6.4 Creating and Evaluating Algebraic Expressions
Key Takeaways
- Algebraic expressions accounts for 2 to 3 of the 20 QAS questions.
- An expression has no equals sign and is evaluated; an equation has one and is solved.
- Like terms share identical variables raised to identical powers, so 3x squared and 5x cannot be combined.
- Translating 'less than' reverses the order of subtraction, making 'five less than x' equal to x minus 5.
- Substituting a test value into two expressions is a fast, reliable check for equivalence.
The Foundation Area
Algebraic expressions: Creating and evaluating expressions to represent situations, and using properties of operations to combine like terms and identify equivalent expressions.
This area carries 2 to 3 of the 20 QAS questions directly, but its real weight is larger — the Linear equations and Linear applications and graphs areas both depend on the translation skills built here.
Expression Versus Equation
- An expression has no equals sign: $3x + 7$. You evaluate or simplify it.
- An equation has one: $3x + 7 = 22$. You solve it.
Questions asking you to "write an expression" want no equals sign, and an answer choice containing one is wrong on form alone.
Translating Words to Algebra
| Phrase | Operation |
|---|---|
| sum, total, increased by, more than | + |
| difference, decreased by, less than, fewer than | − |
| product, times, of, twice, triple | × |
| quotient, per, divided by, ratio of | ÷ |
| is, equals, results in, yields | = |
| at most / no more than | ≤ |
| at least / no less than | ≥ |
The Reversal Traps
Two phrases reverse the order of their operands, and both are tested deliberately.
"Less than"
Five less than x → $x - 5$, not $5 - x$.
"Subtracted from"
Five subtracted from x → $x - 5$.
But "x less than five" → $5 - x$. The order in the phrase is the reverse of the order in the expression, so read carefully.
Compare with a phrase that does not reverse:
x decreased by five → $x - 5$ (same order)
Division phrasing behaves similarly:
The quotient of x and 4 → $x/4$ 4 divided into x → $x/4$ x divided by 4 → $x/4$ 4 divided by x → $4/x$
Building Expressions From Context
A technician charges a $65 service fee plus $48 per hour. Write an expression for the total cost of a job lasting h hours.
The fee is fixed; the hourly charge scales with $h$:
$65 + 48h$
The structure constant + rate × variable describes a large share of QAS application items. Identifying which quantity is fixed and which scales is the entire modeling step.
A tank holds 400 gallons and drains at 12 gallons per minute. Write an expression for the volume remaining after m minutes.
$400 - 12m$
Draining means the rate is subtracted.
Consecutive Integers
| Description | Expressions |
|---|---|
| Consecutive integers | $n$, $n+1$, $n+2$ |
| Consecutive even integers | $n$, $n+2$, $n+4$ |
| Consecutive odd integers | $n$, $n+2$, $n+4$ |
Consecutive even and odd integers use the same pattern — the difference is whether $n$ starts even or odd. Students often write $n+1$, $n+3$ for odd integers, which produces integers of alternating parity.
Evaluating Expressions
Substitute and apply the order of operations. Parentheses around substituted values prevent sign errors.
Evaluate $-2x^2 + 5x - 3$ at $x = -3$.
$-2(-3)^2 + 5(-3) - 3$ $= -2(9) - 15 - 3$ $= -18 - 15 - 3 = -36$
The critical step is $(-3)^2 = 9$, not $-9$. Exponentiation precedes the coefficient's multiplication, and the parentheses make that visible.
Contrast: $-3^2 = -9$, because without parentheses the exponent binds only to the 3 and the negative sign is applied afterward. Distinguishing $(-3)^2$ from $-3^2$ is one of the highest-frequency arithmetic errors on the test.
Combining Like Terms
Like terms have identical variable parts — same variables, same exponents.
$3x^2 + 5x - 2x^2 + 7x = (3-2)x^2 + (5+7)x = x^2 + 12x$
$x^2$ and $x$ are not like terms and never combine. Neither are $xy$ and $x^2$.
The Distributive Property
$3(2x - 5) = 6x - 15$
With a negative multiplier, every sign flips:
$-4(3x - 7) = -12x + 28$
A leading minus sign before a parenthesis distributes $-1$:
$8 - (3x - 2) = 8 - 3x + 2 = 10 - 3x$
Producing $8 - 3x - 2$ is among the most common algebra errors on the QAS test.
Identifying Equivalent Expressions
Two expressions are equivalent if they produce the same value for every input.
The Substitution Check
The fastest reliable method: pick a convenient test value and evaluate both.
Is $2(3x - 4) + 5$ equivalent to $6x - 3$?
Try $x = 2$:
- $2(6 - 4) + 5 = 2(2) + 5 = 9$
- $6(2) - 3 = 9$ ✓
Try $x = 0$ to confirm:
- $2(0 - 4) + 5 = -8 + 5 = -3$
- $6(0) - 3 = -3$ ✓
Equivalent. Use two test values, avoiding 0 and 1 alone, since those can make different expressions coincide by accident.
Algebraic Verification
$2(3x - 4) + 5 = 6x - 8 + 5 = 6x - 3$ ✓
Substitution is faster under test conditions; algebraic expansion is more certain. Use substitution to screen the choices, then expand to confirm the survivor.
A gym charges a one-time $40 registration fee plus $29 per month. Which expression gives the total cost for m months?
Evaluate $-x^2 + 4x - 6$ when $x = -2$.
Simplify $7 - (4x - 3)$.