Integers & Directed Numbers

On a number line, numbers further to the left are smaller. $-5$ is further left than $-2$, so $-5$ is smaller. (With negative numbers, the one with the larger digit is actually the smaller value.)

Start at $-3$ and move 8 to the right: $-3 + 8 = 5$. (The positive 8 outweighs the negative 3.)

Substitute carefully: $a^2 = (-3)^2 = 9$ and $-2b = -2\times(-4) = +8$. So $9 + 8 = 17$. (Writing $(-3)^2$ as $-9$, or $-2\times(-4)$ as $-8$, are the common traps.)

Subtracting moves further to the left: $-2 - 5 = -7$. (Two "downward" moves add together.)

Subtracting a negative is the same as adding: $-6 - (-4) = -6 + 4 = -2$. (Two minus signs together become a plus.)

The two known integers multiply to $-2 \times 3 = -6$. The third must give $-6 \times (\text{third}) = -24$, so it is $\dfrac{-24}{-6} = 4$. (Check: $-2 \times 3 \times 4 = -24$.)

Multiply the fourth term by $-2$: $-24 \times (-2) = 48$. The sign alternates and the size doubles each step, so the fifth term is $48$.

Work backwards. Undo the $+7$: $-8 - 7 = -15$. Undo the $\times(-3)$: $\dfrac{-15}{-3} = 5$. (Check: $5 \times (-3) + 7 = -15 + 7 = -8$.)

The five temperatures must total $5 \times (-1) = -5$°C. The four known ones total $-5 + 2 - 3 + 0 = -6$°C. So the fifth is $-5 - (-6) = 1$°C.

The most negative number is smallest, so the order is $-7,\ -3,\ 0,\ 2,\ 5$. ($-7$ is smaller than $-3$ even though 7 is a bigger digit.)

Multiplication first: $3 \times (-4) = -12$. Then $-2 - (-12) + (-5) = -2 + 12 - 5 = 5$. (Working strictly left to right without multiplying first gives a wrong answer.)

$(-2)^2 = 4$, $(-3)^2 = 9$ and $(-1)^3 = -1$. So $4 - 9 + (-1) = 4 - 9 - 1 = -6$. (Both squares are positive; the odd power $(-1)^3$ stays negative.)

The brackets mean the whole $-3$ is squared: $(-3)^2 = (-3)\times(-3) = 9$ (negative times negative is positive).

Without brackets, only the 3 is squared: $3^2 = 9$, then apply the negative to get $-9$. (Compare with $(-3)^2 = 9$: the brackets are the trap.)

Range = highest − lowest = $4 - (-9) = 4 + 9 = 13$°C. (Subtracting a negative adds; doing $4 - 9$ with the wrong sign gives 5.)

$-3^2 = -(3\times 3) = -9$; $(-3)^2 = 9$; $(-2)^3 = -8$; $3^2 = 9$; $(-1)^{10} = 1$. The smallest is $-3^2 = -9$. (The brackets and where the negative sign sits change the value completely.)

Descending 3 m each second for 4 seconds is $4 \times (-3) = -12$, so the diver is at $-12$ m (12 m below the surface). (Adding 3 and 4 instead of multiplying gives the wrong $-7$.)

Follow the steps in order: $-25 + 40 = 15$, then $15 - 30 = -15$. So the account ends $15 overdrawn, a balance of -$15.

The multiples of 3 in that range are $-9,\ -6,\ -3,\ 0,\ 3,\ 6,\ 9$, which is 7 numbers. (Remember 0 is a multiple of 3, and count the negatives too.)

The student treated $-3 - 5$ as $-3 + 5$. But subtracting 5 from $-3$ moves further left on the number line: $-3 - 5 = -8$.

Let the middle integer be $n$; the three are $n-1,\ n,\ n+1$ and their sum is $3n = -6$, so $n = -2$. The integers are $-3, -2, -1$, and the largest is $-1$.

Call them $x$ and $y$ with $x + y = -2$ and $x - y = 8$. Adding the two facts gives $2x = 6$, so $x = 3$ and $y = -5$. (Check: $3 + (-5) = -2$ and $3 - (-5) = 8$.)

$(-2)^3 = (-2)\times(-2)\times(-2)$. The first two give $+4$, then $4 \times (-2) = -8$. (An odd power of a negative stays negative.)

A positive product means the two integers share the same sign (both positive or both negative). For the sum to be negative as well, they must both be negative. (Two positives would give a positive sum.)

The midpoint is the average: $\dfrac{-6 + 4}{2} = \dfrac{-2}{2} = -1$. (Check: $-1$ is 5 steps from both $-6$ and $4$.)