ACT Prep: Pre‑Algebra and Algebra (Fractions, Exponents, Linear Equations, Inequalities)

ACT – Pre‑Algebra and Algebra (Fractions, Exponents, Linear Equations, Inequalities)

ACT Pre-Algebra & Algebra Study Guide: Fractions, Exponents, Linear Equations, Inequalities

What This Is

Pre-algebra and algebra form the backbone of the ACT Math section, appearing in ~30-40% of questions (18-24 problems). These concepts test your ability to manipulate fractions, exponents, linear equations, and inequalities—skills essential for higher math and real-world problem-solving (e.g., calculating discounts, adjusting recipes, or interpreting data trends). A typical ACT question might ask: "If 3x + 5 = 2x – 7, what is the value of x?" Mastering these topics ensures you can quickly solve equations, avoid careless errors, and recognize common traps like sign errors or misapplied exponent rules.

Key Terms & Rules

Fractions

• Simplifying Fractions: Divide numerator and denominator by their greatest common divisor (GCD). Example: 12/18 simplifies to 2/3.
• Adding/Subtracting Fractions: Requires a common denominator. Example: 1/4 + 1/6 = (3/12) + (2/12) = 5/12.
• Multiplying Fractions: Multiply numerators and denominators. Example: (2/3) × (4/5) = 8/15.
• Dividing Fractions: Multiply by the reciprocal of the divisor. Example: (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 = 5/6.
• Mixed Numbers: Convert to improper fractions before operations. Example: 2 1/3 = (2×3 + 1)/3 = 7/3.

Exponents

• Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ. Example: x² × x³ = x⁵.
• Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Example: y⁷ ÷ y² = y⁵.
• Power Rule: (aᵐ)ⁿ = aᵐⁿ. Example: (z³)² = z⁶.
• Negative Exponents: a⁻ⁿ = 1/aⁿ. Example: 5⁻² = 1/25.
• Zero Exponent: a⁰ = 1 (for a ≠ 0). Example: 7⁰ = 1.
• Fractional Exponents: a^(m/n) = (ⁿ√a)ᵐ. Example: 8^(2/3) = (³√8)² = 4.

Linear Equations

• Slope-Intercept Form: y = mx + b, where m = slope (rise/run) and b = y-intercept.
• Point-Slope Form: y – y₁ = m(x – x₁), where (x₁, y₁) is a point on the line.
• Standard Form: Ax + By = C (A, B, C are integers; A ≥ 0).
• Slope Formula: m = (y₂ – y₁)/(x₂ – x₁).
• Parallel Lines: Same slope (m₁ = m₂).
• Perpendicular Lines: Slopes are negative reciprocals (m₁ × m₂ = –1).

Inequalities

• Solving Inequalities: Similar to equations, but reverse the inequality sign when multiplying/dividing by a negative number.
  Example: –2x > 6 → x < –3.
• Graphing Inequalities:
• > or <: Dashed line (not included).
• ≥ or ≤: Solid line (included).
• Shade above for > or ≥, below for < or ≤.
• Compound Inequalities: Combine two inequalities (e.g., –3 < x ≤ 5).

Step-by-Step / Process Flow

Solving a Linear Equation (e.g., 3x + 5 = 2x – 7)

1. Isolate the variable term: Subtract 2x from both sides → x + 5 = –7.
2. Isolate the variable: Subtract 5 from both sides → x = –12.
3. Check your answer: Plug x = –12 back into the original equation → 3(–12) + 5 = –31 and 2(–12) – 7 = –31. ✔️

Solving an Inequality (e.g., –4x + 3 ≤ 19)

1. Isolate the variable term: Subtract 3 from both sides → –4x ≤ 16.
2. Divide by a negative number: Reverse the inequality sign → x ≥ –4.
3. Graph the solution: Draw a number line with a closed circle at –4 and shade to the right.

Simplifying Exponents (e.g., (2x³y⁻²)⁴)

1. Apply the power rule to each term: (2⁴)(x³)⁴(y⁻²)⁴ = 16x¹²y⁻⁸.
2. Simplify negative exponents: 16x¹²/y⁸.

Adding Fractions with Variables (e.g., 3/(x+2) + 4/(x–1))

1. Find a common denominator: (x+2)(x–1).
2. Rewrite each fraction: [3(x–1) + 4(x+2)] / [(x+2)(x–1)].
3. Combine like terms: (3x – 3 + 4x + 8) / (x² + x – 2) = (7x + 5)/(x² + x – 2).

Common Mistakes

Mistake 1: Forgetting to Reverse the Inequality Sign

• Example: Solving –2x > 6 → x > –3 (incorrect).
• Correction: Reverse the sign when dividing by –2 → x < –3.
• Why? Multiplying/dividing by a negative number flips the inequality direction.

Mistake 2: Misapplying Exponent Rules

• Example: (x²)³ = x⁵ (incorrect).
• Correction: (x²)³ = x⁶ (power rule: multiply exponents).
• Why? The power rule applies to exponents, not addition.

Mistake 3: Incorrectly Combining Like Terms

• Example: 3x + 5x² = 8x³ (incorrect).
• Correction: 3x + 5x² cannot be combined (different degrees).
• Why? Only terms with the same variable and exponent can be combined.

Mistake 4: Ignoring Common Denominators in Fractions

• Example: 1/2 + 1/3 = 2/5 (incorrect).
• Correction: 1/2 + 1/3 = 5/6 (common denominator = 6).
• Why? Fractions must share a denominator to add/subtract.

Mistake 5: Sign Errors in Linear Equations

• Example: 4x – 7 = 2x + 5 → 4x – 2x = 5 – 7 → 2x = –2 → x = –1 (correct, but easy to mess up signs).
• Correction: Always move variables to one side and constants to the other, double-checking signs.
• Why? A single sign error can lead to the wrong answer.

Exam Insights

1. Most-Tested Concepts:
2. Linear equations (solving for x, slope, intercepts) appear in ~10-12 questions.
3. Exponent rules (especially negative/fractional exponents) are tested in ~4-6 questions.

4. Inequalities (solving and graphing) appear in ~3-5 questions.

5. Common Distractors:

6. Sign errors: The ACT often includes answer choices with the "wrong sign" (e.g., x = 5 instead of x = –5).
7. Misapplied exponent rules: Look for choices like (x²)³ = x⁵ (incorrect) or x⁻² = –x² (incorrect).

8. Fraction traps: The ACT may include unsimplified fractions (e.g., 4/8 instead of 1/2) as distractors.

9. Tricky Question Types:

10. "Plug-in" questions: The ACT may ask, "Which of the following is equivalent to (2x³)²?" Instead of simplifying, plug in x = 1 to test answer choices.
11. Word problems with inequalities: Phrases like "at least" (≥) or "no more than" (≤) are common.

12. Systems of equations: Often disguised as word problems (e.g., "The sum of two numbers is 10, and their difference is 4. Find the larger number.").

13. Calculator Tips:

14. Use your calculator for fractions (e.g., 3/4 + 5/6) to avoid manual errors.
15. For exponents, use the ^ key (e.g., 2^3 = 8).
16. Graph inequalities on your calculator (e.g., y ≤ 2x + 1) to visualize solutions.

Quick Check Questions

Question 1

If 4(x – 3) = 2x + 6, what is the value of x? A) –3 B) 3 C) 6 D) 9

Answer: D) 9 Explanation: Distribute the 4 → 4x – 12 = 2x + 6 → 2x = 18 → x = 9.

Question 2

Which of the following is equivalent to (3x⁻²y³)²? A) 6x⁻⁴y⁶ B) 9x⁻⁴y⁵ C) 9x⁻⁴y⁶ D) 6x⁴y⁶

Answer: C) 9x⁻⁴y⁶ Explanation: Apply the power rule: (3²)(x⁻²)²(y³)² = 9x⁻⁴y⁶.

Question 3

Solve the inequality: –5x + 7 ≤ 22.
A) x ≤ –3 B) x ≥ –3 C) x ≤ 3 D) x ≥ 3

Answer: B) x ≥ –3 Explanation: Subtract 7 → –5x ≤ 15 → divide by –5 (reverse sign) → x ≥ –3.

Last-Minute Cram Sheet

1. Fractions: Always find a common denominator before adding/subtracting. ⚠️ Never add denominators!
2. Exponents: a⁻ⁿ = 1/aⁿ and (aᵐ)ⁿ = aᵐⁿ. ⚠️ (x²)³ ≠ x⁵!
3. Slope: m = (y₂ – y₁)/(x₂ – x₁). ⚠️ Rise over run, not run over rise!
4. Inequalities: Reverse the sign when multiplying/dividing by a negative. ⚠️ –2x > 6 → x < –3.
5. Linear Equations: Isolate x by moving variables to one side, constants to the other.
6. Parallel Lines: Same slope (m₁ = m₂). ⚠️ Perpendicular lines: m₁ × m₂ = –1.
7. Word Problems: "At least" = ≥, "no more than" = ≤.
8. Plug-In Strategy: For exponent questions, plug in x = 1 to test answer choices.
9. Graphing Inequalities: Solid line for ≥/≤, dashed line for >/<.
10. Common Traps: ⚠️ Sign errors, unsimplified fractions, misapplied exponent rules. Always double-check your work!