Elementary Algebra

Elementary Algebra makes up about 15-20% of the ACT Math section. This area tests your ability to work with variables, solve equations and inequalities, and translate word problems into algebraic expressions.

Solving Linear Equations

To solve an equation, isolate the variable by performing inverse operations on both sides. Always check your answer by substituting back into the original equation.

Verification: 5(10 − 3) + 2 = 5(7) + 2 = 37. And 3(10) + 7 = 37. Both sides match.

Example 2: Solve (2x + 1)/3 = 5
Step 1: Multiply both sides by 3: 2x + 1 = 15
Step 2: Subtract 1: 2x = 14
Step 3: Divide by 2: x = 7
Check: (2(7) + 1)/3 = 15/3 = 5. Correct.

Example 3: Solve 3(2x − 4) = 2(x + 6)
Distribute: 6x − 12 = 2x + 12
Subtract 2x: 4x − 12 = 12
Add 12: 4x = 24 → x = 6

Solving Linear Inequalities

Inequalities are solved like equations, with one critical exception: when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Example 1: Solve −4x + 6 ≥ 22
Step 1: Subtract 6: −4x ≥ 16
Step 2: Divide by −4 (flip the sign!): x ≤ −4

Example 2: Solve 2(x − 3) < 5x + 9
Distribute: 2x − 6 < 5x + 9
Subtract 5x: −3x − 6 < 9
Add 6: −3x < 15
Divide by −3 (flip!): x > −5

Translating Word Problems

Common translations:

• "Is" or "equals" → =
• "More than" or "increased by" → +
• "Less than" or "decreased by" → −
• "Times" or "product of" → ×
• "Quotient of" or "divided by" → ÷
• "A number" → x (or any variable)
• "Consecutive integers" → x, x+1, x+2, ...
• "Consecutive even/odd integers" → x, x+2, x+4, ...

Example 1: "Five less than three times a number is 22." Translate: 3x − 5 = 22. Solve: 3x = 27, so x = 9.

Example 2: "The sum of three consecutive integers is 72." Translate: x + (x+1) + (x+2) = 72. Simplify: 3x + 3 = 72 → 3x = 69 → x = 23. The integers are 23, 24, 25.

Factoring Polynomials

Example: Factor 2x² − 8x − 10.
Step 1 — GCF: Factor out 2: 2(x² − 4x − 5)
Step 2 — Trinomial: Find m, n where m + n = −4 and m × n = −5 → m = −5, n = 1
Result: 2(x − 5)(x + 1)

Operations with Polynomials

Adding/Subtracting: Combine like terms. (3x² + 2x − 5) + (x² − 4x + 1) = 4x² − 2x − 4.

Multiplying with FOIL: For binomials, multiply First, Outer, Inner, Last terms.
Example: (2x + 3)(x − 5) = 2x² − 10x + 3x − 15 = 2x² − 7x − 15

Example: Expand (x + 4)²
This is NOT x² + 16. Use FOIL or the pattern (a + b)² = a² + 2ab + b²:
(x + 4)² = x² + 2(x)(4) + 16 = x² + 8x + 16

Substitution and Evaluation

Replace variables with given values and compute carefully using order of operations.
Example 1: If f(x) = 2x² − 3x + 1 and x = −2, then f(−2) = 2(4) − 3(−2) + 1 = 8 + 6 + 1 = 15.

Example 2: If a = 3 and b = −2, evaluate 2a² − 3ab + b².
2(9) − 3(3)(−2) + (4) = 18 + 18 + 4 = 40.

Systems of Two Equations

Solve using substitution (solve one equation for a variable, plug into the other) or elimination (add/subtract equations to cancel a variable).

Example (Elimination): Solve: 2x + 3y = 12 and 4x − 3y = 6
Add both equations: 6x = 18 → x = 3. Substitute back: 2(3) + 3y = 12 → 3y = 6 → y = 2. Solution: (3, 2).

Example (Substitution): Solve: y = 2x − 1 and 3x + y = 14
Substitute: 3x + (2x − 1) = 14 → 5x − 1 = 14 → 5x = 15 → x = 3. Then y = 2(3) − 1 = 5. Solution: (3, 5).

Exponent Rules in Algebra

• x^a × x^b = x^(a+b)
• x^a / x^b = x^(a−b)
• (x^a)^b = x^(ab)
• (xy)^a = x^a × y^a

Common Mistakes

Quick Reference — Formula Sheet

ACT Strategies for Elementary Algebra

• Balance the equation: Always perform the same operation on both sides.
• Underline keywords: In word problems, underline key phrases and translate them to math before solving.
• Backsolving: Plug answer choices into the equation — start with choice B (the middle value among 4 choices) to save time.
• Substitution vs. Elimination: Use substitution when one variable is already isolated; use elimination when both equations are in standard form.
• GCF first: When factoring, always check for a greatest common factor before trying other methods.
• Smart number picking: Plug in easy numbers like 2, 3, or 5 (avoid 0 and 1) to test expressions and eliminate wrong answers.
• Check sign flips: The most common mistake on inequality problems is forgetting to flip the sign when dividing by a negative.
• Verify with substitution: After solving any equation, plug your answer back in. This takes 10 seconds and catches careless errors.

Practice Walkthrough

ACT-Style Problem: If 3(2x − 4) − 2(x + 5) = 14, what is the value of x?

A. 5    B. 7    C. 9    D. 11

Algebraic approach:
Distribute: 6x − 12 − 2x − 10 = 14
Combine: 4x − 22 = 14
Add 22: 4x = 36
Divide: x = 9 → Answer C.

Backsolving approach: Try choice B (x = 7): 3(14 − 4) − 2(7 + 5) = 3(10) − 2(12) = 30 − 24 = 6. Not 14, and too small. Try choice C (x = 9): 3(18 − 4) − 2(9 + 5) = 3(14) − 2(14) = 42 − 28 = 14. Correct! Answer: C