Chapter 1. Equations and Inequalities. 1.1 Basic Equations

Transcription

1 Chapter 1. Equations and Inequalities 1.1 Basic Equations Properties of Equalities (the Cancellation properties): 1. A = B if and only if A + C = B + C 2. A = B if and only if AC = BC ( C 0 ) The values of the unknown that satisfies the equation are called its solutions, or its roots. Linear Equations A linear (or first degree) equation in one variable is an equation of the form ax + b = 0, where a, b are constants; a 0. All other equations that cannot be written in the above form are, collectively, known as nonlinear equations. Example: 3x + 5 = 2 It is a linear equation 3x + 7 = 0, which has a root x = 7/3. Examples of nonlinear equations: x 2 + 2x = 0, 2 x x = 1, 2 x= x

2 Fractional Equations Put both sides into a common denominator, then equate the numerators. Example: (#26) 1 + t 1 t 3t 2 = 1 3 Solving Simple n-th-degree Equations The equation X n = a has solution(s): X = n a, if n is odd X = ± n a, if n is even and a 0 If n is even and a < 0, the equation has no real solution (but it may have complex number solutions).

3 Example: (#62) (x 1) = 0 Example: (#70) 6x 2/3 216 = 0 Example: (#81) Solving for one variable in terms of others = + ; Solve for R R R R Answer: R 1 = RR2 R R 2

4 1.2 Modeling with Equations That is, word problems! Steps for solving word problems: 1. Identify the variable: what quantity does the problem asks? 2. Express all unknown quantities in term of the said variable. 3. Set up the model: Come up with an equation relating the quantities. 4. Solve the equation. And check your answer! Example: (#20) Helen earns $7.50 an hour at her job, but if she works more than 35 hours in a week she is paid 1.5 times her regular wage for the overtime hours worked. One week her gross pay was $ How many overtime hours did she work that week?

5 Example: (#35) A jeweler has five rings, each weighing 18 g, made of an alloy of 10% silver and 90% gold. He decides to melt down the rings and add enough silver to reduce the gold content to 75%. How much silver should he add?

6 Work-rate-time / distance-speed-time Problems Formulas to know: distance traveled = speed elapsed time d = r t amount of work completed = rate elapsed time Example: (#50) Two cyclists, 90 miles apart, start riding toward each other at the same time. One cycles twice as fast as the other. If they meet 2 hours later, at what average speed is each cyclist traveling? Example: (#51) A pilot flew a jet from Montreal to Los Angeles, a distance of 2500 miles. On the return trip the average speed was 20% faster than the outbound speed. The round-trip took 9 hours 10 minutes. What was the jet s speed from Montreal to Los Angeles?

7 1.3 Quadratic Equations A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a, b, and c are constants; a 0. Zero-Product Property For all real numbers A and B, AB = 0 if and only if A = 0 or B = 0. Using this property, a quadratic equation could be solved if we can express it as a zero product of two linear (first degree) factors. Solving a Quadratic Equation by Factoring Example: x 2 + 5x + 4 = 0 Because x 2 + 5x + 4 = (x + 1)(x + 4) = 0 Therefore, x + 1 = 0 or x + 4 = 0 x = 1 or x = 4 (Check!) Example: 2x 2 7x + 3 = 0 (Answer: x = 1/2 or x = 3)

8 Solving a Quadratic Equation by Completing the Square The expression x 2 + bx or x 2 bx can be made into a perfect square by b adding 2 2, the square of half of x s coefficient, yielding b b x ± bx+ = x±. 2 2 Example: x 2 + 5x + 4 = 0 Example: 2x 2 7x + 3 = 0

9 The Quadratic Formula Given ax 2 + bx + c = 0, a 0, then x b± b 2 4ac =. 2a Why? Comment: The quadratic formula is nothing more than the method of completing the square packaged in a tidy little box with a bow on top! The quantity, D = b 2 4ac, inside the radical sign of the formula is called the discriminant of the quadratic equation. The nature of the equation s solutions depends on the value of its discriminant: When b 2 4ac > 0, the roots are 2 distinct real numbers. When b 2 4ac = 0, the roots are the same real number (double root). When b 2 4ac < 0, the roots are 2 conjugate complex numbers. Example: Without finding them using the quadratic formula, we are nevertheless certain that the equation x 2 + 2x + 5 = 0 has no real roots, because its discriminant D = 2 2 4(1)(5) = 16 < 0. (In fact, the roots are x = 1 ± 2i.)

10 Modeling with Quadratic Equations Example: (#78) A rectangular bedroom is 7 ft longer than it is wide. Its area is 228 ft 2. What is the width of the room? What are the dimensions? Example: (#84) A cylindrical can has a volume of 40π cm 3 and is 10 cm tall. What is its diameter?

11 1.4 Complex Numbers A complex number is a number of the form a + bi, where a and b are real numbers, and i = 1. The number a is called the real part and b is called the imaginary part of the complex number. Comment: Strictly speaking every real number is always a complex number (with its imaginary part b = 0), hence the set of real numbers is a subset of complex numbers. But colloquially speaking, most people refer only those numbers with a nonzero imaginary part as complex numbers. Arithmetic of Complex Numbers Addition / Subtraction (a + bi) ± (c + di) = (a ± c) + (b ± d)i Multiplication (a + bi) (c + di) = (ac bd) + (ab + bc)i Division: ( a+ bi) ( c+ di) = a c + + bi di = a c + + bi di c c di di = ( ac+ bd) + ( bc ad) i 2 2 c + d Notation: Suppose r is a positive real number, then the principal square root of its negative, r, is denoted by r = r i= i r. Its other square root is denoted by i r.

12 Caution: If a and b are both negative, then a b ab!! For instance, 4 9 = 2i 3i= 6 ( 4)( 9) = 36 = 6. Example: Let A = 4 + i, B = 2 5i. Find: A + B = 6 5i B A = 2 6i AB = B / A = Exercise: Use the quadratic formula to verify that x = 1 ± 2i are the roots of the quadratic equation x 2 + 2x + 5 = 0. Comment: Every quadratic equation ax 2 + bx + c = 0, where a, b, and c are real numbers, with negative discriminant (b 2 4ac < 0) will always have a two complex roots that are conjugates. In general, every polynomial of degree 2, and of real number coefficients, if it should have a complex root, the conjugate of the said complex root is always itself a root of the equation. That is, for polynomials with real coefficients whatever complex roots it might have will always come in conjugated pair(s). Food for thought: What is i?

13 1.5 Other Types of Equations Polynomila Equations Any equation of the form p(x) = 0, where p(x) is a polynomial, is called a polynomial equation. (If p is of degree 1, then the equation is just a linear equation; similarly, a degree 2 polynomial gives us a quadratic equation.) Polynomial equations can be solved using the zero-product property. Example: 16x 4 = 1 Example: 9x 5 16x 3 = 0 x 3 (9x 2 16) = 0 x 3 (3x + 4) (3x 4) = 0 x = 0, 4/3, or 4/3 Note: A polynomial equation needs not to have any real root. But it will always have at least one real or complex root. Indeed, an n-th degree polynomial equation can have up to n, and no more than n, different real or complex roots. This fact is given by the Fundamental Theorem of Algebra.

14 Example: Find all roots, real and complex, of the equation x = 0. [Hint: It factors to (x + 2)(x 2 2x + 4) = 0.] Equations involving Radicals Example: (#50) 4 x 4x = 3 2 x 2 2 Example: (#52) 11 x = x Comment: It is especially important to check your answer if you have squared both sides while solving the equation. Always worry about the existence of extraneous solution(s).

15 Quadratic Type Equations An equation of quadratic type is an equation of the form aw 2 + bw + c = 0, where W is an expression in terms of the unknown x; a, b, and c are constants; a 0. Such type of equations can be solved by first solving for W = 0 (treating the equation as a quadratic equation), then solve for x from the intermediate result. Example: x 4 + x 2 = 12 Let W = x 2, the eq. becomes W 2 + W 12 = 0. Factoring, (W + 4)(W 3) = 0, we have W = 4, or W = 3. That is, W = x 2 = 4 or W = x 2 = 3 x = ±2i or x = ± 3 Example: x 4 + 9x 4 = 6 [Hint] Let W = x 2 x 2 3

16 1.6 Inequalities Properties of Inequalities: 1. A B if and only if A ± C B ± C 2. For C > 0, A B if and only if AC BC 3. For C < 0, A B if and only if AC BC 4. For A, B > 0, A B if and only if 1 A 1 B 5. If A B and C D, then A + C B + D (Inequalities can be added, but not, generally, subtracted.) Linear Inequalities Example: (#18) 6 x 2x + 9 x 2x + 3 3x 3 x 1. Example: (#28) 1 < 3x < 3x 12 1 < x 4

17 Nonlinear Inequalities Example: (#50) 16x x 3 First simplify: 16x x 3 0 x(16 x 2 ) 0 x(4 + x)(4 x) Example: (#60) 1 x 1 x

18 1.7 Absolute Value Equations and Inequalities The absolute value function y = x, given by the explicit description x, x 0 y = x =, x. x< 0 is the distance between any point x and the original on the real number line. Fig. For a positive number a, the distance from the point x = a or x = a to the origin is a, which is the absolute value for both a and a. The graph of y = x : Alternatively, the absolute value function can be written as (Think about this for a moment.) y x = 2 = x.

19 Note: 1. It is important to keep in mind that, in general, it is not true that 2 x = x. Rather, 2 x, x 0 x =. x. x< 0 2. This is consistent with the familiar distance functions for the xyplane or the xyz-space. Recall that the distance from the origin to a point (x,y) is d = x y on the xy-plane; and that the distance from the origin to a point (x,y,z) is d = x + y z in the xyz-space. Properties of Absolute Values For any real numbers a, b, and an integer n, a) ab = a b a = a b) b b c) a n = a n d) a + b a + b

20 Solving Equations / Inequalities Containing Absolute Values Suppose a is a positive real number, then a) x = a x = ± a b) x > a x > a or x < a c) x a x a or x a d) x < a a < x < a e) x a a x a Example: x 2 5 = 4 Example: 5x