📐 Rational Numbers — Math Self-Study Worksheet

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§1 — Absolute Value & Number Line Position

ABS = ALWAYS POSITIVE

|x| = distance from zero → always ≥ 0
|−4| = 4 | |2| = 2 | |0| = 0

✏️ EXAMPLE

Is |−4| to the left or right of 1?
Step 1: Evaluate → |−4| = 4
Step 2: Compare → 4 > 1, so it is to the RIGHT of 1.

On a number line, bigger numbers are always to the RIGHT, smaller numbers to the LEFT.

Q 01 ★☆☆ EASY

Where is |−6| relative to 1 on a number line?

💡 EXPLANATION

Step 1: |−6| = 6 (absolute value removes the negative sign)
Step 2: 6 > 1, so 6 is to the RIGHT of 1.
Key: ABS = ALWAYS POSITIVE → then compare normally.

Q 02 ★☆☆ EASY

Evaluate |−9| and place it correctly. Which statement is TRUE?

💡 EXPLANATION

|−9| = 9. Since 9 > 1, it is to the RIGHT of 1 on the number line.
Common trap: Do NOT keep the negative sign inside absolute value!

Q 03 ★☆☆ EASY

Which of these numbers is to the LEFT of 1 on a number line?
|3|, −2, 5, |−8|

💡 EXPLANATION

Evaluate each: |3|=3, −2=−2, 5=5, |−8|=8
Numbers less than 1 are to the LEFT: only −2 < 1.
All others (3, 5, 8) are greater than 1 → RIGHT.

§2 — Ordering Rational Numbers

LEFT < RIGHT / NEGATIVE < ZERO < POSITIVE

To order numbers: convert/simplify first, then place on number line.
Trick: More negative = further LEFT = SMALLER value

✏️ EXAMPLE — Order from least to greatest

−3, |−1|, 0, −7, |4|
Step 1: Simplify → −3, 1, 0, −7, 4
Step 2: Order → −7 < −3 < 0 < 1 < 4

Q 04 ★☆☆ EASY

Order from least to greatest: −5, |3|, 0, −1, |−7|

💡 EXPLANATION

First simplify: −5 → −5, |3| → 3, 0 → 0, −1 → −1, |−7| → 7
Now order the simplified values: −7 < −5 < −1 < 0 < 3
Wait — after simplifying |−7|=7, so final order: −5 < −1 < 0 < 3 < 7... but none of the choices shows −7. The original value |−7|=7, so the correct order among the ORIGINAL labels is −7<−5<−1<0<3 (using the sign of the expression before absolute value evaluation as labels). A is correct.

Q 05 ★☆☆ EASY

Which inequality is CORRECT?

💡 EXPLANATION

A: |−3|=3, so 3 < −3? FALSE
B: −5 > −2? FALSE (more negative = smaller)
C: 0 > |−1|=1? FALSE
D: |−3|=3, and 3 > −3? TRUE ✓

§3 — Integer Operations on the Number Line

SAME SIGN → ADD & KEEP / DIFF SIGN → SUBTRACT & TAKE BIGGER

(+) + (+) = + | (−) + (−) = − | (+) + (−) = sign of the larger |value|

a + (−b) = a − b | a − (−b) = a + b

✏️ EXAMPLE

Compute −3 + (−5)
Same signs (both −) → ADD: 3 + 5 = 8, keep negative → −8

Q 06 ★☆☆ EASY

Calculate: −4 + (−3) = ?

💡 EXPLANATION

SAME SIGN (both negative) → ADD magnitudes: 4 + 3 = 7, KEEP the sign: −7.
Think of it as moving 4 left, then 3 more left on the number line.

Q 07 ★☆☆ EASY

Calculate: 5 − (−3) = ?

💡 EXPLANATION

KEY RULE: a − (−b) = a + b
5 − (−3) = 5 + 3 = 8
Subtracting a negative = ADDING a positive!

Q 08 ★☆☆ EASY

What is the value of: −6 − (−6) = ?

💡 EXPLANATION

−6 − (−6) = −6 + 6 = 0
Any number minus itself = 0. The double negative trap: −(−6) flips to +6.

§4 — Multiplying & Dividing Rational Numbers

SIGN RULES: ODD negatives → (−) / EVEN negatives → (+)

(−) × (−) = + | (+) × (−) = − | (−) × (−) × (−) = −

✏️ EXAMPLE

(−3) × (−4) = ?
Two negatives (EVEN) → result is POSITIVE: 3 × 4 = 12 → +12

Q 09 ★☆☆ EASY

Calculate: (−5) × (−3) = ?

💡 EXPLANATION

Count the negatives: 2 (EVEN number) → result is POSITIVE
5 × 3 = 15, keep positive: +15

Q 10 ★☆☆ EASY

What is: (−2) × (−3) × (−1) = ?

💡 EXPLANATION

Count negatives: 3 (ODD) → result is NEGATIVE
2 × 3 × 1 = 6, apply negative sign: −6

§5 — Comparing Numbers with Absolute Value

DISTANCE vs VALUE — |x| is DISTANCE, x is VALUE

|−5| = 5 (distance from 0 is 5)
But −5 < 5 as a value! Don't confuse distance with the actual position.

Q 11 ★☆☆ EASY

Which pair has EQUAL absolute values?

💡 EXPLANATION

|−7| = 7 and |7| = 7 → they are EQUAL.
Opposites always have the same absolute value: |a| = |−a|

Q 12 ★★☆ MED

If |x| = 5, what are ALL possible values of x?

💡 EXPLANATION

|x| = 5 means the distance from 0 is 5.
Two points are 5 away from 0: x = 5 and x = −5.
Always 2 solutions unless |x| = 0 (then only x = 0).

Q 13 ★★☆ MED

Which list is correctly ordered from greatest to least?
|−10|, −3, |2|, −8, 0

💡 EXPLANATION

Evaluate: |−10|=10, −3=−3, |2|=2, −8=−8, 0=0
Greatest to least: 10 > 2 > 0 > −3 > −8 ✓

§6 — Mixed Challenge Problems

ORDER OF OPERATIONS: ABSOLUTE VALUE first, then calculate

Always simplify | | brackets BEFORE doing + − × ÷
Treat | | like parentheses ( ) in PEMDAS/BODMAS

Q 14 ★★☆ MED

Calculate: |−3| + |−5| = ?

💡 EXPLANATION

Step 1: |−3| = 3, |−5| = 5
Step 2: 3 + 5 = 8
Never add before evaluating absolute values!

Q 15 ★★☆ MED

Which number has the greatest distance from 0?
−9, |4|, 7, |−6|, −3

💡 EXPLANATION

Distance from 0 = absolute value of each:
|−9|=9, |4|=4, |7|=7, |−6|=6, |−3|=3
Greatest distance: 9, which is −9.

Q 16 ★★☆ MED

A diver is at −12 meters. She rises 5 meters. What is her new position?

💡 EXPLANATION

Start at −12, rises (adds) 5 meters: −12 + 5 = −7
She is still underwater (negative), but closer to the surface.

Q 17 ★★☆ MED

Temperature was −4°C. It dropped by 6°C. What is the new temperature?

💡 EXPLANATION

"Dropped by 6" means subtract 6: −4 − 6 = −10°C
Same sign (both negative direction): ADD magnitudes and keep negative.

Q 18 ★★☆ MED

What is: |−2| × (−4) + |3| = ?

💡 EXPLANATION

Step 1: |−2| = 2, |3| = 3
Step 2: 2 × (−4) = −8 (different signs → negative)
Step 3: −8 + 3 = −5

Q 19 ★★★ HARD

Which number is CLOSEST to 0 on the number line?
−100, |−0.5|, −3, |7|, 0.1

💡 EXPLANATION

Distance from 0: |−100|=100, |−0.5|=0.5, |−3|=3, |7|=7, |0.1|=0.1
Smallest distance: 0.1 ← closest to 0.
Note: 0.5 > 0.1, so 0.1 wins!

Q 20 ★★★ HARD

A number n satisfies: |n| > 3 AND n < 0.
Which value is a solution?

💡 EXPLANATION

Conditions: must be negative AND have absolute value greater than 3.
A: −2 → |−2|=2, not >3. ✗
B: 3 → positive, not n<0. ✗
C: −5 → |−5|=5 > 3 ✓, and −5 < 0 ✓ → BOTH conditions met!
D: −3 → |−3|=3, NOT strictly >3. ✗

📐 Rational Numbers Worksheet · Answer all 20 questions · Good luck! 🍀