Algebraic Expressions, Equations, and Inequalities

Algebraic Expressions

Writing Expressions

Algebraic expressions are used to represent verbal descriptions of a problem.

When verbal statements communicate how quantities are related to each other, they can be translated to algebraic expressions. It is helpful to know the common words associated with the four fundamental operations and understand how these verbal phrases translate to mathematical expressions.

When a value is referred to as simply "a number," the value is unknown and should be represented by a variable. It is important to remember that there is no commutative property of subtraction or division. For these operations, the order of the values is indicated in the verbal phrase.

Operation Algebraic Expression Verbal Expression
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a+8a+8
  • A number plus 8
  • The sum of a number and 8
  • 8 added to a number
  • A number increased by 8
  • 8 more than a number
k13k-13
  • A number minus 13
  • 13 less than a number
  • The difference of a number and 13
  • 13 subtracted from a number
  • 13 fewer than a number
  • A number decreased by 13
×\times
3t3t
3t3 \cdot t
3×t3 \times t
  • 3 times a number
  • The product of 3 and a number
  • 3 multiplied by a number
  • 3 of a number
÷\div
x5\frac{x}{5}
x÷5x \div 5
  • A number divided by 5
  • The quotient of a number and 5

Step-By-Step Example
Translating Words into Algebraic Expressions

Tim and Ben work at the same store. Last week, Tim worked 4 hours fewer than twice as many hours as Ben.

Write an algebraic expression that represents the total number of hours that both Tim and Ben worked.

Step 1

Choose a variable.

The number of hours Tim worked is described in relation to the number of hours Ben worked. So, let bb represent the number of hours Ben worked.
b=Number of hours Ben workedb=\text{Number of hours Ben worked}
Step 2
Write an expression for the number of hours Tim worked. The phrase "twice as many hours" means to multiply by 2, which is 2b2b. The phrase "4 hours fewer than" means to subtract 4. So, the number of hours Tim worked is:
2b4=Number of hours Tim worked2b-4=\text{Number of hours Tim worked}
Step 3
The total number of hours is the sum of the hours that Tim worked and the hours that Ben worked:
b+2b4b+2b-4
Simplify by combining like terms:
3b43b-4
Solution
The total number of hours that Tim and Ben worked last week is:
3b43b-4
The variable bb represents the number of hours that Ben worked.

Evaluating Expressions

To evaluate an algebraic expression, substitute the given values of the variables in the expression and simplify.
When simplifying an expression with variables, the result is usually another expression with variables. For example:
3x+5+2x2x+7\begin{gathered}3x+5+2-x\\2x+7\end{gathered}
When evaluating an expression, values of the variables are given, and the result is usually a number. To evaluate an algebraic expression, substitute the given value for each occurence of the variable, and simplify the resulting numerical expression. If the expression has more than one operation, follow the order of operations.
Step-By-Step Example
Calculating Expressions
Evaluate the expression for x=1x=-1:
x23x+7x^2-3x+7
Step 1
Replace each occurrence of xx with –1.
x23x+7(1)23(1)+7\begin{gathered}x^2-3x+7\\({\color{#c42126}{-1}})^2-3({\color{#c42126}{-1}})+7\end{gathered}
Step 2
Simplify according to the order of operations. There are no grouping symbols. So, simplify the exponents.
(1)23(1)+713(1)+7\begin{gathered}{\color{#c42126}{(-1)^{2}}}-3(-1)+7\\{\color{#c42126}{1}}-3(-1)+7\end{gathered}
Step 3
Perform the multiplication.
1(3)+71+3+7\begin{gathered}1-(-3)+7\\1+3+7\end{gathered}
Solution
Add from left to right.
4+711\begin{gathered}4+7\\11\end{gathered}