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Precalculus

Algebra Review

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1
The Real Number System

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2
Exponents, Roots, and Radicals

Simplify each expression and write it using positive exponents. Assume that variables represent nonzero real numbers.

\(\dfrac{16x^{10}y^4}{4x^{10}y^8}\)

\(\dfrac{\left(3x^{-2}\right)^3}{x^{-5}}\)

\(\left(\dfrac{54s^2t^{-3}}{6zt}\right)^{-2}\)

In 2012, Japan had a population os \(1.27\times 10^8\) and a land area of \(3.64\times 10^5\) square kilometers. What is the population density of Japan, that is, the number of people per square kilometer of land? Express your answer in scientific notation.

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3
Polynomials and Factoring

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4
Rational Expressions

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5
Linear and Quadratic Equations

The height of a ball after being thrown vertically upward from a point \(80\) feet above the ground with a velocity of \(40\) feet per second is given by \(h=-16t^2+40t+80\), where \(t\) is the time in seconds since the ball was thrown and \(h\) is in feet.

When will the ball be \(50\) feet above the ground?

When will the ball reach the ground?

For what values of \(t\) does this problem make sense (from a physical sense)?

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6
Linear Inequalities

The Verizon phone company in New Jersey has the following two plans for local toll calls:

Plan A charges \(\$4.00\) per month plus 8 cents per minute for every local toll call.

Plan B charges a flat rate of \(\$20\) per month for local toll calls regardless of the number of minutes of use.

a. Express the monthly cost for Plan A in terms of the number of minutes used.

b. Express the monthly cost for Plan B in terms of the number of minutes used.

c. How many minutes would you have to use per month for Plan B to be cheaper than Plan A?

To operate a gourmet coffee booth in a shopping mall, it costs \(\$500\) (the fixed cost) plus \(\$6\) for each pound of coffee bought at wholesale price. The coffee is then sold to customers for \(\$10\) per pound.

Find an expression for the operating cost of selling \(q\) pounds of coffee.

Find an expression for the revenue earned by selling \(q\) pounds of coffee.

Find the break-even point.

How many pounds of coffee must be sold for the revenue to be greater than the total cost?

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7
Equations and Inequalities Involving Absolute Value

A thermometer measures temperature with an uncertainty of \(0.25^\circ\text{F}\). If a personÃ¢â‚¬â„¢s body temperature is measured at \(98.3^\circ\text{F}\), use absolute value notation to write an inequality for the range of possible body temperatures.

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8
Other Types of Equations

A theatre club arranged for a chartered bus trip to a play at a cost of \(\$350\). To lower costs, \(10\) nonmembers were invited to join the trip. The bus fare per person then decreased by \(\$4\). How many theatre club members were going on the trip?

Jennifer is standing on one side of a river that is \(3\) kilometers wide. Her bus is located on the opposite side of the river. Jennifer plans to cross the river by rowboat and then jog the rest of the way to reach the bus, which is \(10\) kilometers along the river from point \(B\) directly across the river from her current location (point \(A\)). If she can row \(5\) kilometers per hour and jog \(7\) kilometers per hour, at which point on the other side of the river should she dock her boat so that it will take her a total of exactly two hours to reach the bus? Assume that JenniferÃ¢â‚¬â„¢s path on each leg of the trip is a straight line, and that there is no river current or wind speed.