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Transcript
CHAPTER 1 • Equations & Inequalities 1.1 Graphs & Graphing Utilities Objectives • Plot points in the rectangular coordinate system • Graph equations in the rectangular coordinate system • Interpret information about a graphing utility’s viewing rectangle or table • Use a graph to determine intercepts • Interpret information from graphs Rectangular coordinate system • • • • X-axis: horizontal (right pos., left neg.) Y-axis: vertical (up pos., down neg.) Ordered pairs: (x,y) Graph of an equation: infinitely many ordered pairs that make a true statement • X-intercept: point where y=0, (x,0) • Y-intercept: point where x=0, (0,y) Graphing an Equation • List ordered pairs that make your equation true (plug values in for x and find the resulting y’s). Include the x-intercept & yintercept among your points. • Plot several points and look for a trend. • Use the graphing utility on your calculator and compare graphs. Do they always match? Why or why not? Intercepts • Intercepts are key points to plot when graphing an equation. Remember, an intercepts is a POINT (x,y), and not just a number! • Look at the x-axis: What is true of EVERY point on the axis? (the y-value’s always 0) • Look at the y-axis: What is true of EVERY point on the axis? (the x-value’s always 0) Horizontal and Vertical Lines • What if the y-value is ALWAYS the same, regardless of the x-value (it could be anything!). (i.e. (4,2), (5,2), (-14,2), (17,2)) It’s a horizontal line! The x isn’t part of the equation, because its value is irrelevant: y=k (k is a constant) • What if the x-value is ALWAYS the same, regardless of the y-value (it could be anything!). (i.e. (2,4), (2,5), (2,-14), (2,17)) It’s a vertical line! The y isn’t part of the equation, because its value is irrelevant: x=c (c is a constant) 1.2 • Linear Equations and Rational Equations Objectives • Solve linear equations in one variable • Solve linear equations containing fractions • Solve rational equations with variables in the denominators • Recognize identities, conditional equations, inconsistent equations What is a linear equation in one variable and what is “solving it”? Only one variable (x or y, generally) is in the equation and it is NOT squared or raised to a power other than 1. To “solve” the equation means to find the value (or values) that would make the equation true. How do we solve an equation? • • • • Eliminate parentheses (distribute!) Collect like terms (additive identity) Isolate the variable (multiplicative identity) Remember: it’s an EQUATION to start with, meaning the left equals the right. It will no longer be equal, if something is done to one side and not the other! • CHECK your solution in the original equation: does it make it true? EXAMPLE • 4(2x-3) = 2(x+3) • 1)Distribute to eliminate parentheses 8x-12 = 2x + 6 2)Collect x’s on one side & constants on the other (use additive identity) 8x(-2x) – 12(+ 12) = 2x(-2x) + 6(+ 12) 6x = 18 3)Isolate the x (use multiplicative identity) 4) Check your solution in the original 4(2(3)-3) = 2(3+3) 4(3) = 2(6) YES!! 6 x 18 6 6 x3 Rational Equations • Equations that involve fractions! • The variable (x) could be in the numerator of the denominator. IF the x is found in the denominator, we must consider values x canNOT take on. (i.e. zero denominator) • EVEN after you’ve simplified an equation to eliminate the fractions, you haven’t eliminated the original restriction that may have been present. • With fractions, EITHER eliminate the fraction OR get a common denominator (if denominators are EQUAL, so are numerators) Solve by getting a common denominator 3 1 1 , x 2,...WHY ?? x2 8 2 3(8) 1( x 2) 1( x 2)( 4) ( x 2)(8) 8( x 2) 2( x 2)( 4) 24 x 2 4x 8 8( x 2) 8( x 2) 26 x 4 x 8 26 x x 8 4 x 8 x 8 18 3 x, x 6 CHECK!! Types of Equations • Conditional: True under certain conditions (could be one or several solutions) • Inconsistent: Inconsistencies between the 2 sides (never true – NO solutions) • Identity: One side of the equation is identical to the other (doesn’t matter what x is, infinitely many solutions) Example • Solve 3x – 6 = 3(x – 2) Notice, after distributing on the right, 3(x – 2) = 3x – 6 The left side is identical to the right. No matter what values you plug in for x, it will always be true. The solution set is: {all reals} THIS IS AN IDENTITY. Example • Solve: 4x – 8 = 4(x – 5) • Distribute on right = 4x – 20 • Think: Can 4 times a number minus 8 possibly equal 4 times the same number minus 20??? NO!! • If you continue to solve, you get: • 0x = -12 (Can 0 times a number ever equal -12? NO! • INCONSISTENCIES!! Solution: { } 1.3 • Models & Applications Objectives • Use linear equations to solve problems • Solve a formula for a variable Solving Word Problems • 1) Carefully read the problem • 2) Determine what do you know and what do you want to know • 3) Identify variables • 4) Develop equation relating what you know & what you want to know • 5) Solve the equation & check (correct?) • 6) Make certain you answered the question you were being asked! EXAMPLE • You need to drive from Chicago to your cousin’s house in Omaha (a distance of 550 miles) at an average 65 mph on the Interstate highway. What time should you leave if you have to be at your cousin’s at 3:30 pm? • What do you want to know? – How long will it take you to drive? (x = time) – What time must you leave? • What do you know? – Total distance you’ll travel (550 miles) – Speed (65 miles per hour) • What is the relationship between known & unknown? – Distance = Rate x Time – 550 miles = 65 mph x (X) (cont. on next slide) 550mi x 8.5hr mi 65 hr Did you answer the question? NO – WHEN should you leave? In order to arrive at 3:30pm, you leave 8.5 hrs earlier, which would be at 8:00 am. EXAMPLE • You have been asked to make an aluminum can (cylindrical shape) to hold 300 ml of your product. The can is to be 10 cm high. How much aluminum (in square cm) do you need? • What do you know? – Can holds 300 ml (the volume!) – Height = 10 cm What do you want to know? -How much material you will need (surface area). What relates the known & unknown? For cylinders: V r h 2 SA 2r 2rh 2 (example continued) V 300ml 300cm r (10cm) 3 3 300cm 2 r 9.55cm 10cm r 3.1cm 2 2 Now find surface area! (answer the question!) (remember, a cylinder is just 2 circles and a rectangle) SA 2r 2rh 2 SA 2 (3.1cm) 2 (3.1cm)(10cm) 2 SA 255cm 2 Ava purchased a new ski jacket, on sale for $66.50. The coat had been advertised as 30% off! What was the original cost? 1. 2. 3. 4. $95 $86.50 $90 $96.50 1.4 • Complex Numbers Objectives • • • • Add & subtract complex numbers Multiply complex numbers Divide complex numbers Perform operations with square roots of negative numbers i = the square root of negative 1 • In the real number system, we can’t take the square root of negatives, therefore the complex number system was created. • Complex numbers are of the form, a+bi, where a=real part & bi=imaginary part • If b=0, a+bi = a, therefore a real number (thus reals are a subset of complex #) • If a=0, a+bi=bi, therefore an imaginary # (imaginary # are a subset of complex #) Adding & Subtracting Complex # • Add real to real, add imaginary to imaginary (same for subtraction) • Example: (6+7i) + (3-2i) • (6+3) + (7i-2i) = 9+5i • When subtracting, DON’T FORGET to distribute the negative sign! • (3+2i) – (5 – i) • (3 – 5) + (2i – (-i)) = -2 + 3i Multiplying complex # • Treat as a binomial x binomial, BUT what is i*i? It’s -1!! Why?? • Let’s consider i raised to the following powers: i ( 1) 1 2 2 i i i (1)i i 3 2 i (i ) (1) 1 4 2 2 2 EXAMPLE (2 3i ) (3 6i ) 6 12i 9i 18i 6 3i 18 (1) 2 6 3i 18 24 3i Dividing Complex # • It is not standard to have a complex # in a denominator. To eliminate it, multiply be a wellchosen one: ( conjugate/conjugate) • The conjugate of a+bi=a-bi • We use the following fact: (a bi) (a bi) a abi abi b i 2 a b (1) a b 2 2 2 2 2 2 EXAMPLE 3 8i (4 3i ) (3 8i ) (4 3i ) 4 3i (4 3i ) 12 9i 32i 24i 16 9 12 41i 12 41i 25 25 25 2 1.5 • Quadratic Equations Objectives Solve quadratic equations by: a) Factoring b) Using the square root property c) Completing the square d) Using the quadratic formula (WHEN TO USE WHICH METHOD?) Use discriminant to determine # & type of solutions Solve application problems involving quadratics. What is a quadratic equation? ax bx c 0 a, b, c {Re als} 2 Zero-Product Rule • If the product of two or more numbers is zero, at least one of the numbers must equal zero! • If AB=0, then A=0 and/or B=0 – One or both of the terms must equal zero Why is this important? It allows us an easy way to solve an equation, but FIRST make certain the expression is a product that equals zero. • A product involves FACTORS • (2x-3)(x+2)=0 • 2x – 3 is a factor of the expression, as is 2+x • Set each factor = 0 • 2x – 3 = 0, thus x = 3/2 • x + 2 = 0, thus x = -2 • SO, if EITHER x = 3/2 or x = -2, the original expression = 0 • SO, solve by FACTORING if equation, once equal to 0, is FACTORABLE Often, you must get expression into factored form FIRST: Solve : 6 x 11x 4 2 6 x 11x 4 0 2 Factor : (3x 4)( 2 x 1) 0 3x 4 0 2x 1 0 4 1 x , 3 2 Solving with square root property • When would you use this approach? – When one side of the equation is a perfect square EXAMPLE: (3 2 x) 2 5 (3 2 x) 2 5 (Why ??) 3 2 x 5i (WHY ??) 2 x 3 5i 3 5i 3 5 x i 2 2 2 Solve by Completing the Square • When can you use this method? ALWAYS – However, if the expression is factorable or is already a perfect square, those methods may be more desirable HOW does it work? If you don’t have a perfect square, you create one by adding a “well-chosen” zero (adding the same term to both sides) Decide what to add by determining what additional term would create a perfect square EXAMPLE Solve : 2 x 2 4 x 7 0 7 2( x 2 2 x ) 0 2 7 2 x 2x 2 ( Note : ( x 1) 2 x 2 2 x 1) 7 5 x2 2x 1 1 2 2 5 2 ( x 1) 2 5 ( x 1) 2 2 5 5 x 1 i, x 1 i 2 2 Completing the square generalized to any quadratic equation results in the quadratic formula. • When can you use it? ALWAYS. (However, it still may be easier to factor & use zero-factor property or take the square root if it’s already a perfect square.) 2 ax bx c 0 b b 4ac x 2a 2 Solve: 2x 4x 6 2 1. 2. 3. 4. x = -1, 3 x = 1, -3 x = 2,3 x=2 What is the discriminant and why is it useful to us? • The discriminant is the part of the quadratic equation that is under the radical. • Based on what is under that radical, we can determine if our solution will be an integer (is what’s under there a perfect square?), an irrational (is what’s under there a positive number that is NOT a perfect square), or complex (is what’s under there a negative number?) 1.6 • Other Types of Equations Objectives • • • • • Solve polynomial equations by factoring. Solve radical equations. Solve equations with rational exponents. Solve equations that are quadratic in form. Solve equations involving absolute value. Solving by factoring • First: Set equation equal to zero • Next: Check for a common term to factor out of all terms • Next: Proceed with factoring, as with quadratics • Remember: The degree of the equation will indicate the maximum number of solutions. (if you now have a 4th degree polynomial, you may have 4 distinct solutions) What if your equation involves a variable under a radical? • In order to eliminate an nth root, you must raise both sides of the equation to the nth power. • Be CERTAIN that you isolate the radical (have it on one side of the equation by itself) before you raise both sides to the nth power. What if the variable is found under a radical twice in an equation? • Isolate one radical and raise both sides to the nth power. • Then, isolate the other radical (it will not have disappeared from the other side), and raise both sides to the nth power again. What is x is raised to an exponent that is NOT an integer? • If the variable (or expression involving a variable) is raised to the (m/n) exponent, you must isolate that expression and then raise BOTH sides to the (n/m) power. • WHY?? When you raise one exponent to another, you multiply the 2 exponents. ( a b) m n m n n m (( a b) ) a b What if the equation involves an expression inside absolute value brackets? • Recall what absolute value means: What is within those brackets could be positive or negative and still have the same overall value. 3 x 7 15 I )3 x 7 15 22 3 x 22, x 3 II ) (3 x 7) 15 3 x 7 15 8 3 x 8, x 3 Solve: 3x 7 2 1 1. 2. 3. 4. No solution. {7/3} {10/3, 4/3} {-7/3, 7/3} 1.7 • Linear Inequalities and Absolute Value Inequalities Objectives • • • • Use interval notation. Find intersections & unions of intervals. Solve linear inequalities. Recognize inequalities with no solution or all numbers as solutions. • Solve compound inequalities. • Solve absolute value inequalities. Linear inequalities • For equalities, you are finding specific values that will make your expression EQUAL something. For inequalities, you are looking for values that will make your expression LESS THAN (or equal to), or MORE THAN (or equal to) something. • In general, your solution set will involve an interval of values that will make the equation true, not just specific points. What if you have more than one inequality? • If two inequalities are joined by the word “AND”, you are looking for values that will make BOTH true at the same time. (the INTERSECTION of the 2 sets) • If two inequalities are joined by the word “OR”, you are looking for values that will make one inequality OR the other true (not necessarily both), therefore it is the UNION of the 2 sets. What IS an absolute value inequality? • Recall that absolute value refers to the expression inside the brackets being either positive or negative, therefore the absolute value inequality involves 2 separate inequalities • IF absolute value expression is LESS THAN a value, you’re looking for values that are WITHIN that distance (intersection of the 2 inequalities) • IF absolute value expression is MORE THAN a value, you’re looking for values that are getting further away in both directions (union of the 2 inequalities) If the absolute value is greater than a number, you’re considering getting further away in both directions, therefore an OR. (get further away left OR right) • See next slide for example: 2x 4 3 2x 4 3 7 x 2 OR ( 2 x 4) 3 2x 8 3 5 x 2 5 7 x x 2 2 • If, however, the absolute value was LESS than a number (think of this as a distance problem), you’re getting closer to your value and staying WITHIN a certain range. Therefore, this is an intersection problem (AND) • Same problem as before, but solved as a LESS than inequality. (next slide) 2x 4 3 2x 4 3 7 x 2 AND (2 x 4) 3 2x 8 3 5 x 2 5 7 x x 2 2 Don’t leave common sense at the door! • Remember to use logic! • Can an absolute value ever be less than or equal to a negative value?? NO! (therefore if such an inequality were presented, the solution would be the empty set) • Can an absolute value ever be more than or equal to a negative value?? YES! ALWAYS! (therefore if such an inequality were given, the solution would be all reals)