all the primes used in either method, which should give the same result.

- 2.10: Prime Factorization and the Least Common Multiple (Part 2)
- The least common multiple (LCM) is the smallest number that is a multiple of two numbers. The LCM of two numbers can be found by listing their multiples or using the prime factors method. The listing method involves writing out the multiples of each number until the first multiple common to both lists is found. The prime factors method involves writing each number as a product of primes, matching primes vertically when possible, and then multiplying the factors together to obtain the LCM.

*Figure 2.1 - Algebra has a language of its own. The picture shows just some of the words you may see and use in your study of Prealgebra.*

## Contributors

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."