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# System prepended metadata

title: Sample midterm exam D

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**Problem 1**
Perform the following operation by combining the fractional expressions.

$$
\frac{1}{4x-5y} - \frac{3}{4x+5y}
$$

**Problem 2**
Solve the following equation over the set of real numbers.

$$
2(2x-2) + 7(x+2) = 7
$$

**Problem 3**
Determine the solution set of the following system of equations.

$$
6x = 10 - 7y, \quad xy = -8
$$

**Problem 4**
Solve the following inequality over the set of real numbers and determine the solution in interval notation.

$$
|6x-4|  \leq -7
$$

**Problem 5**
Perform the following operation and express the result as a power of x with a fractional exponent.

$$
\frac{\sqrt[5]{x^4 \cdot \sqrt[4]{x}}}{(\sqrt[5]{x})^{-7}}
$$

**Problem 6**
Solve the following equation over the set of real numbers.

$$
\sqrt{x-1} \cdot \sqrt{6x-5} = 1
$$

**Problem 7**
Graph the function given by the following formula and determine its largest possible domain and range in interval notation.

$$
f(x) = -(x-7)(x+2)
$$

**Problem 8**
The 4th term of a geometric sequence is 108, and its 8th term is 8748. The common ratio is a positive number. What is the first term of the sequence?

**Problem 9**
The equation of a circle is

$$
x^2 - 18x + y^2 + 10y + 102 = 0.
$$

Transform the equation into a form from which the coordinates of the center and the radius of the circle can be directly read. What are the coordinates of the center and what is the radius?

**Problem 10**
A right triangle has one angle of 67°, and the leg adjacent to the angle is 13 cm long. Calculate the area of the triangle!