Algebraic Expressions: Building and Reading Expressions with Variables

Expression Anatomy: What an Algebraic Expression Is Made Of

An algebraic expression is a mathematical “phrase” built from numbers, variables, and operations. Unlike an equation, it does not include an equals sign. To build and read expressions correctly, focus on these parts:

Variables, numbers, and operations

• Variable: a letter that stands for a number (such as x, n, t).
• Constant: a fixed number (such as 5, -2, 1/3).
• Operations: addition (+), subtraction (-), multiplication (× or implied), division (/ or a fraction bar), exponents (^).

Terms and how they are separated

A term is a piece of an expression separated by + or -. For example, in 4x - 7 + 2x^2, the terms are 4x, -7, and 2x^2.

Coefficients and factors

A coefficient is the number multiplying a variable. In 3x, the coefficient is 3. In -x, the coefficient is -1 (because -x = -1·x).

Factors are things being multiplied. In 2(x + 4), the factors are 2 and (x + 4).

Multiplication Without the “×” Symbol (Juxtaposition)

In algebra, multiplication is often written without a multiplication sign. This is called juxtaposition. These all mean multiplication:

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• 3x means 3 × x
• ab means a × b
• 2(x + 4) means 2 × (x + 4)
• x(x - 1) means x × (x - 1)

Tip for clarity: Avoid writing a variable right next to a number in a way that looks like a two-digit number. For example, write 2x (not 2 x with confusing spacing), and avoid using l (lowercase L) as a variable because it can look like 1.

Division and Fractions: Writing and Reading Clearly

Division can be written using a slash or a fraction bar. Parentheses often matter.

• x/5 means “x divided by 5” (or “one-fifth of x”).
• 5/x means “5 divided by x.”
• (x+4)/2 means “the quantity x+4 divided by 2.”
• x+4/2 means “x plus 4/2,” because division happens before addition.

When you want the entire top part to be divided, use parentheses: (x + 4)/2.

Exponents: Compact Repeated Multiplication

An exponent tells how many times a base is multiplied by itself.

• x^2 means x · x (read “x squared”).
• x^3 means x · x · x (read “x cubed”).
• 3x^2 means 3 · x^2, not (3x)^2.

Parentheses change meaning with exponents:

• x^2 + 1 means “square x, then add 1.”
• (x + 1)^2 means “add 1 to x, then square the result.”

Parentheses: Grouping That Changes Meaning

Parentheses tell you to treat something as a single group. This affects multiplication, division, and exponents.

Example: Multiplying a group

Compare:

• 2x + 4 means “two times x, then add 4.”
• 2(x + 4) means “two times the quantity x + 4.”

These are not the same expression. Parentheses make x + 4 act like one unit.

Example: Dividing a group

• x + 4/2 means x + (4/2).
• (x + 4)/2 means the entire sum x + 4 is divided by 2.

Guided Conversions: Verbal Phrases ↔ Algebraic Expressions

When converting words to symbols, identify (1) the variable, (2) the operation words, and (3) what is grouped together.

Common phrase translations

Step-by-step examples

1) “Five more than a number”

• Let the number be n.
• “More than” indicates addition.
• Expression: n + 5.

2) “Three times a number, then subtract 7”

• Let the number be x.
• “Three times a number” → 3x.
• “Then subtract 7” → 3x - 7.

3) “A number divided by 5, plus 2”

• Let the number be x.
• “Divided by 5” → x/5.
• “Plus 2” → x/5 + 2.

4) “Two times the quantity (a number plus 4)”

• Let the number be x.
• “Quantity (a number plus 4)” means parentheses: (x + 4).
• “Two times” means multiply by 2: 2(x + 4).

5) “The square of the sum of a number and 4”

• Let the number be x.
• “Sum of a number and 4” → (x + 4).
• “Square of” means exponent 2 applied to the whole group: (x + 4)^2.

Symbolic to verbal (reading expressions)

Practice reading expressions in a way that makes grouping clear.

• 3x: “three times x.”
• x/5: “x divided by 5.”
• x^2: “x squared.”
• 2(x + 4): “two times the quantity x + 4.”
• (x + 4)/2: “the quantity x + 4 divided by 2.”

Practice Set A: Build Expressions from Words

Write an algebraic expression for each phrase. Use x as the variable unless another letter is given.

• 1) seven more than a number
• 2) nine less than a number
• 3) four times a number
• 4) a number divided by 8
• 5) the square of a number
• 6) three more than twice a number
• 7) half of a number, plus 6
• 8) the sum of a number and 4, multiplied by 2

Practice Set B: Translate Symbols into Words

Write each expression in words. Make grouping clear by saying “the quantity” when needed.

• 1) 5x
• 2) x/3
• 3) x^2 + 4
• 4) (x + 4)^2
• 5) 2(x + 4)
• 6) (x + 4)/2

Practice Set C: Mixed and More Complex

Convert each verbal phrase into an expression. Pay attention to “sum,” “difference,” and grouping.

• 1) the difference between a number and 12
• 2) the sum of a number and 12, divided by 3
• 3) three times the square of a number
• 4) the square of three times a number
• 5) twice the difference between a number and 5
• 6) a number divided by the quantity 5 plus 1

Error-Spotting: What’s Wrong and How to Fix It

Each item shows a common mistake. Identify the issue and write a corrected expression.

1) Missing multiplication symbol or unclear grouping

• Incorrect: 3(x)4
• What’s wrong: It’s unclear whether 4 is being multiplied and how the factors are intended.
• Correct (if the intent is multiply all factors): 3x·4 or 12x.

2) Using × like a variable or mixing symbols

• Incorrect: 3× (intended to mean “three times x”)
• What’s wrong: × is an operation symbol, not a variable.
• Correct: 3x.

3) Parentheses needed for “the quantity”

• Incorrect (for “two times the sum of x and 4”): 2x + 4
• What’s wrong: This means “two times x, plus 4,” not “two times (x + 4).”
• Correct: 2(x + 4).

4) Confusing exponent placement

• Incorrect (for “the square of x plus 4”): (x + 4)^2
• What’s wrong: This squares the entire sum. “Square of x plus 4” means square first, then add 4.
• Correct: x^2 + 4.

5) Ambiguous division without parentheses

• Incorrect (for “the sum of x and 4 divided by 2”): x + 4/2
• What’s wrong: Only the 4 is divided by 2 due to operation order.
• Correct: (x + 4)/2.

6) Writing a variable next to a variable when addition is intended

• Incorrect (for “x plus y”): xy
• What’s wrong: xy means multiplication, not addition.
• Correct: x + y.