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# System prepended metadata

title: Sample midterm exam C

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**Problem 1.** Simplify the following expression.

$$
\frac{x^2 + 6x}{x^2 + 12x + 36}
$$

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

$$
7 \cdot (-7x - 2) = 8 - 5 \cdot (x + 4)
$$

**Problem 3.** A bistro sells 5 dl of orange juice and 2 dl of mango juice for 3075 Ft, while 3 dl of orange juice and 6 dl of mango juice cost 4725 Ft. Determine the price of 1 dl of orange juice and 1 dl of mango juice.

**Problem 4.** Determine the largest possible domain of the function given by the following formula in interval notation.

$$
f(x) = \log_{6}(3x^2 + 6x - 9)
$$

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

$$
\frac{\sqrt[5]{x^2 \cdot \sqrt[3]{x}}}{(\sqrt[3]{x})^{-6}}
$$

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

$$
\frac{\sqrt{-6x - 9}}{\sqrt{-3x - 7}} = 9
$$

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

$$
f(x) = -6 \cdot 5^x + 6
$$

**Problem 8.** The price of a product is first increased by 5%, and then decreased 4 times in a row, each time by 4%. By what percentage did the price of the product change overall?

**Problem 9.** A circle has center at $(4, -6)$ and radius 5. Determine whether the point $(9, 6)$ lies inside, on, or outside the circle. Justify your answer with calculation.

**Problem 10.** One side of a triangle has length $c = 17$ cm, and the adjacent angles are $12^\circ$ and $129^\circ$. What is the length of the longest side of the triangle?

