Solved Exercises on Multi-Step Inequalities in Integrated Math 1

Master multi-step inequalities: combining like terms, distributive property, variables on both sides, and complex inequalities through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Combining like terms
Exercise 1
Solve for x:
\(3x + 2x - 5 > 20\)
Definition:

Multi-step inequality: An inequality requiring multiple operations to isolate the variable

Like terms: Terms with the same variable raised to the same power

Combining like terms: Adding or subtracting coefficients of like terms

Multi-step inequality method:

To solve inequalities with multiple operations:

  1. Combine like terms: Simplify both sides by combining similar terms
  2. Move variables to one side: Use inverse operations to collect all variables
  3. Move constants to other side: Use inverse operations to collect all constants
  4. Isolate the variable: Divide by the coefficient
  5. Check for sign reversal: Flip inequality symbol if dividing by negative
Original
\(3x + 2x - 5 > 20\)
Combine like terms
\(5x - 5 > 20\)
Add 5
\(5x > 25\)
Divide by 5
\(x > 5\)
Step 1: Write the inequality

\(3x + 2x - 5 > 20\)

Step 2: Combine like terms on the left side

\(3x + 2x = 5x\), so: \(5x - 5 > 20\)

Step 3: Undo subtraction by adding 5 to both sides

\(5x - 5 + 5 > 20 + 5\)

\(5x > 25\)

Step 4: Undo multiplication by dividing both sides by 5

\(\frac{5x}{5} > \frac{25}{5}\)

\(x > 5\)

Step 5: Express the solution

The solution is all real numbers greater than 5

\(x > 5\)
Final answer:

The solution is \(x > 5\)

Applied rules:

• Combining Like Terms: Add coefficients of terms with the same variable

• Addition/Subtraction Property: Adding or subtracting the same number from both sides preserves inequality direction

• Multiplication/Division Property: Dividing both sides by the same positive number preserves inequality direction

Tip: Always combine like terms first to simplify the inequality!
Tip: Like terms have the same variable part: \(3x\) and \(2x\) are like terms, but \(3x\) and \(3\) are not.
2 Distributive property
Exercise 2
Solve for x:
\(2(x + 3) + 4 \leq 18\)
Definition:

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

Multi-step inequality with distribution: Requires distributing before other operations

Original
\(2(x + 3) + 4 \leq 18\)
Distribute
\(2x + 6 + 4 \leq 18\)
Combine like terms
\(2x + 10 \leq 18\)
Subtract 10
\(2x \leq 8\)
Divide by 2
\(x \leq 4\)
Step 1: Write the inequality

\(2(x + 3) + 4 \leq 18\)

Step 2: Apply the distributive property

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

So: \(2x + 6 + 4 \leq 18\)

Step 3: Combine like terms

\(6 + 4 = 10\), so: \(2x + 10 \leq 18\)

Step 4: Undo addition by subtracting 10 from both sides

\(2x + 10 - 10 \leq 18 - 10\)

\(2x \leq 8\)

Step 5: Undo multiplication by dividing both sides by 2

\(\frac{2x}{2} \leq \frac{8}{2}\)

\(x \leq 4\)

Step 6: Express the solution

The solution is all real numbers less than or equal to 4

\(x \leq 4\)
Final answer:

The solution is \(x \leq 4\)

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 preserves inequality direction

• Multiplication/Division Property: Dividing both sides by the same positive number preserves inequality direction

Tip: Always distribute before combining like terms!
Tip: Don't forget to multiply the outside number by EVERY term inside the parentheses.
3 Variables on both sides
Exercise 3
Solve for x:
\(5x + 3 > 2x + 12\)
Definition:

Variables on both sides: An inequality where the variable appears on both sides of the inequality symbol

Collecting variables: Moving all variable terms to one side and constants to the other

Original
\(5x + 3 > 2x + 12\)
Subtract 2x
\(3x + 3 > 12\)
Subtract 3
\(3x > 9\)
Divide by 3
\(x > 3\)
Step 1: Write the inequality

\(5x + 3 > 2x + 12\)

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

\(5x + 3 - 2x > 2x + 12 - 2x\)

\(3x + 3 > 12\)

Step 3: Move constants to the right side by subtracting 3 from both sides

\(3x + 3 - 3 > 12 - 3\)

\(3x > 9\)

Step 4: Isolate the variable by dividing both sides by 3

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

\(x > 3\)

Step 5: Express the solution

The solution is all real numbers greater than 3

\(x > 3\)
Final answer:

The solution is \(x > 3\)

Applied rules:

• Variable Collection: Move all variable terms to one side and constants to the other

• Addition/Subtraction Property: Adding or subtracting the same number from both sides preserves inequality direction

• Multiplication/Division Property: Dividing both sides by the same positive number preserves inequality direction

Tip: It doesn't matter which side you move variables to - just be consistent!
Tip: Always remember to keep the inequality symbol pointing in the same direction when adding/subtracting!
Rules and methods, laws,...
\(a(x + b) + c > d \Rightarrow x > \frac{d - c}{a} - b\)
Distributive Form
\(ax + b > cx + d \Rightarrow x > \frac{d - b}{a - c} \text{ (when } a > c)\)
Both Sides Form
\(ax + bx + c > d \Rightarrow x > \frac{d - c}{a + b}\)
Like Terms Form
Like Terms
\(ax + bx + c > d\)
Combine: \((a+b)x + c > d\)
Distribution
\(a(x + b) + c > d\)
Distribute: \(ax + ab + c > d\)
Both Sides
\(ax + b > cx + d\)
Collect: \((a-c)x > d - b\)
Complex
\(a(x + b) + c > d(x + e) + f\)
Distribute then collect
Inequality Property: Both sides of an inequality must remain balanced when performing operations.
Sign Reversal: Multiplying or dividing by a negative number flips the inequality symbol.
Solution: Exercises 4 to 5
4 Complex distribution with negative
Exercise 4
Solve for x:
\(-3(x - 2) + 4x \geq 2(x + 1) + 8\)
Definition:

Complex multi-step inequality: An inequality requiring distribution on both sides and collecting variables

Multiple distributions: Apply distributive property to each set of parentheses

Original
\(-3(x - 2) + 4x \geq 2(x + 1) + 8\)
Distribute both sides
\(-3x + 6 + 4x \geq 2x + 2 + 8\)
Combine like terms
\(x + 6 \geq 2x + 10\)
Subtract 2x
\(-x + 6 \geq 10\)
Subtract 6
\(-x \geq 4\)
Divide by -1 (flip symbol)
\(x \leq -4\)
Step 1: Write the inequality

\(-3(x - 2) + 4x \geq 2(x + 1) + 8\)

Step 2: Distribute on the left side

\(-3(x - 2) = -3x + 6\), so: \(-3x + 6 + 4x \geq 2(x + 1) + 8\)

Step 3: Distribute on the right side

\(2(x + 1) = 2x + 2\), so: \(-3x + 6 + 4x \geq 2x + 2 + 8\)

Step 4: Combine like terms on both sides

Left: \(-3x + 4x = x\), so \(x + 6\)

Right: \(2 + 8 = 10\), so \(2x + 10\)

Result: \(x + 6 \geq 2x + 10\)

Step 5: Move variables to the left by subtracting 2x from both sides

\(x + 6 - 2x \geq 2x + 10 - 2x\)

\(-x + 6 \geq 10\)

Step 6: Move constants to the right by subtracting 6 from both sides

\(-x + 6 - 6 \geq 10 - 6\)

\(-x \geq 4\)

Step 7: Isolate the variable by dividing both sides by -1

IMPORTANT: Dividing by a negative number flips the inequality symbol!

\(\frac{-x}{-1} \leq \frac{4}{-1}\)

\(x \leq -4\)

Step 8: Express the solution

The solution is all real numbers less than or equal to -4

\(x \leq -4\)
Final answer:

The solution is \(x \leq -4\)

Applied rules:

• Distributive Property: Apply to each set of parentheses separately

• Combining Like Terms: Only combine terms with the same variable part

• Variable Collection: Move all variable terms to one side

• Sign Reversal Rule: Dividing by a negative number flips the inequality symbol!

Tip: When dividing by a negative number, ALWAYS flip the inequality symbol!
Tip: When subtracting a group like \(-3(x - 2)\), distribute the negative: \(-3x + 6\).
5 Complex multi-step inequality
Exercise 5
Solve for x:
\(4(2x - 3) - 3(x + 1) < 5x - 2(3x + 4)\)
Definition:

Most complex multi-step inequality: An inequality with multiple distributions on both sides and complex variable collection

Systematic approach: Distribute completely, then collect like terms and variables

Original
\(4(2x - 3) - 3(x + 1) < 5x - 2(3x + 4)\)
Distribute all
\(8x - 12 - 3x - 3 < 5x - 6x - 8\)
Combine like terms
\(5x - 15 < -x - 8\)
Add x
\(6x - 15 < -8\)
Add 15
\(6x < 7\)
Divide by 6
\(x < \frac{7}{6}\)
Step 1: Write the inequality

\(4(2x - 3) - 3(x + 1) < 5x - 2(3x + 4)\)

Step 2: Distribute on the left side

\(4(2x - 3) = 8x - 12\)

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

So left side: \(8x - 12 - 3x - 3\)

Step 3: Distribute on the right side

\(-2(3x + 4) = -6x - 8\)

So right side: \(5x - 6x - 8\)

Step 4: Simplify both sides

Left: \(8x - 3x = 5x\), \(-12 - 3 = -15\), so: \(5x - 15\)

Right: \(5x - 6x = -x\), so: \(-x - 8\)

Result: \(5x - 15 < -x - 8\)

Step 5: Move variables to the left by adding x to both sides

\(5x - 15 + x < -x - 8 + x\)

\(6x - 15 < -8\)

Step 6: Move constants to the right by adding 15 to both sides

\(6x - 15 + 15 < -8 + 15\)

\(6x < 7\)

Step 7: Isolate the variable by dividing both sides by 6

\(\frac{6x}{6} < \frac{7}{6}\)

\(x < \frac{7}{6}\)

Step 8: Express the solution

The solution is all real numbers less than \(\frac{7}{6}\) (approximately 1.17)

\(x < \frac{7}{6}\)
Final answer:

The solution is \(x < \frac{7}{6}\)

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 combining like terms

• Sign Preservation: Dividing by a positive number keeps the inequality symbol the same

Tip: For complex inequalities, 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 Multi-Step Inequalities
\(\text{General approach: } ax + b > cx + d \Rightarrow x > \frac{d - b}{a - c} \text{ (when } a > c)\)
Multi-Step Strategy
Key definitions:

Multi-step inequality: An inequality requiring multiple operations to isolate the variable, typically involving 3 or more steps

Like terms: Terms that have identical variable parts (same variables raised to the same powers)

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

Variable collection: Moving all variable terms to one side and constants to the other

Sign reversal rule: When multiplying or dividing by a negative number, flip the inequality symbol

Complete methodology:
  1. Distribute if necessary: Apply distributive property to eliminate parentheses
  2. Combine like terms: Simplify both sides by combining similar terms
  3. Move variables to one side: Use inverse operations to collect all variables
  4. Move constants to other side: Use inverse operations to collect all constants
  5. Isolate the variable: Divide by the coefficient
  6. Check for sign reversal: If dividing by negative, flip the inequality symbol
  7. Express the solution: Write the final inequality
Tip 1: Always distribute before combining like terms!
Tip 2: When subtracting a group like \(-3(x + 2)\), distribute the negative: \(-3x - 6\).
Tip 3: Keep your work organized - each step should flow logically to the next.
Tip 4: The BIGGEST RULE: When multiplying or dividing by a negative number, flip the inequality symbol!
Tip 5: Fractional solutions are perfectly acceptable - don't panic if you get one!
Common errors: Forgetting to flip the inequality symbol when dividing by negative numbers, forgetting to distribute to all terms, making sign errors, combining unlike terms, not verifying solutions.
Exam preparation: Practice inequalities with distribution, variables on both sides, and complex combinations. Focus on the sign reversal rule.
Formulas to know by heart:

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

• Like Terms: \(ax + bx = (a + b)x\)

• Variables on Both Sides: \(ax + b > cx + d \Rightarrow x > \frac{d - b}{a - c}\) (when \(a > c\))

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

• Sign Reversal: When multiplying/dividing by negative, flip inequality symbol

Multi-Step Inequalities Workflow

📊
Solving Process
1
Distribute
2
Combine Like Terms
3
Collect Variables
4
Collect Constants
5
Isolate Variable
6
Check Sign
Multi-Step Inequality Forms
Like Terms: ax + bx + c > d → combine then solve
Distribution: a(x + b) + c > d → distribute first
Both Sides: ax + b > cx + d → collect variables
Complex: Multiple distributions → systematic approach
Master This Systematic Approach to Excel in Algebra!

Questions & Answers

Question: When I have an inequality like \(-3(x + 2) \geq 5\), do I need to flip the inequality symbol when I distribute the -3?

Answer: No, you only flip the inequality symbol when you multiply or divide both sides by a negative number, not when you're just distributing.

For \(-3(x + 2) \geq 5\):

  • \(-3 \times x = -3x\)
  • \(-3 \times 2 = -6\)
  • So: \(-3x - 6 \geq 5\)

The inequality symbol stays the same during distribution. You would only flip it if you later divide both sides by -3 to solve for x.

For example, continuing: \(-3x - 6 \geq 5\), then \(-3x \geq 11\), then dividing by -3: \(x \leq -\frac{11}{3}\) (flipped symbol).

Question: How do I know when to combine like terms versus when to move variables to one side?

Answer: Follow this systematic order:

  1. First: Distribute to eliminate parentheses if needed
  2. Second: Combine like terms on each side separately
  3. Third: Move all variable terms to one side (typically the left)
  4. Fourth: Move all constant terms to the other side (typically the right)
  5. Fifth: Isolate the variable

For example, in \(3x + 2x - 5 > 20\), combine like terms first: \(5x - 5 > 20\), then move the constant.

In \(5x + 3 > 2x + 12\), first move variables: \(5x - 2x + 3 > 12\), which simplifies to \(3x + 3 > 12\).

This systematic approach prevents errors and makes the process predictable!

Question: What should I do if I get a fractional answer? Is that normal for inequalities?

Answer: Yes, fractional answers are completely normal and valid! Many multi-step inequalities result in fractional solutions.

For example, solving \(6x < 7\) gives \(x < \frac{7}{6}\), which is a perfectly valid solution.

To verify a fractional solution:

  • Substitute a value that satisfies the inequality back into the original
  • For \(x < \frac{7}{6}\), try \(x = 1\) (since \(1 < \frac{7}{6}\))
  • Both sides should maintain the correct relationship

Don't try to force a whole number answer - let the math guide you to the correct solution, whether it's a whole number, fraction, or decimal.

If the fraction seems unusual, double-check your work, but don't assume it's wrong just because it's not a whole number!