1.2 Use the language of algebra  (Page 7/18)

Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.

Key concepts

• Notation                      The result is…
  
  $\begin{array}{cccc}\circ a+b\hfill & & & \text{the sum of}a\text{and}b\hfill \\ \circ a-b\hfill & & & \text{the difference of}a\text{and}b\hfill \\ \circ a·b,ab,\left(a\right)\left(b\right)\left(a\right)b,a\left(b\right)\hfill & & & \text{the product of}a\text{and}b\hfill \\ \circ a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}\hfill & & & \text{the quotient of}a\text{and}b\hfill \end{array}$
• Inequality
  
  $\begin{array}{cccc}\circ a<b\text{is read}\text{“}a\text{is less than}b\text{”}\hfill & & & a\text{is to the left of}b\text{on the number line}\hfill \\ \circ a>b\text{is read}\text{“}a\text{is greater than}b\text{”}\hfill & & & a\text{is to the right of}b\text{on the number line}\hfill \end{array}$
• Inequality Symbols                 Words
  
  $\begin{array}{cccc}\circ a\ne b\hfill & & & a\text{is}\mathbf{\text{not equal to}}b\hfill \\ \circ a<b\hfill & & & a\text{is}\mathbf{\text{less than}}b\hfill \\ \circ a\le b\hfill & & & a\text{is}\mathbf{\text{less than or equal to}}b\hfill \\ \circ a>b\hfill & & & a\text{is}\mathbf{\text{greater than}}b\hfill \\ \circ a\ge b\hfill & & & a\text{is}\mathbf{\text{greater than or equal to}}b\hfill \end{array}$
• Grouping Symbols
  • Parentheses $\left(\right)$
  • Brackets $\left[\right]$
  • Braces $\left\{\right\}$
• Exponential Notation
  • $a^n$ means multiply $a$ by itself, $n$ times. The expression $a^n$ is read $a$ to the $n^{th}$ power.
• Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
  1. Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
  2. Exponents: Simplify all expressions with exponents.
  3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
  4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
• Combine Like Terms
  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add or subtract the coefficients and keep the same variable for each group of like terms.