Solved Exercises on Finding Number of Solutions in Grade 4

Master finding number of solutions: equations, inequalities, and patterns through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Simple Equation Solutions
Exercise 1
How many solutions does this equation have? x + 5 = 8
Definition:

Equation Solution: A value that makes the equation true when substituted for the variable.

Method:
  1. Isolate the variable by performing inverse operations
  2. Check if the resulting value satisfies the original equation
  3. Count the number of valid solutions found
Original Equation
x + 5 = 8
Subtract 5
x = 8 - 5
Solution
x = 3
Step 1: Subtract 5 from both sides

x + 5 - 5 = 8 - 5

x = 3

Step 2: Verify the solution

Replace x with 3 in original equation:

3 + 5 = 8 ✓

Step 3: Count solutions

We found exactly one value: x = 3

x = 3
1
Number of solutions: 1
Final answer:

This equation has 1 solution: x = 3

Applied rules:

• Inverse Operations: Addition and subtraction are inverses

• Equality Property: What you do to one side, do to the other

• Verification: Always check your solution in original equation

2 Inequality Solutions
Exercise 2
How many solutions does this inequality have? x > 5 (x is a whole number)
Definition:

Inequality Solution: Values that make the inequality true. These often form a set of numbers.

Inequality
x > 5
Whole numbers
6,7,8,9...
Solutions
Infinite
Step 1: Understand the constraint

x > 5 means x is greater than 5

Since x is a whole number: 6, 7, 8, 9, 10...

Step 2: List possible values

6 > 5 ✓

7 > 5 ✓

8 > 5 ✓

And so on forever...

Step 3: Count solutions

There are infinitely many whole numbers greater than 5

x = 6
x = 7
x = 8
x = 9
...
6
7
8
9
...
Number of solutions: ∞ (infinite)
Final answer:

This inequality has infinite solutions: 6, 7, 8, 9, 10, ...

Applied rules:

• Inequality: Shows a range of possible values

• Domain restriction: Specifies allowed values (whole numbers)

• Infinite sets: Some inequalities have infinitely many solutions

3 Multiple Solutions
Exercise 3
How many solutions does this equation have? 2 × x = 12
Definition:

Multiplication Equation: An equation where the variable is multiplied by a number.

Original Equation
2 × x = 12
Divide by 2
x = 12 ÷ 2
Solution
x = 6
Step 1: Divide both sides by 2

(2 × x) ÷ 2 = 12 ÷ 2

x = 6

Step 2: Verify the solution

Replace x with 6 in original equation:

2 × 6 = 12 ✓

Step 3: Check for other solutions

Try x = 5: 2 × 5 = 10 ≠ 12 ✗

Try x = 7: 2 × 7 = 14 ≠ 12 ✗

Only x = 6 works

x = 6
1
Number of solutions: 1
Final answer:

This equation has 1 solution: x = 6

Applied rules:

• Inverse Operations: Division undoes multiplication

• One-to-one relationship: Linear equations have one solution

• Verification: Always substitute back to check

Rules and methods, laws,...
x + a = b \Rightarrow x = b - a
Addition/Subtraction Equations
Rule 1
x + a = b \Rightarrow x = b - a
Addition equation has 1 solution
Rule 2
a × x = b \Rightarrow x = b ÷ a
Multiplication equation has 1 solution
Rule 3
x > a \Rightarrow infinite\ solutions
Inequality may have infinite solutions
Key definitions:

Equation: Statement showing two expressions are equal

Solution: Value that makes an equation or inequality true

Number of Solutions: Count of valid values for the variable

Complete methodology:
  1. Analyze the problem: Identify if it's an equation or inequality
  2. Solve systematically: Use inverse operations to isolate the variable
  3. Verify solutions: Substitute back into original statement
  4. Count solutions: Determine total number of valid answers
Tip 1: Linear equations (with x) have exactly 1 solution.
Tip 2: Inequalities often have infinite solutions.
Tip 3: Always verify your solution by substitution.
Tip 4: Domain restrictions affect the number of solutions.
Common patterns: Addition/subtraction equations: 1 solution, Multiplication/division equations: 1 solution, Inequalities: often infinite solutions.
Key insight: The operation determines the method and number of solutions.
Formulas to know by heart:

• x + a = b → x = b - a (1 solution)

• x - a = b → x = b + a (1 solution)

• a × x = b → x = b ÷ a (1 solution)

• a ÷ x = b → x = a ÷ b (1 solution)

Solution: Exercises 4 to 5
4 Two-Step Equations
Exercise 4
How many solutions does this equation have? 2x + 3 = 11
Definition:

Two-Step Equation: An equation requiring two inverse operations to solve.

Original
2x + 3 = 11
Subtract 3
2x = 8
Divide by 2
x = 4
Step 1: Subtract 3 from both sides

2x + 3 - 3 = 11 - 3

2x = 8

Step 2: Divide both sides by 2

(2x) ÷ 2 = 8 ÷ 2

x = 4

Step 3: Verify the solution

Replace x with 4 in original equation:

2(4) + 3 = 8 + 3 = 11 ✓

Step 4: Check for other solutions

Try x = 3: 2(3) + 3 = 6 + 3 = 9 ≠ 11 ✗

Try x = 5: 2(5) + 3 = 10 + 3 = 13 ≠ 11 ✗

Only x = 4 works

x = 4
1
Number of solutions: 1
Final answer:

This equation has 1 solution: x = 4

Applied rules:

• Order of operations: Undo operations in reverse order

• Two-step process: First undo addition/subtraction, then multiplication/division

• Verification: Always substitute back to confirm

5 Domain-Restricted Solutions
Exercise 5
How many solutions does this inequality have? x < 10 where x is a positive even number?
Definition:

Domain Restriction: Additional conditions that limit possible values for the variable.

Condition 1
x < 10
Condition 2
x is even
Solutions
2,4,6,8
Step 1: List all positive even numbers

Positive even numbers: 2, 4, 6, 8, 10, 12, 14...

Step 2: Apply the inequality constraint

Which of these are less than 10?

2 < 10 ✓, 4 < 10 ✓, 6 < 10 ✓, 8 < 10 ✓

10 < 10 ✗ (not strictly less than)

Step 3: List valid solutions

Solutions: 2, 4, 6, 8

Step 4: Count solutions

There are 4 solutions

x = 2
x = 4
x = 6
x = 8
2
4
6
8
Number of solutions: 4
Final answer:

This inequality has 4 solutions: 2, 4, 6, 8

Applied rules:

• Domain restriction: Additional constraints reduce solution set

• Systematic listing: Enumerate possibilities within constraints

• Verification: Check each potential solution against all conditions

Finding Number of Solutions - Complete Guide
Solutions = \{x | condition\}
Solution Set Notation
Key definitions:

Solution: A value that makes an equation or inequality true when substituted for the variable

Solution Set: The collection of all possible solutions

Domain: The set of allowed values for the variable

Finding Number of Solutions Methods:
  1. Solve the equation/inequality: Find all potential solutions
  2. Apply domain restrictions: Filter solutions based on constraints
  3. Verify each solution: Substitute back into original statement
  4. Count valid solutions: Determine the final number
Tip 1: Linear equations in one variable have exactly 1 solution.
Tip 2: Inequalities often have infinite solutions unless bounded.
Tip 3: Domain restrictions can turn infinite solutions into finite ones.
Tip 4: Always check if your solutions satisfy all conditions.
Tip 5: For word problems, solutions must make sense in context.
Solution types: Exactly one solution (linear equations), zero solutions (contradictions), infinite solutions (identities), finite set (with restrictions).
Verification importance: Never assume a solution is correct without checking.
Important Properties:

• Adding/subtracting same value to both sides preserves equality

• Multiplying/dividing both sides by non-zero value preserves equality

• Domain restrictions limit the solution set

• The number of solutions depends on the equation type and constraints

Solutions Patterns Infographic
📊
Addition Equations
x + a = b → 1 solution
1
Multiplication Equations
a × x = b → 1 solution
1
Inequalities
x > a → ∞ solutions
∞
How to Find Number of Solutions
1
Analyze Type
2
Solve Equation
3
Apply Restrictions
4
Count Solutions
Solution Types Summary
Linear equations: 1 solution
Inequalities: Infinite or finite set
Contradictions: 0 solutions
Identities: Infinite solutions
Remember: Always Verify Your Solutions!

Questions & Answers

Question: How do I know if an equation has one solution, no solution, or infinite solutions?

Answer: At Grade 4 level, most equations you'll encounter have exactly one solution. Here's how to identify different cases:

  • One solution: Standard equations like x + 5 = 8 or 3x = 12
  • No solution: Contradictions like x + 1 = x + 3 (impossible!)
  • Infinite solutions: Identities like x = x or 2x = 2x (always true)

For now, focus on solving equations step-by-step. Most linear equations (with x to the first power) have exactly one solution.

Always check your answer by substituting back into the original equation!

Question: Why do some inequalities have infinite solutions while others have finite ones?

Answer: It depends on the type of inequality and any restrictions:

  • Infinite solutions: Unbounded inequalities like x > 5 (if x can be any number)
  • Finite solutions: Bounded inequalities like 3 < x < 7 with domain restrictions

Example of infinite: x > 5 (x can be 6, 7, 8, 9, 10, ... forever)

Example of finite: x < 10 AND x is a positive even number (solutions: 2, 4, 6, 8)

The domain restrictions (like "x is a positive even number") limit the solution set.

Question: What should I do if I get stuck when trying to find the number of solutions?

Answer: Here's a step-by-step approach when you're stuck:

  1. Read carefully: Make sure you understand what the problem is asking
  2. Solve systematically: Use inverse operations to isolate the variable
  3. Check your work: Substitute your solution back into the original equation
  4. Consider restrictions: Look for any domain limitations
  5. Count carefully: List all valid solutions before counting

If you're still stuck, try simpler numbers or draw a picture to visualize the problem. Practice with different types of equations will build your confidence!

Remember: Every problem has a logical solution path. Take it one step at a time!