Chapter 1

Perform the indicated operations and simplify. $$ \left(3 m^{2}-1\right)\left(2 m^{2}+3 m-4\right) $$

Determine whether the statement is true for all real numbers \(a\) and \(b\). $$ |a-4|=|4-a| $$

Solve the equation. $$ 1-\frac{5}{2 y}-\frac{6}{y^{2}}=0 $$

Write the expression in simplest radical form. $$ \sqrt[3]{\sqrt{9}} $$

Suppose an asset has an original value of \(\$ C\) and is depreciated linearly over \(N\) yr with a scrap value of \(\$ S\). Then the book value \(V\) (in dollars) of the asset at the end of \(t\) yr is given by $$V=C-\left(\frac{C-S}{N}\right) t$$ a. Solve for \(C\) in terms of \(V, S, N\), and \(t\). b. A speed boat is being depreciated linearly over 5 yr. If the scrap value of the boat is $$\$ 40,000$$ and the book value of the boat at the end of 3 yr is $$\$ 70,000$$, what was its original value?

Simplify the expression, writing your answer using positive exponents only. $$ \frac{x^{-1}-1}{x^{-1}+1} $$

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ 4 x^{3}-9 x y^{2}+4 x^{2} y-9 y^{3} $$

Perform the indicated operations and simplify. $$ 2 x-\\{3 x-[x-(2 x-1)]\\} $$

Determine whether the statement is true for all real numbers \(a\) and \(b\). $$ |a+1|=|a|+1 $$

Solve the equation. $$ 6+\frac{1}{k}-\frac{2}{k^{2}}=0 $$

Write the expression in simplest radical form. $$ \sqrt[5]{\sqrt[3]{9}} $$

The distance \(s\) (in feet) covered by a car traveling along a straight road is related to its initial speed \(u\) (in \(\mathrm{ft} / \mathrm{sec})\), its final speed \(v\) (in \(\mathrm{ft} / \mathrm{sec})\), and its (constant) acceleration \(a\) (in \(\mathrm{ft} / \mathrm{sec}^{2}\) ) by the equation \(v^{2}=u^{2}+2 a s\). a. Solve the equation for \(a\) in terms of the other variables. b. A car starting from rest and accelerating at a constant rate reaches a speed of \(88 \mathrm{ft} / \mathrm{sec}\) after traveling \(\frac{1}{4}\) mile \((1320 \mathrm{ft})\). What is its acceleration?

Simplify the expression, writing your answer using positive exponents only. $$ \frac{x^{-1}-y^{-1}}{x^{-1}+y^{-1}} $$

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ 4 u^{4}+11 u^{2} v^{2}-3 v^{4} $$

Perform the indicated operations and simplify. $$ 3 m-2\\{m-3[2 m-(m-5)]+4\\} $$

Solve the equation. $$ \frac{3}{x^{2}-1}+\frac{2 x}{x+1}=\frac{7}{3} $$

Cowling's rule is a method for calculating pediatric drug dosages. If \(a\) denotes the adult dosage (in milligrams) and if \(t\) is the child's age (in years), then the child's dosage is given by $$c=\left(\frac{t+1}{24}\right) a$$ a. Solve the equation for \(t\) in terms of \(a\) and \(c\). b. If the adult dose of a drug is \(500 \mathrm{mg}\) and a child received a dose of \(125 \mathrm{mg}\), how old was the child?

Simplify the expression, writing your answer using positive exponents only. $$ \frac{u^{-1}-v^{-1}}{v-u} $$

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ x^{4}+3 x^{3}-2 x-6 $$

Perform the indicated operations and simplify. $$ x-\\{2 x-[-x-(1-x)]\\} $$

Solve the equation. $$ \frac{m}{m-2}-\frac{27}{7}=\frac{2}{m^{2}-m-2} $$

Write the expression in simplest radical form. $$ \sqrt[3]{-\sqrt[4]{x^{3}}} $$

Simplify the expression, writing your answer using positive exponents only. $$ \frac{(u v)^{-1}}{u^{-1}+v^{-1}} $$

Perform the indicated operations and simplify. $$ 3 x^{2}-\left\\{x^{2}+1-x[x-(2 x-1)]\right\\}+2 $$

Find the minimum cost \(C\) (in dollars) given that $$ 5(C-25) \geq 1.75+2.5 C $$

Solve the equation. $$ \frac{3 x}{x-2}+\frac{4}{x+2}=\frac{24}{x^{2}-4} $$

Rationalize the denominator of the expression. $$ \frac{2}{\sqrt{3}} $$

Simplify the expression, writing your answer using positive exponents only. $$ \left(\frac{a^{-1}-b^{-1}}{a^{-1}+b^{-1}}\right)^{-1} $$

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ a u^{2}+(a+c) u+c $$

Perform the indicated operations and simplify. $$ (2 x-3)^{2}-3(x+4)(x-4)+2(x-4)+1 $$

Find the maximum profit \(P\) (in dollars) given that $$ 6(P-2500) \leq 4(P+2400) $$

Solve the equation. $$ \frac{3 x}{x+1}+\frac{2}{x}+5=\frac{3}{x^{2}+x} $$

Rationalize the denominator of the expression. $$ \frac{3}{\sqrt{5}} $$

Simplify the expression, writing your answer using positive exponents only. $$ \left[\left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right)\right]^{-2} $$

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ a x^{2}-(1+a b) x y+b y^{2} $$

Perform the indicated operations and simplify. $$ (x-2 y)^{2}+2(x+y)(x-3 y)+x(2 x+3 y+2) $$

DRIVING RANGE OF A CAR An advertisement for a certain car states that the EPA fuel economy is \(20 \mathrm{mpg}\) city and 27 mpg highway and that the car's fuel-tank capacity is 18.1 gal. Assuming ideal driving conditions, determine the driving range for the car from the foregoing data.

Solve the equation. $$ \frac{2 t+1}{t-2}-\frac{t}{t+1}=-1 $$

Rationalize the denominator of the expression. $$ \frac{3}{2 \sqrt{x}} $$

The accumulated amount after \(t\) yr when a deposit of \(P\) dollars is made in a bank and earning interest at the rate of \(r /\) year is \(A=P+P r t .\) Factor the expression on the right-hand side of this equation.

CELSIUS AND FAHRENHEIT TEMPERATURES The relationship between Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) temperatures is given by the formula $$ C=\frac{5}{9}(F-32) $$ a. If the temperature range for Montreal during the month of January is \(-15^{\circ}<\mathrm{C}^{\circ}<-5^{\circ}\), find the range in degrees Fahrenheit in Montreal for the same period. b. If the temperature range for New York City during the month of June is \(63^{\circ}<\mathrm{F}^{\circ}<80^{\circ}\), find the range in degrees Celsius in New York City for the same period.

Solve the equation. $$ \frac{x}{x+1}-\frac{3}{x-2}+\frac{2}{x^{2}-x-2}=0 $$

Rationalize the denominator of the expression. $$ \frac{3}{\sqrt{x y}} $$

The incidence (number of new cases/day) of a contagious disease spreading in a population of \(M\) people, where \(k\) is a positive constant and \(x\) denotes the number of people already infected, is given by \(k M x-k x^{2} .\) Factor this expression.

Perform the indicated operations and simplify. $$ -3\left[(x+2 y)^{2}-(3 x-2 y)^{2}+(2 x-y)(2 x+y)\right] $$

MEETING SALES TARGETS A salesman's monthly commission is \(15 \%\) on all sales over \(\$ 12,000\). If his goal is to make a commission of at least \(\$ 3000 / \mathrm{mo}\), what minimum monthly sales figures must he attain?

Solve the equation. $$ \sqrt{u^{2}+u-5}=1 $$

Rationalize the denominator of the expression. $$ \frac{2 y}{\sqrt{3 y}} $$

The strength of a human body's reaction to a dosage \(D\) of a certain drug, where \(k\) is a positive constant, is given by $$\frac{k D^{2}}{2}-\frac{D^{3}}{3}$$ Factor this expression.

MANUFACTURING PROFIT The total revenue realized in the sale of \(x\) units of the LectroCopy photocopying machine is $$ -0.04 x^{2}+2000 x $$ dollars/week, and the total cost incurred in manufacturing \(x\) units of the machines is $$ 0.000002 x^{3}-0.02 x^{2}+1000 x+120,000 $$ dollars/week \((0 \leq x \leq 50,000)\). Find an expression giving the total weekly profit of the company.