Chapter 1: Fundamentals

Factor the expression completely. Begin by factoring out the lowest power of each common factor.

$3x^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+4x^{1/2}+x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Find the real solutions of the equation $\frac{x}{2x+7}-\frac{x+1}{x+3}=1$.

Read the biographical notes on Pythagoras (page 241), Euclid (page 542), and Archimedes (page 787). Choose one of these mathematicians, and find out more about him from the library or on the Internet. Write a short essay on your findings. Include both biographical information and a description of the mathematics for which he is famous.

Use the scientific notation, the Laws of exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data.

$\frac{\left(73.1\right)\left(1.6341\right)\times {10}^{28}}{0.0000000019}$

Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at ${70}^{\circ }F$ and 168 chirps per minute at ${80}^{\circ }F.$

(a)Find the linear equation that relates the temperature tand the number of chirps per minute n

(b)If the crickets are chirping at 150 chirps per minute, estimate the temperature.

Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line.

$\frac{5}{x^3-x^2-4x+4}<0$

A phrase describing a set of real numbers is given. Express the phase as an inequality involving an absolute value.

All real numbers x more than 2 units from 0

A phrase describing a set of real numbers is given. Express the phase as an inequality involving an absolute value.

All real numbers x at least 5 units from 7

A small business buys a computer for \(4000. After 4 years the value of the computer is expected to be \)200. For accounting purposes the business uses, linear depreciationto assess the value of the computer at a given time. This means that if Vis the value of the computer at time t, then a linear equation is used to relate Vand t.

(a)Find a linear equation that relates V andt

(b)Sketch a graph of this linear equation.

(c)What do the slope and V-intercept of the graph represent?

(d)Find the depreciated value of the computer 3 years from the date of purchase.

Solve the inequality $\left|x-5\right|\le \mathbf{3}$, express the solution using interval notation and graph the solution set on real number line.