Two negatives do not always make a positive. It depends entirely on the operation.
Evaluate
−5 − (−8)
Subtracting a negative is the same as adding: −5 − (−8) becomes −5 + 8 = 3. Picture it on a number line — you start at −5 and move 8 to the right. A adds the two negatives as though the second minus were part of the number. B keeps the answer negative out of habit; 8 is bigger than 5, so the result must be positive.
|x| = how far x is from zero, ignoring direction.
|−12| = 12
distance only
−|−12| = −12
the minus is outside the bars
|3 − 10| = |−7| = 7
finish inside the bars first
Work out everything inside before taking the absolute value. |3 − 10| is 7, not |3| − |10| = −7.
| Signs | Do this | Example |
|---|---|---|
| same | add the sizes, keep the sign | −6 + (−4) = −10 |
| different | subtract, keep the bigger one's sign | −9 + 4 = −5 |
| different | same rule again | −3 + 11 = 8 |
Money is the easiest picture: owing ₱6 and then owing ₱4 more leaves you ₱10 down. Owing ₱9 and earning ₱4 leaves you ₱5 down.
a − b becomes a + (−b), every time.
−5 − (−8)
the opening problem
= −5 + 8
the opposite of −8 is 8
= 3
different signs, so subtract
Rewrite every subtraction as an addition before applying the sign rules. Trying to keep both in your head at once is what produces −13.
An even number of negative factors gives a positive; an odd number gives a negative.
(−2)(3)(−4)
two negatives
sizes: 2 × 3 × 4 = 24
ignore signs first
= +24
even count → positive
"Two negatives make a positive" is true for multiplying and dividing, and for subtracting a negative. It is not true for adding: −6 + (−4) is −10, not 10.
Gains, rises and deposits are +. Losses, drops and payments are −.
A tricycle driver starts the day owing his operator ₱120. He earns ₱450 in fares and then pays ₱280 for the day's boundary. Where does he stand?
−120 + 450 − 280
owing is negative; earning adds; paying subtracts
= 330 − 280
different signs: 450 − 120, keep the + sign
= 50
positive, so he is ₱50 ahead
Check by reading it back: ₱450 clears the ₱120 debt with ₱330 left over, and paying ₱280 leaves ₱50 ✓. The sign of the answer is the answer to "ahead or behind".
Writing the debt as +120 gives 120 + 450 − 280 = 290, as though he started with money in his pocket. Owing is below zero, so it starts at −120 and the true result is 50.
The same sign rules; these are the three places a careful student still slips.
Evaluate (−3)³ + (−1)⁸.
(−3)³ = (−3)(−3)(−3) = −27
three negative factors: odd, so negative
(−1)⁸ = 1
eight negative factors: even, so positive
−27 + 1 = −26
different signs, so subtract
A negative base raised to an odd power stays negative; to an even power it turns positive.
Evaluate (−48) ÷ (−6) × (−2).
(−48) ÷ (−6) = 8
left to right: the division comes first here
8 × (−2) = −16
one negative factor left
Multiplying first gives (−6) × (−2) = 12 and then (−48) ÷ 12 = −4. Multiplication and division share one level, so they are done in order from left to right, which gives −16.
Evaluate −|4 − 9| × (−2).
|4 − 9| = |−5| = 5
finish inside the bars
−5 × (−2) = 10
the minus outside the bars stays
The bars make the inside positive; a minus written outside them still counts as a negative factor.
Brackets, exponents, multiply and divide, then add and subtract.
−3 + 4(−2)² ÷ 8
the expression
= −3 + 4(4) ÷ 8
(−2)² = 4, the bracket is squared
= −3 + 16 ÷ 8
multiply and divide left to right
= −3 + 2 = −1
add last
Integer items are usually one long expression mixing signs, powers and bars, or a short story about money, depth or temperature. The fastest route is to rewrite every subtraction as adding the opposite and then count negatives; the 10-second check is the sign alone — decide whether the answer must be positive or negative before you trust the size.
Evaluate
|−7 + 3| − (−2)(−5)
Inside the bars first: −7 + 3 = −4, so |−4| = 4. Then (−2)(−5) = 10, two negatives multiplied. Finally 4 − 10 = −6. C adds instead of subtracting the 10. B gets the size right but not the sign — 10 is bigger than 4, so the answer must be negative.
Hint: three separate jobs — finish inside the bars, do the multiplication, then subtract. Write each one on its own line.
