The Number System

1. a) $-8 + 5 = -3$. When adding numbers with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.

b) $-3 + (-7) = -10$. When adding two negative numbers, add their absolute values and keep the negative sign.

c) $6 - (-4) = 6 + 4 = 10$. Subtracting a negative is the same as adding its opposite.

d) $-15 - (-9) = -15 + 9 = -6$. Subtracting $-9$ is the same as adding $9$.

2. Starting depth: $-250$ feet.

After rising $80$ feet:

$-250 + 80 = -170 \text{ feet}$

After diving $120$ feet:

$-170 + (-120) = -170 - 120 = -290 \text{ feet}$

The submarine's final depth is $-290$ feet (or $290$ feet below sea level).

3. a) $(-4) \times 7 = -28$. A negative times a positive gives a negative result.

b) $(-9) \times (-6) = 54$. A negative times a negative gives a positive result.

c) $(-48) \div 8 = -6$. A negative divided by a positive gives a negative result.

d) $(-56) \div (-7) = 8$. A negative divided by a negative gives a positive result.

4. Follow order of operations — multiplication before addition.

First, compute the multiplication:

$\left(-\frac{2}{3}\right) \times \frac{6}{5} = \frac{(-2) \times 6}{3 \times 5} = \frac{-12}{15} = -\frac{4}{5}$

Now add:

$\frac{3}{4} + \left(-\frac{4}{5}\right) = \frac{3}{4} - \frac{4}{5}$

Find a common denominator (LCD $= 20$):

$\frac{3}{4} = \frac{15}{20}, \quad \frac{4}{5} = \frac{16}{20}$

$\frac{15}{20} - \frac{16}{20} = -\frac{1}{20}$

5. Temperature at noon: $-3.5°$F.

At 6 PM (dropped by $8.2°$):

$-3.5 - 8.2 = -11.7°\text{F}$

At midnight (rose by $4.7°$):

$-11.7 + 4.7 = -7.0°\text{F}$

The temperature at midnight was $-7.0°$F.