Solved Exercises on Integer Number Line in Grade 7

Master integer number line: plotting integers, comparing integers, absolute values, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Plotting Integers on Number Line
Exercise 1
Plot the integers -4, 0, 3, -2, and 5 on a number line. Which integer is furthest to the right? Which is furthest to the left?
Definition:

Integer Number Line: A horizontal line with zero at the center, positive integers to the right, and negative integers to the left.

Method for plotting integers:
  1. Draw a horizontal line with arrowheads on both ends
  2. Mark zero at the center
  3. Mark positive integers to the right of zero
  4. Mark negative integers to the left of zero
  5. Place points at the correct positions
Left to Right
-4, -2, 0, 3, 5
Furthest Left
-4
Furthest Right
5
Step 1: Draw the number line

Draw a horizontal line with equally spaced marks

Step 2: Mark zero at the center

Place 0 in the middle of the line

Step 3: Plot each integer

• -4: Four units to the left of zero

• -2: Two units to the left of zero

• 0: At the center

• 3: Three units to the right of zero

• 5: Five units to the right of zero

Step 4: Identify furthest positions

• Furthest to the right: 5

• Furthest to the left: -4

-4
-2
0
3
5
Furthest left: -4 | Furthest right: 5
Final answer:

The integers -4, -2, 0, 3, and 5 have been plotted. The integer furthest to the right is 5, and the integer furthest to the left is -4.

Applied rules:

• Number Line Direction: Numbers increase as you move right

• Negative Values: Negative numbers are to the left of zero

• Positive Values: Positive numbers are to the right of zero

Key Concept:

On a number line, larger numbers are positioned to the right of smaller numbers. The further right a number is, the larger its value.

2 Comparing Integers
Exercise 2
Use a number line to compare -3 and 2. Which is greater? Explain your reasoning. Also compare -5 and -1.
Definition:

Comparing Integers: Determining which integer is greater by their position on the number line. The number to the right is always greater.

-3 vs 2
2 > -3
-5 vs -1
-1 > -5
Step 1: Plot -3 and 2 on the number line

-3 is to the left of zero, 2 is to the right of zero

Step 2: Compare positions

Since 2 is to the right of -3, 2 is greater than -3

Therefore: 2 > -3

Step 3: Plot -5 and -1 on the number line

Both are to the left of zero, but -1 is closer to zero

Step 4: Compare negative positions

Since -1 is to the right of -5, -1 is greater than -5

Therefore: -1 > -5

2 > -3 | -1 > -5
Final answer:

2 is greater than -3 because 2 is positioned to the right of -3 on the number line. -1 is greater than -5 because -1 is positioned to the right of -5 on the number line.

Applied rules:

• Position Rule: Numbers to the right are greater than numbers to the left

• Positive vs Negative: All positive numbers are greater than negative numbers

• Negative Comparison: Among negatives, the one closer to zero is greater

⬅️
-5
Further from zero
➡️
-1
Closer to zero
🌟
Result
-1 > -5
3 Absolute Value
Exercise 3
Find the absolute value of -7, 4, and -2. Plot these numbers and their absolute values on a number line. What do you notice about the relationship between a number and its absolute value?
Definition:

Absolute Value: The distance of a number from zero on the number line, denoted by |n|. The absolute value is always non-negative.

|−7|
7
|4|
4
|−2|
2
Step 1: Find absolute values

|−7| = 7 (distance from zero)

|4| = 4 (distance from zero)

|−2| = 2 (distance from zero)

Step 2: Plot original numbers and their absolute values

Original: -7, 4, -2

Absolute values: 7, 4, 2

Step 3: Observe the pattern

• Absolute values are always positive

• Absolute values represent distance from zero

• Opposite numbers have the same absolute value

-7
Original
→
7
Absolute Value
→
7 units
Distance from 0
|−7| = 7 | |4| = 4 | |−2| = 2
Final answer:

The absolute values are |−7| = 7, |4| = 4, and |−2| = 2. The absolute value of a number is its distance from zero on the number line, always positive.

Applied rules:

• Absolute Value Rule: |n| = n if n ≥ 0, |n| = -n if n < 0

• Distance Rule: Absolute value represents distance from zero

• Non-Negative Rule: Absolute values are always ≥ 0

Key Concept:

The absolute value of a number represents its distance from zero, regardless of direction. Therefore, |n| = |-n| for any integer n.

Integer Number Line: Rules and Methods
\(\text{Absolute Value}: |n| = \begin{cases} n & \text{if } n \geq 0 \\ -n & \text{if } n < 0 \end{cases}\)
Absolute Value Definition
Number Line
Numbers increase rightward
Positive to right, negative to left
Comparison
Right > Left
Greater numbers to the right
Absolute Value
Distance from zero
Always non-negative
Key definitions:

Integer: A whole number that can be positive, negative, or zero (..., -2, -1, 0, 1, 2, ...).

Number Line: A visual representation of numbers arranged in order along a straight line.

Absolute Value: The distance of a number from zero on the number line, always positive.

Opposite Numbers: Two numbers that are the same distance from zero but on opposite sides.

Integer number line methodology:
  1. Draw Line: Create a horizontal line with arrows
  2. Mark Zero: Place zero at the center
  3. Mark Intervals: Equal spaces to the left and right
  4. Label Numbers: Positive to right, negative to left
  5. Plot Points: Place dots at correct positions
  6. Compare: Use position to determine greater/less
Tip 1: Remember that numbers increase as you move right on the number line.
Tip 2: For negative numbers, the one closer to zero is greater.
Tip 3: Absolute value is always positive - it's distance from zero.
Tip 4: Plot points accurately to avoid comparison errors.
Real-Life Applications: Temperature scales, elevation above/below sea level, debt/credit, time before/after.
Common Pitfalls: Confusing negative signs, misreading number line positions, forgetting absolute value is always positive.
Integer Number Line Rules:

• Direction Rule: Right is greater than left

• Zero Rule: Zero is neither positive nor negative

• Absolute Value Rule: |n| ≥ 0 for all integers n

• Opposite Rule: Numbers equidistant from zero are opposites

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
The temperature at noon was -3°C. It dropped by 5°C by evening. What was the evening temperature? Plot both temperatures on a number line and find the absolute difference between them.
Definition:

Temperature Change: A real-world application of integers where negative temperatures are below freezing and positive temperatures are above freezing.

Evening Temp
-3 - 5 = -8°C
Absolute Difference
|-3 - (-8)| = |5| = 5°C
Step 1: Calculate evening temperature

Starting temperature: -3°C

Temperature drop: 5°C

Evening temperature: -3 - 5 = -8°C

Step 2: Plot temperatures on number line

• Noon: -3°C (3 units left of zero)

• Evening: -8°C (8 units left of zero)

Step 3: Find absolute difference

Absolute difference = |Noon temp - Evening temp|

Absolute difference = |-3 - (-8)| = |-3 + 8| = |5| = 5°C

Evening temperature: -8°C | Absolute difference: 5°C
Final answer:

The evening temperature was -8°C. The absolute difference between noon and evening temperatures is 5°C.

Applied rules:

• Subtraction Rule: Dropping temperature means subtracting

• Absolute Difference: Distance between two values

• Real-World Context: Negative temperatures are below freezing

☀️
Noon
-3°C
🌙
Evening
-8°C
🌡️
Change
-5°C
5 Elevation Problem
Exercise 5
A hiker starts at an elevation of -200 feet (below sea level). She climbs up 450 feet, then descends 150 feet. What is her final elevation? How far is she from sea level? Use a number line to model her journey.
Definition:

Elevation: Height above or below sea level, where sea level is represented by zero.

Start
-200 ft
After Climb
-200 + 450 = 250 ft
Final Elevation
250 - 150 = 100 ft
Step 1: Start at initial elevation

Starting elevation: -200 feet (below sea level)

Step 2: Calculate elevation after climbing

Climbing up 450 feet: -200 + 450 = 250 feet

Step 3: Calculate final elevation after descending

Descending 150 feet: 250 - 150 = 100 feet

Step 4: Find distance from sea level

Distance from sea level = |Final elevation| = |100| = 100 feet

Final elevation: 100 ft | Distance from sea level: 100 ft
Final answer:

The hiker's final elevation is 100 feet above sea level. She is 100 feet away from sea level.

Applied rules:

• Positive Movement: Climbing up increases elevation

• Negative Movement: Descending decreases elevation

• Distance Rule: Distance from zero is absolute value

Comprehensive Summary: Integer Number Line
\(\text{Absolute Value}: |n| = \text{distance from zero}, \quad \text{Comparison}: a > b \text{ if } a \text{ is to the right of } b\)
Core Integer Number Line Formulas
Core Definitions:

Integer Number Line: A visual representation of integers arranged in order along a straight line with zero at the center.

Absolute Value: The distance of a number from zero on the number line, always non-negative.

Opposite Numbers: Two numbers that are the same distance from zero but on opposite sides of zero.

Integer Number Line Problem-Solving Steps:
  1. Draw Number Line: Create horizontal line with zero at center
  2. Mark Intervals: Equally spaced marks on both sides
  3. Label Positions: Positive numbers to right, negative to left
  4. Plot Points: Place dots at correct integer positions
  5. Compare Values: Use position to determine greater/less
  6. Calculate Distance: Use absolute value for distance from zero
Quick Tip: Numbers to the right are always greater than numbers to the left.
Memory Aid: Larger absolute values are further from zero.
Strategy: Plot numbers first, then compare their positions.
Verification: Check that your plotted positions match the actual values.
Real-Life Applications: Temperature scales, elevation above/below sea level, financial gains/losses, sports scores, time before/after.
Common Scenarios: Bank balances, altitude changes, temperature changes, scoring differences.
Key Rules and Properties:

• Direction Rule: Numbers increase as you move right on the number line

• Comparison Rule: A number to the right is greater than a number to the left

• Absolute Value Rule: |n| represents distance from zero, always non-negative

• Zero Rule: Zero is neither positive nor negative

• Opposite Rule: Numbers equidistant from zero are opposites (a and -a)

➡️
Right
Greater values
0️⃣
Zero
Center point
⬅️
Left
Smaller values

Questions & Answers

Question: I get confused about which direction is greater on the number line. How do I remember?

Answer: Great question! Here are some memory aids:

Direction Rule: Numbers increase as you move right on the number line. Think of reading a book from left to right - the numbers get larger.

Visual Memory:

  • Right = "Rise" = Greater values
  • Left = "Less" = Smaller values

Practical Example: If you're moving along the ground, going forward (right) takes you to higher numbers, going backward (left) takes you to lower numbers.

Remember: On a number line, RIGHT = BIGGER, LEFT = SMALLER.

Question: Why is the absolute value always positive? What's the point of it?

Answer: The absolute value represents DISTANCE from zero, and distance is always positive!

Why it's always positive:

  • Distance cannot be negative - you can't walk -5 feet
  • It measures how far, not which direction
  • Both 5 and -5 are 5 units away from zero

Practical applications:

  • Measuring error: How far off was your guess?
  • Temperature difference: How many degrees apart?
  • Financial: How much money was gained or lost?

The absolute value strips away direction (positive/negative) and focuses only on magnitude (size).

Question: When comparing negative numbers, I sometimes get confused. Is -2 greater than -5 or is -5 greater than -2?

Answer: -2 is greater than -5! Here's how to remember:

Number Line Method: Plot both numbers on the number line. The number to the right is greater.

Real-World Analogy: Think of debt. Would you rather owe $2 or $5? Owing $2 (-2) is better than owing $5 (-5), so -2 > -5.

Distance from Zero: Among negative numbers, the one closer to zero is greater.

Rule for Negatives: For negative numbers, the one with the smaller absolute value is greater.

Example:

  • |-2| = 2, |-5| = 5
  • Since 2 < 5, we have -2 > -5

Remember: With negative numbers, the "smaller" number (closer to zero) is actually greater!