How to Simplify Algebraic Expressions | ACT Math Guide for Grades 9-12
Simplifying algebraic expressions is one of the most fundamental skills you’ll need for the ACT Math section. Whether you’re dealing with polynomials, fractions, or complex equations, the ability to combine like terms and apply the distributive property efficiently can save you precious time and help you avoid careless mistakes. This skill appears in approximately 15-20% of ACT Math questions, making it absolutely essential for achieving your target score.
ACT SCORE BOOSTER: Master This Topic for 3-5 Extra Points!
This topic appears in most ACT tests (8-12 questions) on the ACT Math section. Understanding it thoroughly can add 3-5 points to your composite score. Let’s break it down with proven strategies that work!
🚀 Jump to ACT Strategy →📚 Understanding Algebraic Simplification
Simplifying algebraic expressions means reducing them to their most compact and manageable form without changing their value. This process involves two primary techniques that you’ll use constantly on the ACT:
🔹 Combining Like Terms: Grouping and adding or subtracting terms that have the same variable(s) raised to the same power(s).
🔹 Distributive Property: Multiplying a term outside parentheses by each term inside the parentheses: $$a(b + c) = ab + ac$$
Why This Matters for the ACT: These skills form the foundation for solving equations, working with polynomials, and tackling word problems. On the ACT, you’ll encounter these concepts in approximately 8-12 questions per test, often embedded within more complex problems. Mastering simplification helps you work faster and more accurately, giving you more time for challenging questions.
Score Impact: Students who master algebraic simplification typically see a 3-5 point improvement in their ACT Math score, as it reduces calculation errors and speeds up problem-solving across multiple question types.
📐 Key Rules & Properties
1️⃣ Identifying Like Terms
Like terms have identical variable parts (same variables with same exponents):
- ✅ Like terms: $$3x$$ and $$7x$$ (both have $$x$$)
- ✅ Like terms: $$5x^2$$ and $$-2x^2$$ (both have $$x^2$$)
- ✅ Like terms: $$4xy$$ and $$9xy$$ (both have $$xy$$)
- ❌ NOT like terms: $$3x$$ and $$3x^2$$ (different exponents)
- ❌ NOT like terms: $$5x$$ and $$5y$$ (different variables)
2️⃣ Combining Like Terms
Add or subtract the coefficients (numbers in front) and keep the variable part unchanged:
$$3x + 7x = 10x$$
$$5x^2 – 2x^2 = 3x^2$$
$$4xy + 9xy – 2xy = 11xy$$
3️⃣ Distributive Property
Multiply the term outside by each term inside the parentheses:
Basic Form: $$a(b + c) = ab + ac$$
Example: $$3(x + 5) = 3x + 15$$
With Subtraction: $$2(3x – 4) = 6x – 8$$
Negative Distribution: $$-4(2x + 3) = -8x – 12$$
4️⃣ Order of Operations (PEMDAS)
When simplifying, always follow this order:
- Parentheses (use distributive property if needed)
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
✅ Step-by-Step Examples
1 Example 1: Combining Like Terms
Problem: Simplify $$5x + 3y – 2x + 7y – 4$$
Step 1: Identify like terms
Group terms with the same variables together:
Terms with $$x$$: $$5x$$ and $$-2x$$
Terms with $$y$$: $$3y$$ and $$7y$$
Constant term: $$-4$$
Step 2: Rearrange to group like terms
$$(5x – 2x) + (3y + 7y) – 4$$
Step 3: Combine coefficients
$$5x – 2x = 3x$$
$$3y + 7y = 10y$$
Step 4: Write the final answer
✓ Final Answer: $$3x + 10y – 4$$
⏱️ ACT Time Tip: This should take 20-30 seconds on the ACT. Practice identifying like terms at a glance!
2 Example 2: Distributive Property
Problem: Simplify $$3(2x – 5) + 4x$$
Step 1: Apply the distributive property
Multiply 3 by each term inside the parentheses:
$$3 \times 2x = 6x$$
$$3 \times (-5) = -15$$
Result: $$6x – 15 + 4x$$
Step 2: Identify like terms
Terms with $$x$$: $$6x$$ and $$4x$$
Constant: $$-15$$
Step 3: Combine like terms
$$6x + 4x = 10x$$
Step 4: Write the final answer
✓ Final Answer: $$10x – 15$$
⏱️ ACT Time Tip: Distribute first, then combine. This should take 30-40 seconds.
3 Example 3: Complex Expression (ACT-Style)
Problem: Simplify $$2(3x + 4) – 5(x – 2) + 7$$
Step 1: Distribute both terms
$$2(3x + 4) = 6x + 8$$
$$-5(x – 2) = -5x + 10$$ (watch the signs!)
Step 2: Rewrite the expression
$$6x + 8 – 5x + 10 + 7$$
Step 3: Group like terms
$$(6x – 5x) + (8 + 10 + 7)$$
Step 4: Combine like terms
$$6x – 5x = x$$
$$8 + 10 + 7 = 25$$
✓ Final Answer: $$x + 25$$
⏱️ ACT Time Tip: Complex problems like this should take 45-60 seconds. Practice makes perfect!
📝 ACT-Style Practice Questions
Test your understanding with these ACT-style problems. Try solving them on your own before checking the solutions!
Question 1 ⭐ Basic
Which of the following is equivalent to $$7x – 3 + 2x + 9$$?
📖 Show Detailed Solution
Step 1: Identify like terms: $$7x$$ and $$2x$$ are like terms; $$-3$$ and $$9$$ are constants.
Step 2: Combine $$x$$ terms: $$7x + 2x = 9x$$
Step 3: Combine constants: $$-3 + 9 = 6$$
✓ Correct Answer: A) $$9x + 6$$
Question 2 ⭐⭐ Intermediate
Simplify: $$4(2x – 3) + 5x$$
📖 Show Detailed Solution
Step 1: Apply distributive property: $$4(2x – 3) = 8x – 12$$
Step 2: Rewrite: $$8x – 12 + 5x$$
Step 3: Combine like terms: $$8x + 5x = 13x$$
✓ Correct Answer: A) $$13x – 12$$
Question 3 ⭐⭐ Intermediate
What is the simplified form of $$3x^2 + 5x – 2x^2 + 7 – 3x$$?
📖 Show Detailed Solution
Step 1: Group like terms: $$(3x^2 – 2x^2) + (5x – 3x) + 7$$
Step 2: Combine $$x^2$$ terms: $$3x^2 – 2x^2 = x^2$$
Step 3: Combine $$x$$ terms: $$5x – 3x = 2x$$
Step 4: Constant remains: $$7$$
✓ Correct Answer: A) $$x^2 + 2x + 7$$
Question 4 ⭐⭐⭐ Advanced
Simplify: $$-2(3x – 4) + 5(2x + 1) – 7x$$
📖 Show Detailed Solution
Step 1: Distribute $$-2$$: $$-2(3x – 4) = -6x + 8$$
Step 2: Distribute $$5$$: $$5(2x + 1) = 10x + 5$$
Step 3: Rewrite: $$-6x + 8 + 10x + 5 – 7x$$
Step 4: Combine $$x$$ terms: $$-6x + 10x – 7x = -3x$$
Step 5: Combine constants: $$8 + 5 = 13$$
✓ Correct Answer: C) $$-3x + 13$$
⚠️ Common Mistake: Watch the negative signs when distributing! $$-2 \times (-4) = +8$$, not $$-8$$.
💡 ACT Pro Tips & Tricks
🎯 Tip #1: Circle Like Terms
On test day, quickly circle or underline like terms in different colors (if allowed) or mentally group them. This prevents you from missing terms and speeds up your work.
⚠️ Tip #2: Watch Negative Signs
The most common error is mishandling negative signs during distribution. Remember: $$-a(b – c) = -ab + ac$$. The negative flips both signs inside!
⏱️ Tip #3: Work Left to Right
Process the expression systematically from left to right. Don’t jump around—this leads to missed terms and calculation errors under time pressure.
✓ Tip #4: Quick Mental Check
After simplifying, plug in a simple number (like $$x = 1$$) into both the original and simplified expressions. If they don’t match, you made an error!
🚀 Tip #5: Eliminate Wrong Answers
On multiple choice, eliminate answers with wrong degrees (e.g., if the problem has $$x^2$$, the answer must too) or obviously wrong coefficients. This narrows your options quickly.
📝 Tip #6: Show Your Work (Even Briefly)
Write down at least one intermediate step in your test booklet. This helps you catch errors and makes it easier to pick up where you left off if you need to return to a question.
⚠️ Common Mistakes to Avoid
❌ Mistake #1: Combining Unlike Terms
Wrong: $$3x + 4y = 7xy$$ ← You CANNOT combine different variables!
Correct: $$3x + 4y$$ stays as is (already simplified)
❌ Mistake #2: Forgetting to Distribute to ALL Terms
Wrong: $$3(x + 5) = 3x + 5$$ ← You forgot to multiply 3 by 5!
Correct: $$3(x + 5) = 3x + 15$$
❌ Mistake #3: Sign Errors with Negative Distribution
Wrong: $$-2(x – 3) = -2x – 6$$ ← Wrong sign on the 6!
Correct: $$-2(x – 3) = -2x + 6$$ (negative times negative = positive)
❌ Mistake #4: Combining Terms with Different Exponents
Wrong: $$2x + 3x^2 = 5x^3$$ ← Completely wrong!
Correct: $$2x + 3x^2$$ stays as is (different exponents = not like terms)
🎯 ACT Test-Taking Strategy for Algebraic Simplification
⏱️ Time Management
- Basic simplification: 20-30 seconds
- With distribution: 30-45 seconds
- Complex multi-step: 45-60 seconds
- If you’re taking longer than 60 seconds, mark it and move on—you can return later
🎲 When to Skip
- If you see more than 3 sets of parentheses and you’re running low on time, skip it initially
- If the expression has fractions with variables in denominators, it might be a harder problem—save for later
- Trust your gut: if it looks overwhelming at first glance, mark it and come back with fresh eyes
✂️ Process of Elimination Strategy
- Check the degree: If the problem has $$x^2$$, eliminate answers without $$x^2$$
- Check the constant: Quickly add up all constant terms—eliminate answers with wrong constants
- Check signs: If all terms in the problem are positive, the answer shouldn’t have many negatives
- Plug in $$x = 0$$: This eliminates all variable terms, leaving just constants—a quick check!
🔍 Quick Verification Technique
After simplifying, use the “plug in 1” method:
Example: You simplified $$2(x + 3) + 4x$$ to $$6x + 6$$
Check: Let $$x = 1$$
Original: $$2(1 + 3) + 4(1) = 2(4) + 4 = 12$$
Simplified: $$6(1) + 6 = 12$$ ✓ Match!
🎯 Common ACT Trap Answers
- Sign flip trap: They’ll offer an answer with one sign wrong (e.g., $$6x – 8$$ instead of $$6x + 8$$)
- Incomplete distribution: An answer where distribution was only partially applied
- Combined unlike terms: An answer that incorrectly combines $$x$$ and $$x^2$$ terms
- Forgot a term: An answer missing one of the terms from the original expression
💪 Score Boost Tip: Master these simplification techniques and you’ll not only answer these questions correctly, but you’ll also solve equations, factor polynomials, and tackle word problems much faster—potentially adding 3-5 points to your ACT Math score!
🌍 Real-World Applications
You might wonder, “When will I ever use this?” Here’s the truth: algebraic simplification is everywhere!
💰 Finance & Business
Simplifying profit formulas, combining revenue streams, and calculating compound interest all use these exact skills. Financial analysts simplify complex expressions daily.
🔬 Science & Engineering
Physics formulas, chemical equations, and engineering calculations require constant simplification. Engineers simplify complex systems to make them workable.
💻 Computer Programming
Code optimization involves simplifying algorithms and expressions. Programmers constantly refactor code to make it more efficient—just like simplifying algebra!
📊 Data Analysis
Statistical models and data formulas need simplification for interpretation. Data scientists simplify complex relationships to find meaningful patterns.
🎓 College Connection: These skills are foundational for college courses in mathematics, economics, physics, chemistry, computer science, and engineering. Mastering them now gives you a huge advantage in your first year of college!
❓ Frequently Asked Questions
✍️ Written by Irfan Mansuri
ACT Test Prep Specialist & Educator
IrfanEdu.com • United States
Irfan Mansuri is a dedicated ACT test preparation specialist with over 15 years of experience helping high school students achieve their target scores. As the founder of IrfanEdu.com, he has guided thousands of students through the ACT journey, with many achieving scores of 30+ and gaining admission to their dream colleges. His teaching methodology combines deep content knowledge with proven test-taking strategies, making complex concepts accessible and helping students build confidence. Irfan’s approach focuses not just on memorization, but on true understanding and strategic thinking that translates to higher scores.
📚 Continue Your ACT Math Journey
Now that you’ve mastered simplifying algebraic expressions, take your skills to the next level with these related topics:
- Solving Linear Equations: Use your simplification skills to solve for variables
- Factoring Polynomials: The reverse of distribution—breaking expressions apart
- Working with Quadratic Expressions: Apply these techniques to more complex problems
- Systems of Equations: Simplification is crucial for elimination and substitution methods
- Rational Expressions: Simplify fractions with variables
💡 Study Tip: Practice 5-10 simplification problems daily for two weeks. This builds muscle memory and dramatically reduces errors on test day. Mix basic and complex problems to build confidence at all levels!
🎉 You’ve Got This!
Simplifying algebraic expressions is a foundational skill that will serve you throughout the ACT Math section and beyond. With consistent practice and the strategies you’ve learned today, you’re well on your way to mastering this topic and boosting your score. Remember: every expert was once a beginner. Keep practicing, stay confident, and watch your skills grow!
Quick Reference: Algebraic Simplification Rules
Master these fundamental rules to simplify any algebraic expression with confidence.
1. Distributive Property
a(b + c) = ab + ac
Multiply the term outside the parentheses by each term inside the parentheses.
Example: 3(x + 4) = 3x + 12
2. Like Terms
Terms with identical variable parts
Terms that have the same variables raised to the same powers. Only like terms can be combined.
Example: 5x² and 3x² are like terms
5x² and 3x are NOT like terms
3. Combining Like Terms
ax + bx = (a + b)x
Add or subtract the coefficients of like terms while keeping the variable part unchanged.
Example: 7x + 3x = 10x
9y² − 4y² = 5y²
4. Commutative Property of Addition
a + b = b + a
The order in which you add terms doesn’t matter; the result is the same.
Example: x + 5 = 5 + x
3y + 2x = 2x + 3y
5. Commutative Property of Multiplication
a × b = b × a
The order in which you multiply factors doesn’t matter; the result is the same.
Example: 5 × x = x × 5
3(x + 2) = (x + 2)3
6. Associative Property of Addition
(a + b) + c = a + (b + c)
When adding three or more terms, the grouping doesn’t affect the sum.
Example: (2 + x) + 3 = 2 + (x + 3)
7. Associative Property of Multiplication
(a × b) × c = a × (b × c)
When multiplying three or more factors, the grouping doesn’t affect the product.
Example: (2 × x) × 3 = 2 × (x × 3) = 6x
8. Identity Property of Addition
a + 0 = a
Adding zero to any expression doesn’t change its value.
Example: x + 0 = x
5y² + 0 = 5y²
9. Identity Property of Multiplication
a × 1 = a
Multiplying any expression by 1 doesn’t change its value.
Example: 1 × x = x
1(3x + 2) = 3x + 2
10. Inverse Property of Addition
a + (−a) = 0
Adding a number and its opposite (negative) equals zero.
Example: 5x + (−5x) = 0
3y − 3y = 0
11. Multiplication by Zero
a × 0 = 0
Any expression multiplied by zero equals zero.
Example: 0 × x = 0
0(5x + 3) = 0
12. Distributing a Negative Sign
−(a + b) = −a − b
A negative sign before parentheses means multiply each term inside by −1, changing all signs.
Example: −(x + 3) = −x − 3
−(2y − 5) = −2y + 5
13. Coefficient
In 5x, the coefficient is 5
The numerical factor in a term. If no number is written, the coefficient is 1 or −1.
Example: In 7xy, coefficient = 7
In −x, coefficient = −1
14. Constant Term
A term with no variable
A number by itself without any variables attached. All constants are like terms.
Example: In 3x + 7, the constant is 7
5 + (−2) = 3
15. Variable
A letter representing an unknown number
A symbol (usually a letter) that stands for a number we don’t know yet or that can change.
Example: x, y, z, a, b
In 5x, x is the variable
16. Term
A single number, variable, or product
A part of an expression separated by + or − signs. Can be a number, variable, or their product.
Example: In 3x² + 5x − 7
Terms are: 3x², 5x, and −7
17. Expression
A combination of terms
A mathematical phrase containing numbers, variables, and operations but no equal sign.
Example: 2x + 3
5y² − 4y + 1
18. Simplify
Reduce to simplest form
Combine all like terms and perform all possible operations to write an expression in its shortest form.
Example: 2x + 3x + 5 simplifies to 5x + 5
19. Order of Operations (PEMDAS)
Parentheses, Exponents, Multiply/Divide, Add/Subtract
The sequence in which operations must be performed: parentheses first, then exponents, then multiplication and division (left to right), finally addition and subtraction (left to right).
Example: 2 + 3(4) = 2 + 12 = 14
NOT 5(4) = 20
20. Exponent
x² means x × x
A small number written above and to the right of a base number, indicating how many times to multiply the base by itself.
Example: x³ = x × x × x
2⁴ = 2 × 2 × 2 × 2 = 16
21. Base
In x², x is the base
The number or variable that is being raised to a power (multiplied by itself).
Example: In 5³, base = 5
In y⁴, base = y
22. Monomial
An expression with one term
A single term consisting of a number, variable, or product of numbers and variables.
Example: 5x
−3xy²
7
23. Binomial
An expression with two terms
An algebraic expression containing exactly two unlike terms separated by + or −.
Example: x + 5
3y² − 2y
2a + 3b
24. Trinomial
An expression with three terms
An algebraic expression containing exactly three unlike terms.
Example: x² + 5x + 6
2a² − 3a + 1
25. Polynomial
An expression with one or more terms
An expression consisting of variables and coefficients using only addition, subtraction, and multiplication with non-negative integer exponents.
Example: 3x² + 2x − 5
y⁴ − 2y² + 1
26. Removing Parentheses
+(a + b) = a + b
When a positive sign precedes parentheses, simply remove them. When negative, change all signs inside.
Example: +(x + 3) = x + 3
−(x + 3) = −x − 3
27. Grouping Symbols
( ), [ ], { }
Symbols used to group terms together. Operations inside grouping symbols are performed first.
Example: 2(x + 3)
5[2x − (y + 1)]
28. Opposite (Additive Inverse)
The opposite of a is −a
Two numbers that are the same distance from zero but on opposite sides. Their sum is zero.
Example: Opposite of 5 is −5
Opposite of −3x is 3x
29. Reciprocal (Multiplicative Inverse)
The reciprocal of a is 1/a
Two numbers whose product is 1. Flip the numerator and denominator.
Example: Reciprocal of 5 is 1/5
Reciprocal of 2/3 is 3/2
30. Factoring Out
ab + ac = a(b + c)
The reverse of the distributive property; finding a common factor in terms and writing it outside parentheses.
Example: 6x + 9 = 3(2x + 3)
5x² + 5x = 5x(x + 1)
💡 Memory Tips
- Distributive Property: Think “distribute the gift” – give the outside number to everyone inside!
- Like Terms: “Like attracts like” – only terms that look alike can combine
- Negative Distribution: “Negative changes everything” – all signs flip when distributing a negative
- Order of Operations: Remember PEMDAS – “Please Excuse My Dear Aunt Sally”
- Combining Terms: “Same variables, same powers” – that’s when you can combine!
✓ Simplification Checklist
- Remove parentheses using the distributive property
- Identify all like terms in the expression
- Combine like terms by adding/subtracting coefficients
- Arrange terms in standard form (highest power first)
- Check that no further simplification is possible
Simplifying Algebraic Expressions
A Guide to the Distributive Property and Combining Like Terms
Learning Objectives
- Apply the distributive property to simplify algebraic expressions
- Identify and combine like terms
The Distributive Property
The distributive property is a fundamental concept in algebra that states: for any real numbers a, b, and c:
a(b + c) = ab + ac
This property allows us to multiply a number by a sum by multiplying the number by each term in the sum separately.
Example 1: Basic Distribution
Problem: Simplify 5(7y + 2)
Solution:
- Multiply 5 times each term inside the parentheses
- 5 · 7y + 5 · 2
- = 35y + 10
Answer: 35y + 10
Example 2: Distributing Negative Numbers
Problem: Simplify −3(2x² + 5x + 1)
Solution:
- Multiply −3 times each coefficient inside the parentheses
- −3 · 2x² + (−3) · 5x + (−3) · 1
- = −6x² − 15x − 3
Answer: −6x² − 15x − 3
Example 3: Partial Distribution
Problem: Simplify 5(−2a + 5b) − 2c
Solution:
- Apply the distributive property only to terms within parentheses
- 5 · (−2a) + 5 · 5b − 2c
- = −10a + 25b − 2c
Answer: −10a + 25b − 2c
Distribution with Division
Division can be rewritten as multiplication by a fraction, allowing us to apply the distributive property:
Example 4: Dividing Expressions
Problem: Divide (25x² − 5x + 10) ÷ 5
Solution:
- Rewrite as: (1/5)(25x² − 5x + 10)
- Multiply each term by 1/5
- (1/5) · 25x² − (1/5) · 5x + (1/5) · 10
- = 5x² − x + 2
Answer: 5x² − x + 2
Combining Like Terms
Like terms are terms that have the same variable parts with the same exponents. When simplifying expressions, we can combine like terms by adding or subtracting their coefficients.
What Are Like Terms?
- 2a and 3a are like terms (same variable)
- 7xy and −5xy are like terms (same variables)
- 10x² and 4x² are like terms (same variable and exponent)
- 3x² and 3x are NOT like terms (different exponents)
Example 5: Simple Like Terms
Problem: Simplify 3a + 2b − 4a + 9b
Solution:
- Identify like terms: (3a − 4a) and (2b + 9b)
- Combine coefficients: −1a + 11b
- = −a + 11b
Answer: −a + 11b
Example 6: Multiple Types of Terms
Problem: Simplify x² + 3x + 2 + 4x² − 5x − 7
Solution:
- Group like terms:
- x² terms: x² + 4x² = 5x²
- x terms: 3x − 5x = −2x
- Constant terms: 2 − 7 = −5
- = 5x² − 2x − 5
Answer: 5x² − 2x − 5
Example 7: Two-Variable Terms
Problem: Simplify 5x²y − 3xy² + 4x²y − 2xy²
Solution:
- x²y terms: 5x²y + 4x²y = 9x²y
- xy² terms: −3xy² − 2xy² = −5xy²
- = 9x²y − 5xy²
Answer: 9x²y − 5xy²
Example 8: Fractional Coefficients
Problem: Simplify (1/2)a − (1/3)b + (3/4)a + b
Solution:
- For a terms: 1/2 + 3/4 = 2/4 + 3/4 = 5/4
- For b terms: −1/3 + 1 = −1/3 + 3/3 = 2/3
- = (5/4)a + (2/3)b
Answer: (5/4)a + (2/3)b
Using Both: Distributive Property and Combining Like Terms
Many problems require both distributing and combining like terms. Always follow the order of operations: multiply first, then add or subtract.
Example 9: Distribution Then Combining
Problem: Simplify 2(3a − b) − 7(−2a + 3b)
Solution:
- Step 1: Distribute both numbers
- 2(3a) + 2(−b) − 7(−2a) − 7(3b)
- = 6a − 2b + 14a − 21b
- Step 2: Combine like terms
- a terms: 6a + 14a = 20a
- b terms: −2b − 21b = −23b
- = 20a − 23b
Answer: 20a − 23b
Example 10: Distributing Negative One
Problem: Simplify 5x − (−2x² + 3x − 1)
Solution:
- The negative sign means multiply by −1
- 5x + (−1)(−2x²) + (−1)(3x) + (−1)(−1)
- = 5x + 2x² − 3x + 1
- Combine like terms: 2x² + 2x + 1
Answer: 2x² + 2x + 1
⚠️ Important Note
When you see a negative sign before parentheses, it means multiply everything inside by −1. This changes all the signs inside the parentheses!
Example 11: Order of Operations
Problem: Simplify 5 − 2(x² − 4x − 3)
Solution:
- Incorrect: 5 − 2 = 3, then 3(x² − 4x − 3) ✗
- Correct: Distribute −2 first (multiplication before subtraction)
- 5 − 2x² + 8x + 6
- Combine constants: 5 + 6 = 11
- = −2x² + 8x + 11
Answer: −2x² + 8x + 11
Example 12: Word Problem Translation
Problem: Subtract 3x − 2 from twice the quantity (−4x² + 2x − 8)
Solution:
- Step 1: Translate to algebra
- “Twice the quantity” = 2(−4x² + 2x − 8)
- “Subtract 3x − 2 from” = [result] − (3x − 2)
- Expression: 2(−4x² + 2x − 8) − (3x − 2)
- Step 2: Distribute
- −8x² + 4x − 16 − 3x + 2
- Step 3: Combine like terms
- = −8x² + x − 14
Answer: −8x² + x − 14
Key Takeaways
- The distributive property: a(b + c) = ab + ac
- Like terms have identical variable parts (same variables with same exponents)
- Combine like terms by adding or subtracting their coefficients
- The variable part stays unchanged when combining like terms
- Always follow order of operations: distribute first, then combine
- A negative sign before parentheses means multiply by −1
- When distributing a negative number, all signs inside change
Practice Problems
Distributive Property
- 3(3x − 2)
- −2(x + 1)
- (2x + 3) · 2
- −(2a − 3b)
- 5(y² − 6y − 9)
Combining Like Terms
- 2x − 3x
- 5x − 7x + 8y + 2y
- 4xy − 6 + 2xy + 8
- x² − y² + 2x² − 3y
- 6x²y − 3xy² + 2x²y − 5xy²
Mixed Practice
- 5(2x − 3) + 7
- 5x − 2(4x − 5)
- 3 − (2x + 7)
- 2(3a − 4b) + 4(−2a + 3b)
- 10 − 5(x² − 3x − 1)
Click to Show Answers
- 9x − 6
- −2x − 2
- 4x + 6
- −2a + 3b
- 5y² − 30y − 45
- −x
- −2x + 10y
- 6xy + 2
- 3x² − y² − 3y
- 8x²y − 8xy²
- 10x − 8
- −3x + 10
- −2x − 4
- −2a + 4b
- −5x² + 15x + 15
Common Mistakes to Avoid
- Mistake: Forgetting to distribute to all terms
Wrong: 3(x + 2) = 3x + 2
Right: 3(x + 2) = 3x + 6 - Mistake: Not changing signs when distributing a negative
Wrong: −2(x − 3) = −2x − 6
Right: −2(x − 3) = −2x + 6 - Mistake: Combining unlike terms
Wrong: 2x + 3x² = 5x³
Right: 2x + 3x² cannot be combined - Mistake: Changing the variable part when combining
Wrong: 3x² + 4x² = 7x⁴
Right: 3x² + 4x² = 7x² - Mistake: Ignoring order of operations
Wrong: 5 − 2(x + 1) = 3(x + 1)
Right: 5 − 2(x + 1) = 5 − 2x − 2 = 3 − 2x
Simplifying Algebraic Expressions ACT Pre Algebra Math Guide
Simplifying Algebraic Expressions ACT Pre Algebra Math Guide
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📺 Simplifying Algebraic Expressions | ACT Elemetry Algebra Math Guide
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