The Density Property states that there is a point between any two points on the number line. Write a rational number in simplest form that is between each pair of numbers. $\frac { 4 } { 5 }$ and $\frac { 7 } { 12 }$

Calculate:

$$\sqrt{\dfrac{25}{64}}$$

A 0.25 -mol sample of a weak acid with an unknown $pK_a$ was combined with 10.0 mL of 3.00 M KOH, and the resulting solution was diluted to 1.500 L. The measured pH of the solution was 3.85. What is the $pK_a$. of the weak acid?

Which of the given values is a solution of the equation? m-3=8 ; 5,11

Determine the values of a for which the system has no solutions, exactly one solution, or infinitely many solutions.

$$\begin{array}{r}{x+\quad 2 y=1} \\ {2 x+\left(a^{2}-5\right) y=a-1}\end{array}$$

Solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$$\left\{\begin{array}{r}{2 x-y=-1} \\ {x+\frac{1}{2} y=\frac{3}{2}}\end{array}\right.$$

Express each proposition as an English sentence and determine whether it is true or false, where r and s are the propositions. r: "the moon is a cube" s: "rain is wet" $r \vee s$

Show that, under these circumstances, the sum of the lengths of the intervals making up each $C _ { n }$ no longer tends to zero as $n \rightarrow \infty .$ What is this limit?

If you had a liquid whose density was half that of mercury, how tall would you need to build a barometer to measure atmospheric pressure in a location where the record pressurfe recorded was $750 \mathrm{~mm} \mathrm{Hg}$ ?

Suppose that two competing television stations, station 1 and station 2, each have 50% of the viewer market at some initial point in time. Assume that over each one-year period station 1 captures 5% of station 2's market share and station 2 captures 10% of station 1's market share. (a) Make a table that shows the market share of each station over a five-year period. (b) Over the long term, how will the market share be distributed between the two stations?

Compute the antiderivative using appropriate substitutions.

$$\int \frac{d t}{\sin ^{-1} t \sqrt{1-t^{2}}}$$

The north pole of a bar magnet points toward a thin circular coil of wire containing 40 turns. The magnet is moved away from the coil, so that the flux through one turn inside the coil decreases by 0.3 $\mathrm{T} \cdot \mathrm{m}^{2}$ in 0.2 s. What is the average emf in the coil during this time interval? Viewed from the right side (opposite the bar magnet), does the induced current run clockwise or counterclockwise? Explain briefly.

Suppose that you are the president of a less developed country. Write a speech to persuade the wealthy elite of your country to invest at home.

Fill in the blank with the correct inequality symbol. If x > -2, then -4x _____ 8.