Suppose a and b are real numbers other than zero and that a>b. State whether the inequality is true or false for all real numbers a and b. $\frac{1}{a}>\frac{1}{b}$

What information do we need to determine the molecular formula of a compound from the empirical formula?

Answer each question. What is the value of

$$\left|\begin{array}{rr} 4 & 0 \\ -2 & 0 \end{array}\right| ?$$

Charlie bought a model airplane kit for $5, three small jars of paint for x dollars each, and a bottle of glue for$2. Write and simplify a polynomial expression to represent the total amount of money he spent.

A drug company is testing 200 patients to see if Painfree (a new headache medicine) is effective. Half the patients receive Painfree and half receive a placebo. The data on the first 50 patients is summarized in this matrix:

$$\begin{matrix} \text{ } & \text{Pain Relief Obtained} & \text{ }\\ \text{ } & \text{Yes} & \text{No}\\ \text{Painfree} & \text{22} & \text{3}\\ \text{Placebo} & \text{8} & \text{17}\\ \end{matrix}$$

(a) Of those who took the placebo, how many got relief? (b) Of those who took the new medication, how many got no relief? (c) The test was repeated on three more groups of 50 patients each, with the results summarized by these matrices.

$$\left[\begin{array}{rr} 21 & 4 \\ 6 & 19 \end{array}\right]\left[\begin{array}{rr} 19 & 6 \\ 10 & 15 \end{array}\right]\left[\begin{array}{rr} 23 & 2 \\ 3 & 22 \end{array}\right]$$

Find the total results for all 200 patients. (d) On the basis of these results, does it appear that Painfree is effective?

Prove each statement for positive integers n and r, with $r \leq n$. $C(0,0)=1$

Find the real roots of each equation by factoring. $\frac{1}{4} x^{2}-x+1=0$

Use the Trapezoid Rule to approximate $\int_{0}^{R} e^{-x^{2}} d x$ with R = 2, 4, and 8. For each value of R, take n = 4, 8, 16, and 32, and compare approximations with successive values of n. Use these approximations to approximate $I=\int_{0}^{\infty} e^{-x^{2}} d x$.

Using either logarithms or a graphing calculator, find the time required for each initial amount to be at least equal to the final amount. $8000, deposited at 3% compounded quarterly, to reach at least$23,000

Solve each equation or inequality. $\left|\frac{1}{2} x+\frac{2}{3}\right|<3$

Find a matrix A such that $Q(\mathbf{x})=\mathbf{x}^{T} A \mathbf{x}$. $Q(x)=-4 x_{1}^{2}+6 x_{2}^{2}-6 x_{1} x_{2}-10 x_{1} x_{3}+4 x_{2} x_{3}$

Determine whether each of the following statements is true or false, and explain why. If A is a $2 \times 3$ matrix and B is a $3 \times 4$ matrix, then A+B is a $2 \times 4$ matrix.

Factor each polynomial. $27 z^{9}+64 y^{12}$

One of the following statements is not equivalent to all the others. Which one is it? (a) r only if s. (b) r implies s. (c) If r, then s. (d) r is necessary for s.