Explain why each of the following is not a good definition. Vertical angles are two angles whose sides form opposite rays.

Identify which are experiments and which are observational studies. Laboratory rats are fed diets with three different levels of caffeine content, and observers record how much time the rats spend each day running on their treadmills.

Explain why each statement is not a good definition. Supplementary angles are angles that form a straight line.

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

$$A=\left[\begin{array}{ll} 1 & 3 \\ 0 & 7 \end{array}\right], B=\left[\begin{array}{rr} 2 & 14 \\ 22 & 6 \end{array}\right], C=\left[\begin{array}{rr} 1 & 5 \\ 8 & 92 \\ 12 & 6 \end{array}\right], D=\left[\begin{array}{rr} 10 & 14 \\ 7 & 2 \\ 5 & 61 \end{array}\right], E=\left[\begin{array}{rr} 6 & 12 \\ 14 & 5 \end{array}\right], F=\left[\begin{array}{rr} 0 & 9 \\ 78 & 17 \\ 15 & 4 \end{array}\right]$$

C + F

For the following exercises, solve the system of three equations using substitution or addition.

$$\begin{array}{r} 6 x-4 y-2 z=2 \\ 3 x+2 y-5 z=4 \\ 6 y-7 z=5 \end{array}$$

The energy gap $E_{\mathrm{g}}$ is 0.72 eV for Ge and 1.10 eV for Si. At what wavelengths will Ge and Si be transparent to radiation? At what wavelengths will they begin to absorb significantly?

Compare rusting of metal to redox reactions in voltaic cells.

Explain why each of the following is not a good definition. A line segment is part of a line.

Let $\left(d_{1}, d_{2}, \ldots, d_{k}\right) \cdot\left(w_{1}, w_{2}, \ldots, w_{k}\right) \equiv 0$ (mod n) be an error detection scheme for the k-digit identification number $d_{1} d_{2} \cdots d_{k}$, where $0 \leq d_{i}<n$. Prove that all transposition errors of two digits $d_{i}$ and $d_{j}$ are detected if and only if $\mathrm{g c d}\left(w_{i}-w_{j}, n\right)=1$ for i and j between 1 and k.

For the following exercises, find the multiplicative inverse of each matrix, if it exists.

$$\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right]$$

Discuss the common ion effect by completing the following paragraph. An ion that is common to two or more ionic compounds is known as a ______. The lowering of the solubility of a substance by the presence of a common ion is called the __________.

List all of the elements in each of the following subgroups. (a) The subgroup of $\mathbb{Z}$ generated by 7, (b) The subgroup of $\mathbb{Z}_{24}$ generated by 15, (c) All subgroups of $\mathbb{Z}_{12}$, (d) All subgroups of $\mathbb{Z}_{60}$, (e) All subgroups of $\mathbb{Z}_{13}$, (f) All subgroups of $\mathbb{Z}_{48}$, (g) The subgroup generated by 3 in U(20), (h) The subgroup generated by 6 in U(18), (i) The subgroup of $\mathbb{R}^{*}$ generated by 7, (j) The subgroup of $\mathbb{C}^{*}$ generated by i where $i^{2}=-1$, (k) The subgroup of $\mathbb{C}^{*}$ generated by 2i, (l) The subgroup of $\mathbb{C}^{*}$ generated by $(1+i) / \sqrt{2}$, (m) The subgroup of $\mathbb{C}^{*}$ generated by $(1+\sqrt{3} i) / 2$.

Graph the function $g(x)=\log (12-6 x)+3.$

For the following exercises, find the partial fraction expansion. $\frac{x^{3}-4 x^{2}+5 x+4}{(x-2)^{3}}$