Solved Exercises on One-Step Inequalities in Integrated Math 1

Master one-step inequalities: addition, subtraction, multiplication, division, and fraction inequalities through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Addition inequality
Exercise 1
Solve for x:
\(x + 5 > 12\)
Definition:

One-step inequality: An inequality requiring only one operation to isolate the variable

Addition inequality: An inequality where the variable is added to a number

Solution set: All values that make the inequality true

Inequality solving method:

To solve addition inequalities:

  1. Identify the operation on the variable side (addition)
  2. Apply the inverse operation to both sides (subtract the same number)
  3. Isolate the variable
  4. Express the solution as an inequality
Original
\(x + 5 > 12\)
Subtract 5
\(x > 7\)
Step 1: Write the inequality

\(x + 5 > 12\)

Step 2: Subtract 5 from both sides

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

Step 3: Simplify both sides

\(x > 7\)

Step 4: Express the solution

The solution is all real numbers greater than 7

\(x > 7\)
Final answer:

The solution is \(x > 7\)

Applied rules:

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

• Inverse Operations: Subtraction undoes addition

• Direction Preservation: Adding or subtracting doesn't change the inequality symbol

Tip: What you do to one side, you must do to the other side!
Tip: Addition/subtraction doesn't change the direction of the inequality symbol.
2 Subtraction inequality
Exercise 2
Solve for x:
\(x - 8 \leq 3\)
Definition:

Subtraction inequality: An inequality where the variable is subtracted by a number

Inverse operation: Addition undoes subtraction

Less than or equal: The solution includes the boundary value

Original
\(x - 8 \leq 3\)
Add 8
\(x \leq 11\)
Step 1: Write the inequality

\(x - 8 \leq 3\)

Step 2: Add 8 to both sides

\(x - 8 + 8 \leq 3 + 8\)

Step 3: Simplify both sides

\(x \leq 11\)

Step 4: Express the solution

The solution is all real numbers less than or equal to 11

\(x \leq 11\)
Final answer:

The solution is \(x \leq 11\)

Applied rules:

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

• Inverse Operations: Addition undoes subtraction

• Direction Preservation: Adding or subtracting doesn't change the inequality symbol

Tip: The ≤ symbol means "less than OR equal to" - include the boundary!
Tip: Subtraction/addition doesn't change the direction of the inequality symbol.
3 Multiplication inequality (positive)
Exercise 3
Solve for x:
\(3x \geq 15\)
Definition:

Multiplication inequality: An inequality where the variable is multiplied by a number

Inverse operation: Division undoes multiplication

Greater than or equal: The solution includes the boundary value

Original
\(3x \geq 15\)
Divide by 3
\(x \geq 5\)
Step 1: Write the inequality

\(3x \geq 15\)

Step 2: Divide both sides by 3

\(\frac{3x}{3} \geq \frac{15}{3}\)

Step 3: Simplify both sides

\(x \geq 5\)

Step 4: Express the solution

The solution is all real numbers greater than or equal to 5

\(x \geq 5\)
Final answer:

The solution is \(x \geq 5\)

Applied rules:

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

• Inverse Operations: Division undoes multiplication

• Direction Preservation: Dividing by a positive number doesn't change the inequality symbol

Tip: When dividing by a POSITIVE number, the inequality symbol stays the same!
Tip: The ≥ symbol means "greater than OR equal to" - include the boundary!
Rules and methods, laws,...
\(x + a > b \Rightarrow x > b - a\)
Addition Inequalities
\(x - a > b \Rightarrow x > b + a\)
Subtraction Inequalities
\(ax > b \Rightarrow x > \frac{b}{a} \text{ (when } a > 0)\)
Multiplication Inequalities (Positive)
\(ax > b \Rightarrow x < \frac{b}{a} \text{ (when } a < 0)\)
Multiplication Inequalities (Negative)
Addition
\(x + a > b\)
Solve by subtracting \(a\) from both sides
Subtraction
\(x - a > b\)
Solve by adding \(a\) to both sides
Multiplication (pos)
\(ax > b\)
Solve by dividing both sides by \(a > 0\)
Multiplication (neg)
\(ax > b\)
Solve by dividing both sides by \(a < 0\), flip symbol
Inequality Property: Both sides of an inequality must remain balanced when performing operations.
Sign Rule: Multiplying or dividing by a negative number reverses the inequality symbol.
Solution: Exercises 4 to 5
4 Division inequality (positive)
Exercise 4
Solve for x:
\(\frac{x}{4} < 6\)
Definition:

Division inequality: An inequality where the variable is divided by a number

Inverse operation: Multiplication undoes division

Less than: The solution excludes the boundary value

Original
\(\frac{x}{4} < 6\)
Multiply by 4
\(x < 24\)
Step 1: Write the inequality

\(\frac{x}{4} < 6\)

Step 2: Multiply both sides by 4

\(\frac{x}{4} \times 4 < 6 \times 4\)

Step 3: Simplify both sides

\(x < 24\)

Step 4: Express the solution

The solution is all real numbers less than 24

\(x < 24\)
Final answer:

The solution is \(x < 24\)

Applied rules:

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

• Inverse Operations: Multiplication undoes division

• Direction Preservation: Multiplying by a positive number doesn't change the inequality symbol

Tip: When multiplying by a POSITIVE number, the inequality symbol stays the same!
5 Multiplication inequality (negative)
Exercise 5
Solve for x:
\(-2x > 8\)
Definition:

Multiplication inequality with negative coefficient: An inequality where the variable is multiplied by a negative number

Sign reversal rule: Dividing by a negative number flips the inequality symbol

Original
\(-2x > 8\)
Divide by -2 (flip symbol)
\(x < -4\)
Step 1: Write the inequality

\(-2x > 8\)

Step 2: Divide both sides by -2

Because we're dividing by a negative number, we FLIP the inequality symbol!

\(\frac{-2x}{-2} < \frac{8}{-2}\)

Step 3: Simplify both sides

\(x < -4\)

Step 4: Express the solution

The solution is all real numbers less than -4

\(x < -4\)
Final answer:

The solution is \(x < -4\)

Applied rules:

• Negative Division: Dividing by a negative number reverses the inequality symbol

• Sign Reversal Rule: When multiplying or dividing by a negative number, flip the inequality symbol

• Verification: Check with a test value to confirm the solution

Tip: When dividing by a NEGATIVE number, ALWAYS flip the inequality symbol!
Tip: Remember: negative × negative = positive, so the sign rule is crucial!
Comprehensive Guide to One-Step Inequalities
\(x + a > b \Rightarrow x > b - a\)
Addition Inequalities
Key definitions:

One-step inequality: An inequality that can be solved in a single step by using inverse operations

Inverse operations: Operations that undo each other (addition/subtraction, multiplication/division)

Solution set: The collection of all values that make the inequality true

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

Complete methodology:
  1. Identify the operation: Determine whether the variable is being added to, subtracted from, multiplied by, or divided by a number
  2. Apply the inverse operation: Perform the opposite operation to both sides of the inequality
  3. Isolate the variable: Simplify to get the variable alone on one side
  4. Check for sign reversal: If you multiplied or divided by a negative number, flip the inequality symbol
  5. Express the solution: Write the final inequality
Tip 1: Think of an inequality as a balance scale - whatever you do to one side, you must do to the other side!
Tip 2: The big rule: MULTIPLYING OR DIVIDING BY A NEGATIVE NUMBER FLIPS THE INEQUALITY SYMBOL!
Tip 3: Always express your answer as an inequality, not as a single value.
Tip 4: Verify your solution by substituting a test value into the original inequality.
Common errors: Forgetting to flip the inequality symbol when multiplying/dividing by negative numbers, treating inequalities like equations, making calculation errors with negatives.
Exam preparation: Practice all four types of inequalities (addition, subtraction, multiplication, division), focus on the sign reversal rule, master verification techniques.
Formulas to know by heart:

• Addition: \(x + a > b \Rightarrow x > b - a\)

• Subtraction: \(x - a > b \Rightarrow x > b + a\)

• Multiplication (positive): \(ax > b \Rightarrow x > \frac{b}{a}\) (when \(a > 0\))

• Multiplication (negative): \(ax > b \Rightarrow x < \frac{b}{a}\) (when \(a < 0\))

• Division: \(\frac{x}{a} > b \Rightarrow x > ab\) (when \(a > 0\))

One-Step Inequalities Workflow

📊
Solving Process
1
Identify Operation
2
Apply Inverse
3
Check Sign
4
Simplify
Inequality Symbol Changes
Addition/Subtraction: > stays >
Multiplication/Division by positive: > stays >
Multiplication/Division by negative: > becomes <
Key: Flip symbol only when multiplying/dividing by negative!
Remember: Flip Symbol When Multiplying/Dividing by Negative!

Questions & Answers

Question: Why do we have to flip the inequality symbol when multiplying or dividing by a negative number? It doesn't happen with equations.

Answer: This is a great observation! The reason is fundamental to how inequalities work:

Let's look at a simple example: Start with \(3 > 2\), which is true.

If we multiply both sides by -1 without flipping the symbol: \(-3 > -2\), which is FALSE!

But if we flip the symbol: \(-3 < -2\), which is TRUE!

Geometrically, multiplying by -1 reflects numbers across zero on the number line. So the relationship between the numbers reverses: what was larger becomes smaller and vice versa.

With equations, both sides remain equal regardless of the sign, so the equals sign doesn't change. With inequalities, the relationship changes when you multiply by a negative number.

Question: How do I know which direction the inequality symbol should point in my final answer?

Answer: The key is to think about the solution in terms of the variable:

< (less than): The variable is smaller than the number

> (greater than): The variable is larger than the number

≤ (less than or equal): The variable is smaller than or equal to the number

≥ (greater than or equal): The variable is larger than or equal to the number

Also, remember the mnemonic: "The mouth opens toward the bigger number."

For example, if you get \(x > 7\), the solution includes all numbers bigger than 7.

Always check with a test value: if \(x > 7\), try \(x = 8\) in the original inequality to verify it works!

Question: What's the difference between < and ≤ in terms of the solution?

Answer: The difference is about the boundary value:

x < 5: The solution includes all numbers less than 5, but NOT 5 itself

x ≤ 5: The solution includes all numbers less than 5 AND 5 itself

Graphically:

  • For \(x < 5\): Draw an open circle at 5 and shade to the left
  • For \(x \leq 5\): Draw a closed circle at 5 and shade to the left

In interval notation:

  • \(x < 5\) is written as \((-\infty, 5)\)
  • \(x \leq 5\) is written as \((-\infty, 5]\)

The brackets indicate whether the endpoint is included: parentheses mean excluded, brackets mean included.