Introduction to The Language of Algebra

Use Variables and Algebraic Symbols

• The sum of [latex]5[/latex] and [latex]3[/latex] means add [latex]5[/latex] plus [latex]3[/latex], which we write as [latex]5+3[/latex].
• The difference of [latex]9[/latex] and [latex]2[/latex] means subtract [latex]9[/latex] minus [latex]2[/latex], which we write as [latex]9 - 2[/latex].
• The product of [latex]4[/latex] and [latex]8[/latex] means multiply [latex]4[/latex] times [latex]8[/latex], which we can write as [latex]4\cdot 8[/latex].
• The quotient of [latex]20[/latex] and [latex]5[/latex] means divide [latex]20[/latex] by [latex]5[/latex], which we can write as [latex]20\div 5[/latex].

[latex]a<b[/latex] is read [latex]a[/latex] is less than [latex]b[/latex] [latex]a[/latex] is to the left of [latex]b[/latex] on the number line

[latex-display]a>b[/latex] is read [latex]a[/latex] is greater than [latex]b[/latex-display] [latex]a[/latex] is to the right of [latex]b[/latex] on the number line

[latex]a<b\text{ is equivalent to }b>a[/latex].

For example, [latex]7<11\text{ is equivalent to }11>7[/latex].

[latex]a>b\text{ is equivalent to }b<a[/latex].

.For example, [latex]17>4\text{ is equivalent to }4<17[/latex]

Example

The information in the table below compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol [latex]\text{=},\text{<},\text{ or }\text{>}[/latex] in each expression to compare the fuel economy of the cars. (credit: modification of work by Bernard Goldbach, Wikimedia Commons)

1. MPG of Prius_____ MPG of Mini Cooper
2. MPG of Versa_____ MPG of Fit
3. MPG of Mini Cooper_____ MPG of Fit
4. MPG of Corolla_____ MPG of Versa
5. MPG of Corolla_____ MPG of Prius

Answer: Solution

1.
MPG of Prius____MPG of Mini Cooper
Find the values in the chart.	48____27
Compare.	48 > 27
MPG of Prius > MPG of Mini Cooper

2.
MPG of Versa____MPG of Fit
Find the values in the chart.	26____27
Compare.	26 < 27
MPG of Versa < MPG of Fit

3.
MPG of Mini Cooper____MPG of Fit
Find the values in the chart.	27____27
Compare.	27 = 27
MPG of Mini Cooper = MPG of Fit

4.
MPG of Corolla____MPG of Versa
Find the values in the chart.	28____26
Compare.	28 > 26
MPG of Corolla > MPG of Versa

5.
MPG of Corolla____MPG of Prius
Find the values in the chart.	28____48
Compare.	28 < 48
MPG of Corolla < MPG of Prius

[latex]\begin{array}{cc}8\left(14 - 8\right)21 - 3\\\left[2+4\left(9 - 8\right)\right]\\24\div \left\{13 - 2\left[1\left(6 - 5\right)+4\right]\right\}\end{array}[/latex]

Identify Expressions and Equations

Simplify Expressions Containing Exponents

[latex]4\cdot 2+1[/latex] [latex-display]8+1[/latex-display] [latex]9[/latex]

Suppose we have the expression [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/latex]. 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 [latex]2\cdot 2\cdot 2[/latex] as [latex]{2}^{3}[/latex] and [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/latex] as [latex]{2}^{9}[/latex]. In expressions such as [latex]{2}^{3}[/latex], the [latex]2[/latex] is called the base and the [latex]3[/latex] is called the exponent. The exponent tells us how many factors of the base we have to multiply.

[latex-display]\text{means multiply three factors of 2}[/latex-display] We say [latex]{2}^{3}[/latex] is in exponential notation and [latex]2\cdot 2\cdot 2[/latex] is in expanded notation.

Exponential Notation

For any expression [latex]{a}^{n},a[/latex] is a factor multiplied by itself [latex]n[/latex] times if [latex]n[/latex] is a positive integer. [latex-display]{a}^{n}\text{ means multiply }n\text{ factors of }a[/latex-display] The expression [latex]{a}^{n}[/latex] is read [latex]a[/latex] to the [latex]{n}^{th}[/latex] power.

[latex]a^2[/latex] is read as "[latex]a[/latex] squared"

[latex]a^3[/latex] is read as "[latex]a[/latex] cubed"

Simplify Expressions Using the Order of Operations

[latex]4+3\cdot 7[/latex]

[latex]\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & & & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }4+3\text{ gives 7.}\hfill & & \hfill 7\cdot 7\hfill \\ \text{And }7\cdot 7\text{ is 49.}\hfill & & \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }3\cdot 7\text{ is 21.}\hfill & & \hfill 4+21\hfill \\ \text{And }21+4\text{ makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}[/latex]

Licenses & Attributions