Solved Exercises on Equations with Parentheses and Distribution in Integrated Math 1

Master equations with parentheses and distribution: simple to complex examples through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Simple distribution
Exercise 1
Solve for x:
\(3(x + 4) = 21\)
Definition:

Distributive property: \(a(b + c) = ab + ac\)

Equation with parentheses: An equation containing grouped terms that need to be distributed

Distribution: Multiplying the outside term by each term inside parentheses

Distribution method:

To solve equations with parentheses:

  1. Distribute: Multiply the outside term by each term inside parentheses
  2. Simplify: Combine like terms if necessary
  3. Isolate the variable: Use inverse operations
  4. Verify: Substitute solution back into original equation
Original
\(3(x + 4) = 21\)
Distribute
\(3x + 12 = 21\)
Subtract 12
\(3x = 9\)
Divide by 3
\(x = 3\)
Step 1: Write the equation

\(3(x + 4) = 21\)

Step 2: Apply the distributive property

\(3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12\)

So: \(3x + 12 = 21\)

Step 3: Undo addition by subtracting 12 from both sides

\(3x + 12 - 12 = 21 - 12\)

\(3x = 9\)

Step 4: Undo multiplication by dividing both sides by 3

\(\frac{3x}{3} = \frac{9}{3}\)

\(x = 3\)

Step 5: Verify the solution

Substitute \(x = 3\) into original: \(3(3 + 4) = 3(7) = 21\) ✓

\(x = 3\)
Final answer:

The solution is \(x = 3\)

Applied rules:

• Distributive Property: Multiply the outside term by each term inside parentheses

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

Tip: Always distribute before performing other operations!
Tip: Don't forget to multiply the outside number by EVERY term inside the parentheses.
2 Distribution with subtraction
Exercise 2
Solve for x:
\(2(x - 5) = 8\)
Definition:

Distribution with subtraction: An equation where parentheses contain a subtraction operation

Key concept: The distributive property applies to subtraction as well as addition

Original
\(2(x - 5) = 8\)
Distribute
\(2x - 10 = 8\)
Add 10
\(2x = 18\)
Divide by 2
\(x = 9\)
Step 1: Write the equation

\(2(x - 5) = 8\)

Step 2: Apply the distributive property

\(2(x - 5) = 2 \cdot x - 2 \cdot 5 = 2x - 10\)

So: \(2x - 10 = 8\)

Step 3: Undo subtraction by adding 10 to both sides

\(2x - 10 + 10 = 8 + 10\)

\(2x = 18\)

Step 4: Undo multiplication by dividing both sides by 2

\(\frac{2x}{2} = \frac{18}{2}\)

\(x = 9\)

Step 5: Verify the solution

Substitute \(x = 9\) into original: \(2(9 - 5) = 2(4) = 8\) ✓

\(x = 9\)
Final answer:

The solution is \(x = 9\)

Applied rules:

• Distributive Property: Applies to both addition and subtraction

• Addition/Subtraction Property: Maintain equality when performing operations

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

Tip: When distributing over subtraction, keep the subtraction sign: \(2(x - 5) = 2x - 10\)
3 Distribution with negative coefficient
Exercise 3
Solve for x:
\(-3(x + 2) = 15\)
Definition:

Distribution with negative coefficient: An equation where the number outside parentheses is negative

Key concept: The negative sign must be distributed to every term inside parentheses

Original
\(-3(x + 2) = 15\)
Distribute
\(-3x - 6 = 15\)
Add 6
\(-3x = 21\)
Divide by -3
\(x = -7\)
Step 1: Write the equation

\(-3(x + 2) = 15\)

Step 2: Apply the distributive property

\(-3(x + 2) = -3 \cdot x + (-3) \cdot 2 = -3x - 6\)

So: \(-3x - 6 = 15\)

Step 3: Undo subtraction by adding 6 to both sides

\(-3x - 6 + 6 = 15 + 6\)

\(-3x = 21\)

Step 4: Undo multiplication by dividing both sides by -3

\(\frac{-3x}{-3} = \frac{21}{-3}\)

\(x = -7\)

Step 5: Verify the solution

Substitute \(x = -7\) into original: \(-3(-7 + 2) = -3(-5) = 15\) ✓

\(x = -7\)
Final answer:

The solution is \(x = -7\)

Applied rules:

• Negative Distribution: Multiply the negative coefficient by each term inside parentheses

• Sign Rules: \(-3 \cdot x = -3x\) and \(-3 \cdot 2 = -6\)

• Division by Negative: Dividing by a negative number changes the sign of the result

Tip: When distributing a negative, every term inside parentheses changes sign!
Tip: Negative coefficients don't change the solving method - just be careful with signs!
Rules and methods, laws,...
\(a(b + c) = ab + ac\)
Distributive Property
\(a(b - c) = ab - ac\)
Distributive Property (Subtraction)
Basic
\(a(b + c)\)
Distribute: \(ab + ac\)
Subtraction
\(a(b - c)\)
Distribute: \(ab - ac\)
Negative
\(-a(b + c)\)
Distribute: \(-ab - ac\)
Complex
\(a(b + c + d)\)
Distribute: \(ab + ac + ad\)
Equality Property: Both sides of an equation must remain equal when performing operations.
Distribution Rule: Multiply the outside term by every term inside parentheses.
Solution: Exercises 4 to 5
4 Multiple distributions
Exercise 4
Solve for x:
\(2(x + 3) = 3(x - 1)\)
Definition:

Equation with distribution on both sides: An equation requiring distribution on both sides before solving

Key concept: Distribute completely on each side, then collect like terms and variables

Original
\(2(x + 3) = 3(x - 1)\)
Distribute both sides
\(2x + 6 = 3x - 3\)
Subtract 2x
\(6 = x - 3\)
Add 3
\(9 = x\)
Step 1: Write the equation

\(2(x + 3) = 3(x - 1)\)

Step 2: Distribute on the left side

\(2(x + 3) = 2 \cdot x + 2 \cdot 3 = 2x + 6\)

Step 3: Distribute on the right side

\(3(x - 1) = 3 \cdot x - 3 \cdot 1 = 3x - 3\)

So: \(2x + 6 = 3x - 3\)

Step 4: Move variables to the right side by subtracting 2x from both sides

\(2x + 6 - 2x = 3x - 3 - 2x\)

\(6 = x - 3\)

Step 5: Move constants to the left side by adding 3 to both sides

\(6 + 3 = x - 3 + 3\)

\(9 = x\)

Or: \(x = 9\)

Step 6: Verify the solution

Substitute \(x = 9\) into original:

Left side: \(2(9 + 3) = 2(12) = 24\)

Right side: \(3(9 - 1) = 3(8) = 24\)

Since both sides equal 24, the solution is correct ✓

\(x = 9\)
Final answer:

The solution is \(x = 9\)

Applied rules:

• Multiple Distributions: Apply distributive property to each set of parentheses

• Variable Collection: Move all variable terms to one side

• Constant Collection: Move all constant terms to the other side

• Order of Operations: Distribute before collecting terms

Tip: Always distribute completely on each side before collecting terms!
5 Complex distribution
Exercise 5
Solve for x:
\(4(x - 2) - 3(x + 1) = 2(x + 3)\)
Definition:

Complex equation with distribution: An equation with multiple distributions and complex variable collection

Key concept: Distribute completely first, then collect like terms and variables

Original
\(4(x - 2) - 3(x + 1) = 2(x + 3)\)
Distribute all
\(4x - 8 - 3x - 3 = 2x + 6\)
Combine like terms
\(x - 11 = 2x + 6\)
Subtract x
\(-11 = x + 6\)
Subtract 6
\(-17 = x\)
Step 1: Write the equation

\(4(x - 2) - 3(x + 1) = 2(x + 3)\)

Step 2: Distribute on the left side - first term

\(4(x - 2) = 4 \cdot x - 4 \cdot 2 = 4x - 8\)

Step 3: Distribute on the left side - second term

\(-3(x + 1) = -3 \cdot x - 3 \cdot 1 = -3x - 3\)

So left side: \(4x - 8 - 3x - 3\)

Step 4: Distribute on the right side

\(2(x + 3) = 2 \cdot x + 2 \cdot 3 = 2x + 6\)

Step 5: Combine like terms on the left side

\(4x - 3x = x\), and \(-8 - 3 = -11\)

So: \(x - 11 = 2x + 6\)

Step 6: Move variables to the right side by subtracting x from both sides

\(x - 11 - x = 2x + 6 - x\)

\(-11 = x + 6\)

Step 7: Move constants to the left side by subtracting 6 from both sides

\(-11 - 6 = x + 6 - 6\)

\(-17 = x\)

Or: \(x = -17\)

Step 8: Verify the solution

Substitute \(x = -17\) into original:

Left side: \(4(-17 - 2) - 3(-17 + 1) = 4(-19) - 3(-16) = -76 + 48 = -28\)

Right side: \(2(-17 + 3) = 2(-14) = -28\)

Since both sides equal -28, the solution is correct ✓

\(x = -17\)
Final answer:

The solution is \(x = -17\)

Applied rules:

• Multiple Distributions: Apply distributive property to every set of parentheses

• Sign Awareness: Pay attention to negative signs when distributing

• Systematic Approach: Distribute completely before collecting terms

• Variable Collection: Move all variables to one side

• Verification: Complex equations require careful checking

Tip: For complex equations, work slowly and check each step to avoid calculation errors.
Tip: When subtracting a group like \(-3(x + 1)\), distribute the negative: \(-3x - 3\).
Comprehensive Guide to Distribution
\(a(b + c) = ab + ac\)
Distributive Property
Key definitions:

Distributive property: \(a(b + c) = ab + ac\) - multiply the outside term by each term inside parentheses

Equation with parentheses: An equation containing grouped terms that need to be distributed

Systematic approach: Following a consistent order of operations to solve efficiently

Sign awareness: Understanding how negative signs affect distribution

Complete methodology:
  1. Distribute completely: Apply distributive property to every set of parentheses
  2. Combine like terms: Simplify each side by combining similar terms
  3. Collect variables: Move all variable terms to one side using inverse operations
  4. Collect constants: Move all constant terms to the other side using inverse operations
  5. Isolate the variable: Divide by the coefficient
  6. Verify: Substitute solution back into original equation
Tip 1: Always distribute before combining like terms or collecting variables!
Tip 2: When distributing a negative, every term inside parentheses changes sign.
Tip 3: Keep your work organized - each step should flow logically to the next.
Tip 4: Always verify your solution by substituting back into the original equation.
Tip 5: Don't forget to multiply the outside number by EVERY term inside the parentheses.
Common errors: Forgetting to distribute to all terms, making sign errors when distributing negatives, not verifying solutions, performing operations on only one side.
Exam preparation: Practice equations with distribution, negative coefficients, and complex combinations. Focus on organization and verification.
Formulas to know by heart:

• Distributive Property: \(a(b + c) = ab + ac\)

• Negative Distribution: \(-a(b + c) = -ab - ac\)

• Subtraction Distribution: \(a(b - c) = ab - ac\)

• Sign distribution: \(-a(b + c) = -ab - ac\)

• Verification: Always substitute back to check

Distribution Workflow

📊
Solving Process
1
Distribute
2
Combine Like Terms
3
Collect Variables
4
Collect Constants
5
Isolate Variable
6
Verify
Distribution Forms
Basic: a(b + c) → distribute to both terms
Subtraction: a(b - c) → distribute to both terms
Negative: -a(b + c) → distribute the negative
Complex: multiple distributions → systematic approach
Master Distribution to Excel in Algebra!

Questions & Answers

Question: When I have an equation like \(-3(x + 2)\), do I distribute the negative sign too?

Answer: Yes, absolutely! The negative sign is part of the coefficient, so you must distribute it to every term inside the parentheses.

For \(-3(x + 2)\):

  • \(-3 \times x = -3x\)
  • \(-3 \times 2 = -6\)
  • So: \(-3(x + 2) = -3x - 6\)

Think of it as \(-3\) being the multiplier for each term inside the parentheses. The negative sign affects both terms!

This is crucial for getting the correct signs in your equation.

Question: How do I know when to distribute versus when to combine like terms?

Answer: Follow this systematic order:

  1. First: Distribute to eliminate ALL parentheses
  2. Second: Combine like terms on each side separately
  3. Third: Move all variable terms to one side
  4. Fourth: Move all constant terms to the other side
  5. Fifth: Isolate the variable

For example, in \(3(x + 2) = 15\), first distribute: \(3x + 6 = 15\), then solve.

In \(2(x + 3) = 3(x - 1)\), first distribute on both sides: \(2x + 6 = 3x - 3\), then solve.

Always distribute completely before combining like terms!

Question: What should I do if I get a negative answer? Is that normal?

Answer: Yes, negative answers are completely normal and valid! Many equations result in negative solutions.

For example, solving \(-3(x + 2) = 15\) gives \(x = -7\), which is a perfectly valid solution.

To verify a negative solution:

  • Substitute the negative number back into the original equation
  • Perform the arithmetic carefully with negative numbers
  • Both sides should equal the same value

Don't try to force a positive answer - let the math guide you to the correct solution, whether it's positive, negative, or zero.

If the negative answer seems unusual, double-check your work, but don't assume it's wrong just because it's negative!