ALGEBRA PDF

# ALGEBRA ## REAL NUMBERS AND ALGEBRAIC EXPRESSIONS - **Real Numbers** correspond to a point on the number line. It is either a rational number or an irrational number. - **Rational Numbers** are fractions that have an integer numerator and a nonzero integer denominator. Every rational number can...

# ALGEBRA ## REAL NUMBERS AND ALGEBRAIC EXPRESSIONS - **Real Numbers** correspond to a point on the number line. It is either a rational number or an irrational number. - **Rational Numbers** are fractions that have an integer numerator and a nonzero integer denominator. Every rational number can be written as a decimal that either repeats in a pattern or terminates. - For example: - 2/3 = 0.66666... = 0.6 - 4/5 = 0.8 - 1/11 = 0.090909... = 0.09 - **Irrational Numbers** written as a decimal neither terminates nor repeats. - For example: - $π ≈ 3.141592...$ - $√2 ≈ 1.414213...$ - **Integers**: {..., -3, -2, -1, 0, 1, 2, 3, ...} - **Whole Numbers**: {0, 1, 2, 3, ...} - **Natural Numbers**: {1, 2, 3, ...} **Algebraic expressions** are a fundamental concept in the study of algebra. They convey mathematical operations and are the building blocks of many equations. Variables and numbers can be combined with the operations of arithmetic to produce algebraic expressions. **Problem 1.** - (a) The sum of *x* and 5 is the same as 3 less than twice *x*. - (b) The sum of *y* and 4 more than the quotient of *z* and 8. **An equation** is a statement indicating that two quantities are equal. Any number that makes an equation true when substituted for its variable is said to satisfy the equation. A number that makes an equation is called a solution or a root of the equation. **Linear equations** are also called first-degree equations since the exponent on the variable is one (1). **Problem 2.** At a stockholders meeting, members representing 4.5 million shares voted in favor of a proposal for a mandatory retirement age for the members of the board of directors. If this represented 75% of the number of shares outstanding, how many shares were outstanding? **Problem 3.** A store manager hires a student to distribute advertising circulars door to door. The student will be paid $24 a day plus 12¢ for every ad she distributes. How many circulars must she distribute to earn $42 in one day? **Problem 4.** At the end of the season, the cost of a snowboard was reduced by 40%. If the snowboard sells for $270 after the decrease, find the original price of the board. ## LINEAR EQUATIONS IN ONE VARIABLE ## LINEAR INEQUALITY A linear inequality is similar to a linear equation except that the equality symbol is replaced with an inequality symbol, such as <, >, ≤, or ≥. | Set Notation | Interval Notation| | ----------- | ----------- | | {x|x < a} | (-∞, a)| | {x|x > a} | (a,+∞)| | {x|x ≤ a} | (-∞, a]| | {x|x ≥ a} | [a, +∞)| | Set Notation | Interval Notation| | ----------- | ----------- | | {x|a < x < b} | (a, b)| | {x|a ≤ x ≤ b} | [a, b]| | {x|a < x ≤ b} | (a, b]| | {x|a ≤ x < b} | [a, b)| **Problem 7.** A salesperson earns $900 a month plus a commission of 15% of sales. Find the minimum amount of sales needed to receive a total income of at least $2400 per month. **Problem 8.** In the United States, the annual consumption of cigarettes is declining. The consumption *c* in billions of cigarettes per year since the year 2000 can be approximated by the formula *c* = −9.4*t* + 431 where *t* is the number of years after 2000. Use this formula to predict the years that the consumption of cigarettes will be less than 200 billion per year. ## ABSOLUTE VALUE EQUATIONS **Solving equations of the form |X| = a:** If *a* is a positive number, then |X| = *a* is equivalent to *X* = *a* or *X* = -*a*. **Problem 9.** Solve |1/2x + 1| = 15. ## SYSTEM OF EQUATIONS ## LINEAR EQUATIONS IN TWO VARIABLES A solution of a system of two equations in two variables is an ordered pair (*x*, *y*) that makes both equations true. **Problem 10.** Solve the system: 5*x* + 3*y* = -1 and *x* - *y* = 1 **Problem 11.** Solve the system: - 3(*x* + 2*y*) - 10 = 6*x* + 4(5 - *y*) - *y* - (*x* + 2*y* + 3)/3 = 3*x* + 5*y* - 1 ## PROBLEM 12 Solve for P and F: - (P* cos 30°)/3 + (F*√5)/4 = 200 = F + (P+100) cos 60° - (P*sin 30°)(3) - 100 = (3*F)/4 = P + 400 ## PROBLEM 13 A farmer raises wheat and soybeans on 215 acres. If he wants to plant 31 more acres in wheat than in soybeans, how many acres of each should he plant? ## PROBLEM 14 The set-up cost of a machine that mills brass plates is $750. After set-up, it costs $0.25 to mill each plate. Management is considering the use of a larger machine that can produce the same plates at a cost of $0.20 per plate. If the set-up cost of the larger machine is $1,200, how many plates would the company have to produce to make the switch worthwhile? ## PROBLEM 15 Investigators discovered that a small business secretly moved $150,000 out of the country to avoid paying income tax. Some of the money was invested in a Swiss bank account that paid 8% interest annually. The remainder was deposited in a Cayman Islands account, paying 7% annual interest. The investigation also revealed that the combined interest earned the first year was $11,500. How much money was invested in each account? ## PROBLEM 16 A boat traveled 30 miles downstream in 3 hours and made the return trip in 5 hours. Find the speed of the boat in still water and the speed of the current. ## PROBLEM 17 A pharmacist needs 500 milliliters of a 20% phenobarbital solution but has only 5% and 25% phenobarbital solutions available. Find how many milliliters of each she should mix to get the desired solution. ## LINEAR EQUATIONS IN THREE VARIABLES A linear equation in three variables has three variables and each variable is raised only to the power 1. A solution of this equation is an ordered triple (*x*, *y*, *z*) that makes the equation a true statement. **Problem 18.** Solve the system: - 3*x* - *y* + *z* = -15 - *x* + 2*y* - *z* = 1 - 2*x* + 3*y* + 2*z* = 0 ## EXPONENT and POLYNOMIALS **Law of Exponents** - **Product Rule**: a^m * a^n = a^(m+n) - **Zero exponent**: a^0 = 1 where a ≠ 0 - **Quotient Rule**: a^m / a^n = a^(m-n) where a ≠ 0 - **Negative exponent**: a^-n = 1/a^n where a ≠ 0 - **Power rules**: - (a^m)^n = a^(m*n) - (a*b)^m = a^m * b^m where *b* ≠ 0 **Problem 19.** Simplify 48*x*(x-2*y*z)^-2 / (2*x -3*z^2)^4(x*y^2) **A polynomial** is a finite sum of terms in which all variables have exponents raised to nonnegative integer powers and no variables appear in a denominator. - 1.3*x*² (monomial) - 1 - 3*y* + 5 (binomial) - 6*z*² - 5*z* + 7 (trinomial) **Special Products** - **Perfect Square Trinomial** - a² + 2ab + b² = (a + b)² - a²-2ab+ b² = (a - b)² - **Difference of two Squares** - a²-b²= (a + b)(a - b) - **Sum and difference of two cubes** - a³ + b³ = (a + b)(a² – ab + b²) - a³ – b³ = (a - b)(a² + ab + b²) ## RATIONAL EXPRESSIONS **Problem 20.** Add 1/x + y where x, y ≠ 0 **Problem 21.** If a number is added to the numerator of 3/4 and twice as much is added to the denominator, the result is 4/7. Find the number. **Problem 22.** An inlet pipe can fill an empty swimming pool in 5 hours, and another inlet pipe can fill the pool in 4 hours. How long will it take both pipes to fill the pool? ## PROPORTIONS A proportion is a statement that two ratios are equal. **Problem 23** A tourist can bicycle 28 miles in the same time as he can walk 8 miles. If he can ride 10 mph faster than he can walk, how much time should he allow to walk a 30-mile trail? **Problem 24.** If 6 apples cost $1.38, how much will 16 apples cost? **Problem 25.** A solution contains 2 quarts of antifreeze and 5 quarts of water. How many quarts of antifreeze must be mixed with 18 quarts of water to have the same concentration? ## VARIATION - **Direct Variation** - y varies directly as x. If *y* is 20 when *x* is 15, find the constant of variation and the direct variation equation. - **Inverse Variation** - y is 20 when x is 15. - **Joint Variation** - **Combined Variation** **Problem 26.** Suppose that y varies directly as x. If y is 20 when x is 15, find the constant of variation and the direct variation equation. **Problem 27.** Boyle's law says that if the temperature stays the same, the pressure P of a gas is inversely proportional to the volume V. If a cylinder in a steam engine has a pressure of 350 kilopascals when the volume is 2.8 cubic meters, find the pressure when the volume decreases to 1.5 cubic meters. **Problem 28.** The maximum weight that a circular column can support is directly proportional to the fourth power of its diameter and is inversely proportional to the square of its height. A 2-meter-diameter column that is 8 meters in height can support 1 ton. Find the weight that a 1-meter-diameter column that is 4 meters in height can support. ## RADICALS A radical function in *x* is a function defined by an expression containing a root of *x*. **Law of Radicals** - $a^{1/n} = \sqrt[n]{a}$ - $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$ - $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ - $\sqrt{a} / \sqrt{b} = \sqrt{a/b}$ **Problem 29.** Simplify $\sqrt[3]{10*x^8 / 27*y^6}$ ## QUADRATIC EQUATIONS A quadratic equation is an equation of the form *ax² + bx + c = 0*, where *a*, *b*, and *c* are real numbers and *a* ≠ 0. **Problem 30.** Solve 2*x² – 3*x*-5 = 0 **Problem 31.** Solve *x* - (4*y - 50) / (5*y - 25) = *y* ## PROBLEM 32 If an object is thrown straight up into the air with an initial velocity of 112 feet per second, its height after *t* seconds is given by the formula *h* = 112*t* - 16*t*² where *h* represents the height of the object in feet. After this object has been thrown, in how many seconds will it hit the ground? ## EXPONENTIAL FUNCTIONS An exponential function with base *b* is defined by the equation $y = b^x$ (*b* > 0, *b* ≠ 1, and *b* is a real number). **Problem 35.** An equation based on the exponential function provides a model for population growth. In the Malthusian model for population growth, the future or past population of a colony is related to the present population by the formula *A* = *Pe^rt*. Earth's population is approximately 6 billion people and is growing at an annual rate of 1.9%. Assuming a Malthusian growth model, find the world population in 30 years. **Problem 36.** Suppose that a country with a population of 1,000 people is growing exponentially according to the formula *P* = 1000*e^(0.02*t*) where *t* is in years. Furthermore, assume that the food supply measured in adequate food per day per person is growing linearly according to the formula *y* = 30.625*x* + 2,000 where *x* is in years. In how many years will the population outstrip the food supply? ## LOGARITHMIC FUNCTIONS If *b* > 0 and *b* ≠ 1, the logarithmic function with base *b* is defined by $y = log_b(x)$ if and only if x = b^y. **Law of logarithm** - $log_b(mn) = log_b(m) + log_b(n)$ - $log_b(m/n) = log_b(m) - log_b(n)$ - $log_b(m^r) = r log_b(m)$ - $log_b (1) = 0$ and $log_b(b) = 1$ - $log_b(m) / log_c(m)$ = (change of base) **Problem 37.** Solve 7^(3*x) = 2^x. **Problem 38.** Solve 6(*x* - 3) = 1.6*(1.03)^*x*. **Problem 39.** Richter Scale: If *R* is the magnitude of an earthquake, *A* is the amplitude (measured in micrometers), and *P* is the period (the time of one oscillation of Earth’s surface, measured in seconds), then *R* = log(A/P). Find the measure on the Richter scale of an earthquake with an amplitude of 10,000 micrometers (1 centimeter) and a period of 0.1 second. ## PROBLEM 40 When a living organism dies, the oxygen/carbon dioxide cycle common to all living things stops and carbon-14, a radioactive isotope with a half-life of 5,700 years, is no longer absorbed. By measuring the amount of carbon-14 present in an ancient object, archaeologists can estimate the object’s age. (a) How old is a wooden statue that retains 25% of its original carbon-14 content? (b) Determine the amount of carbon-14 in a 20,000-year-old wooden statue. ## ARITHMETIC SEQUENCE An arithmetic sequence is a sequence in which each term (after the first) differs from the preceding term by a constant amount *d*. The constant *d* is called the common difference of the sequence. The general term *a_n* of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$ where *a₁* is the first term and *d* is the common difference **Problem 41.** If the third term of an arithmetic sequence is 12 and the eighth term is 27, find the fifth term. ## PROBLEM 42 A starting salary for a consulting company is $57,000 per year with guaranteed annual increases of $2200 for the next 4 years. Write the general term for the arithmetic sequence that models the potential annual salaries and find the salary for the third year. ## PROBLEM 43 A person has a choice between two job offers. Job A has an annual starting salary of $30,000 with guaranteed annual raises of $1200 for the next four years, whereas job B has an annual starting salary of $28,000 with guaranteed annual raises of $2500 for the next four years. Compare the fifth partial sums for each sequence to determine which job would pay more money over the next 5 years. ## GEOMETRIC SEQUENCE A geometric sequence is a sequence in which each term (after the first) is obtained by multiplying the preceding term by a constant *r*. The constant *r* is called the common ratio of the sequence. The general term *a_n* of a geometric sequence is given by $a_n = a_1r^(n-1)$ where *a₁* is the first term and *r* is the common ratio. **Problem 44.** If the third term of a geometric sequence is 4 and the fourth term is -12, find the eighth term. **Problem 45.** After applying a test antibiotic, the population of a bacterial culture is reduced by one-half every day. Predict how large the culture will be at the start of day 7 if it measures 4800 units at the beginning of day 1. **Problem 46.** A grant from an alumnus to a university specified that the university was to receive $800,000 during the first year and 75% of the preceding year's donation during each of the following 5 years. Find the total amount donated during the 6 years. ## PROBLEM 47 On its first pass, a pendulum swings through an arc whose length is 24 inches. On each pass thereafter, the arc length is 75% of the arc length on the preceding pass. Find the total distance the pendulum travels before it comes to rest.