# 2.1 Use the language of algebra  (Page 3/8)

 Page 3 / 8

Use [link] to fill in the appropriate $\text{symbol},\text{=},\text{<},\text{or}\phantom{\rule{0.2em}{0ex}}\text{>}.$

1. MPG of Prius_____MPG of Versa
2. MPG of Mini Cooper_____ MPG of Corolla

1. >
2. >

Use [link] to fill in the appropriate $\text{symbol},\text{=},\text{<},\text{or}\phantom{\rule{0.2em}{0ex}}\text{>}.$

1. MPG of Fit_____ MPG of Prius
2. MPG of Corolla _____ MPG of Fit

1. <
2. <

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. [link] lists three of the most commonly used grouping symbols in algebra.

Common Grouping Symbols
parentheses $\left(\phantom{\rule{0.5em}{0ex}}\right)$
brackets $\left[\phantom{\rule{0.5em}{0ex}}\right]$
braces $\left\{\phantom{\rule{0.5em}{0ex}}\right\}$

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

$8\left(14-8\right)\phantom{\rule{4em}{0ex}}21-3\left[2+4\left(9-8\right)\right]\phantom{\rule{4em}{0ex}}24÷\left\{13-2\left[1\left(6-5\right)+4\right]\right\}$

## Identify expressions and equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.

In algebra, we have expressions and equations . An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:

Expression Words Phrase
$3+5$ $3\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}5$ the sum of three and five
$n-1$ $n$ minus one the difference of $n$ and one
$6·7$ $6\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}7$ the product of six and seven
$\frac{x}{y}$ $x$ divided by $y$ the quotient of $x$ and $y$

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation    is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

Equation Sentence
$3+5=8$ The sum of three and five is equal to eight.
$n-1=14$ $n$ minus one equals fourteen.
$6·7=42$ The product of six and seven is equal to forty-two.
$x=53$ $x$ is equal to fifty-three.
$y+9=2y-3$ $y$ plus nine is equal to two $y$ minus three.

## Expressions and equations

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.

Determine if each is an expression or an equation:

1. $\phantom{\rule{0.2em}{0ex}}16-6=10$
2. $\phantom{\rule{0.2em}{0ex}}4·2+1$
3. $\phantom{\rule{0.2em}{0ex}}x÷25$
4. $\phantom{\rule{0.2em}{0ex}}y+8=40$

## Solution

1. This is an equation—two expressions are connected with an equal sign.
2. This is an expression—no equal sign.
3. This is an expression—no equal sign.
4. This is an equation—two expressions are connected with an equal sign.

Determine if each is an expression or an equation:

1. $\phantom{\rule{0.2em}{0ex}}23+6=29\phantom{\rule{0.4em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}7·3-7$

1. equation
2. expression

Determine if each is an expression or an equation:

1. $\phantom{\rule{0.2em}{0ex}}y÷14\phantom{\rule{0.4em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}x-6=21$

1. expression
2. equation

## Simplify expressions with exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4·2+1$ we’d first multiply $4·2$ to get $8$ and then add the $1$ to get $9.$ A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

$4·2+1$
$8+1$
$9$

Suppose we have the expression $2·2·2·2·2·2·2·2·2.$ We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write $2·2·2$ as ${2}^{3}$ and $2·2·2·2·2·2·2·2·2$ as ${2}^{9}.$ In expressions such as ${2}^{3},$ the $2$ is called the base and the $3$ is called the exponent. The exponent tells us how many factors of the base we have to multiply.

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