Algebra is a precise language for describing quantities and relationships. On the SAT, the hard part is often not calculation—it is deciding exactly what a sentence says and preserving that structure in symbols.
The language of algebra
- Variable
- A symbol, such as \(x\), that represents a number whose value may be unknown or may change.
- Constant
- A fixed numerical value. In \(4x+9\), \(9\) is the constant term.
- Algebraic expression
- A combination of numbers, variables, operations, and grouping symbols. An expression such as \(3(x-2)+5\) does not contain an equality sign.
Translate operation words
| Operation | Common language | Original example | Expression |
|---|---|---|---|
| Addition | sum; more than; increased by | Eight more than \(n\) | \(n+8\) |
| Subtraction | difference; less than; decreased by; minus | Four less than \(p\) | \(p-4\) |
| Multiplication | product; times; of | Three-fourths of \(q\) | \(\frac34q\) |
| Division | quotient; divided by; per | The quotient of \(r\) and \(5\) | \(\frac r5\) |
Read similar phrases precisely
Five less than x
\(x-5\): the amount \(5\) is subtracted from \(x\).
Five minus x
\(5-x\): the minus sign follows the spoken order.
Grouping and multi-step phrases
Translate from the innermost noun phrase outward. ‘Twice the difference of \(x\) and \(3\)’ first creates \(x-3\), then doubles the complete difference: \(2(x-3)\). Without parentheses, \(2x-3\) doubles only \(x\).
Model a two-stage situation
A library orders \(b\) cartons with \(18\) books each, then donates \(7\) books from every carton. Write an expression for the books retained.
- Interpret
Each carton is one repeated group. The change of \(7\) happens once per carton.
- Set up
One carton retains \(18-7\) books, so \(b\) cartons retain \(b(18-7)\).
- Work
Simplify the constant inside the grouping: \(b(11)=11b\).
- Check
The coefficient \(11\) has units of books per carton, so \(11b\) has units of books.
Evaluate and interpret expressions
To evaluate, substitute the given value everywhere the variable appears, using parentheses around negative numbers. For \(3x^2-2x\) at \(x=-2\), write \(3(-2)^2-2(-2)\), not \(3-2^2-2-2\).
Check your understanding
Which expression means one-third of the sum of \(a\) and \(6\)?
- \(a/3+6\)
- \((a+6)/3\)
- \(3(a+6)\)
- \(a+2\)
Show answer and explanation
Answer: \((a+6)/3\)
The complete sum \(a+6\) is multiplied by \(1/3\), so it must be grouped.
Key takeaways
What to remember
- Name the operation before writing symbols.
- Preserve order for subtraction and division; treat ‘less than’ as a reversal warning.
- Group a multi-term quantity before an outside operation.
- Use parentheses for negative substitutions and use units to check models.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.