Solve the following problems: 1) The sum of two numbers exceeds their product by one. Their difference is equal to their product less five. Find the numbers. 2) Solve the inequalities: 3x - 1 > -4 and

Let the two numbers be $x$ and $y$.
"The sum of two numbers exceeds their product by one":

$$x + y = xy + 1 \quad (1)$$

"Their difference is equal to their product less five":

$$x - y = xy - 5 \quad (2)$$

Subtract equation (2) from equation (1):

$$(x + y) - (x - y) = (xy + 1) - (xy - 5)$$ $$x + y - x + y = xy + 1 - xy + 5$$ $$2y = 6$$ $$y = 3$$

$$x + 3 = x(3) + 1$$ $$x + 3 = 3x + 1$$ $$3 - 1 = 3x - x$$ $$2 = 2x$$ $$x = 1$$

The two numbers are 1 and 3.

$$\boxed{1 \text{ and } 3}$$

2)

$$3x - 1 > -4$$

Add 1 to both sides:

$$3x > -4 + 1$$ $$3x > -3$$

Divide by 3:

$$x > -1$$

The problem states $2x + 1 \le 7$.
Subtract 1 from both sides:

$$2x \le 7 - 1$$ $$2x \le 6$$

Divide by 2:

$$x \le 3$$

The values of $x$ must satisfy both $x > -1$ and $x \le 3$.
This can be written as a compound inequality:

$$\boxed{-1 < x \le 3}$$

3)

$$-12 \div (-3) \times 4 - (-15)$$

Perform division first:

$$4 \times 4 - (-15)$$

Perform multiplication:

$$16 - (-15)$$

Simplify subtraction of a negative:

$$16 + 15 = 31$$

$$-5 \times 6 \div 2 + (-5)$$

Perform multiplication first:

$$-30 \div 2 + (-5)$$

Perform division:

$$-15 + (-5)$$

Perform addition:

$$-15 - 5 = -20$$

$$\frac{31}{-20} = -\frac{31}{20}$$ $$\boxed{-\frac{31}{20}}$$

4)

$$(-4)(-2) + (-12) - (+3) + -20 + (+4) + -6$$

Perform multiplication:

$$8 + (-12) - 3 - 20 + 4 - 6$$

Combine terms:

$$8 - 12 - 3 - 20 + 4 - 6$$ $$(8 + 4) - (12 + 3 + 20 + 6)$$ $$12 - 41$$ $$-29$$

The denominator is given as $-9 - (15) \quad 46 - (8+2) - 3$. This implies multiplication between the two parts.
First part:

$$-9 - 15 = -24$$

Second part:

$$46 - (8+2) - 3$$ $$46 - 10 - 3$$ $$36 - 3 = 33$$

Multiply the two parts:

$$(-24) \times (33) = -792$$

$$\frac{-29}{-792} = \frac{29}{792}$$ $$\boxed{\frac{29}{792}}$$

5)

$$-8 - 2 + 12 \times 9 - 4 \times 6$$

Perform multiplications:

$$-8 - 2 + 108 - 24$$

Perform additions and subtractions from left to right:

$$-10 + 108 - 24$$ $$98 - 24$$ $$74$$

$$56 - 7 \times 2$$

Perform multiplication:

$$56 - 14$$

Perform subtraction:

$$42$$

$$\frac{74}{42}$$

Divide both numerator and denominator by their greatest common divisor, which is 2:

$$\frac{74 \div 2}{42 \div 2} = \frac{37}{21}$$ $$\boxed{\frac{37}{21}}$$