. Linear and Rational Equations In this section you will learn to: recognize equations that are identities, conditional, or contradictions solve linear equations in one variable find restrictions on variable values solve rational equations with variables in the denominator solve formulas for a specific value Types of Equations Identity Conditional Contradiction/Inconsistent Solution set consists of all real numbers*. Eample: + = 5 + = (or 0 = 0) (Both sides are identical.) Solution set consists of one or more solutions (but not an identity). Eample: 5 + 6 = 0 = or = The equation has no solutions. Eample: + = 6 + = (Since, there is no solution.) ALWAYS TRUE SOMETIMES TRUE NEVER TRUE *ecept when one or both sides of the equation results in division by 0 (Eample: = ) Solving Linear Equations ( a + b = 0, where a and b are real numbers and a 0 ) Eample : Solve ( 6) = 6 Steps: (if coefficients are integers). Simplify each side of the equation.. Move variable terms to one side and constant terms to the other.. Isolate the variable. (Multiply or divide based on the coefficient.) 4. Check the solution in the original equation. Page (Section.)
Eample : Solve 5 ( + ) = + ( 5) Eample : Solve: + = 5( + ) + 6 (NOTE: a b is equivalent to b a.) Steps: (if coefficients are fractions). Find the LCD (Lowest Common Denominator). Apply distributive property first if necessary.. Multiply both sides by the LCD. 6. Apply Steps -4 in Eample. Finding Variable Restrictions Eample 4: Find the restrictions on the variable a in each of the equations below. (a) a = a 5 a + 5 (b) a = a 4 a 5 a Solving Rational Equations (equations having one or more rational epressions) 5 0 Eample 5: Solve = + 4 Steps: (if variable is in denominator). Eclude any values that cause a zero denominator (restrictions on the variable). Page (Section.). Find the LCD and multiply both sides by the LCD.. Simplify and solve the equation. 4. Check your solution in the original equation.
Page (Section.) Eample 6: Find all values of for which y y = given 6 = y and 4 4 + + = y. Eample 7: Solve for n: 8 6 5 4 = + n n n n Eample 8 (optional): Solve: + = + +
Solving Formulas for a Specific Value Steps: Eample 9: Solve V = Bh for B.. If fractions are involved, multiply by LCD.. Move terms with desired variable to one side. Move all other terms to other side.. If two or more terms contain the desired variable, FACTOR out the variable. 4. Divide both sides by the non variable factor. Eample 0: Solve S = P + Prt for t. Eample : Solve S = P + Prt for P. Eample : Solve a b = for b. Eample : Solve c w = y + for z. z Eample 4: Solve A + B C = for A. Eample 5: Solve + = 5 for. A B y Page 4 (Section.)
. Homework Problems. Find restrictions on the variable in each equation below: (a) = 5 4 (b) ( + 4) = + 4 (c) + = 7 For Problems 8, solve the equations and classify each as an identity, contradiction or conditional.. 4(5 + ) = + 8. 5 = 0 4. + 4 = 5. 4 + 0 = ( + ) + 6. 8 6 0 + = + + 7. 7 = 7( ) 8. + + = + 9. SolveC = π r for r. 0. Solve a = b c for c. a y. Solve = b for.. Solve A + B C = for B. A B. Solve AC + D A + B = for D. 4. Solve: D = for b. a b c 5 5. Solve C = ( F ) for F. 9. Homework Answers:.(a) 0, ± (b) 4, 0, (c) 0, ±. {-}; conditional 77 4. {}; conditional 4. ; conditional 5. φ ; contradiction 6. 5 7 ; conditional C b a ab 7. all real numbers; identity 8. {}; conditional 9. r = 0. c =. = π b + y AC A AC ac 9. B =. D = 4. b = 5. F = C + C + A + B c a 5 Page 5 (Section.)
. Applications of Linear Equations In this section you will learn to: use linear equations to solve word problems Steps/Tips for Solving Word Problems:. Read the problem carefully. Underline key words and phrases. Let (or any variable) represent one of the unknown quantities. Draw a picture or diagram if possible.. If necessary write epressions for other unknowns in terms of.. Write a verbal model of the problem and then replace words with numbers, variables and/or symbols. 4. Solve the equation. Answer the question in the problem. Label answers! 5. Check your answer(s) in the original word problem (not in your equation). Eample : The number of cats in the U. S. eceeds the number of dogs by 7.5 million. The number of cats and dogs combined is 4.7 million. Determine the number of dogs and cats in the U. S. Eample : After a 0% discount, a cell phone sells for $. Find the original price of the cell phone before the discount was applied to the purchase. Eample : Including a 6% sales ta, an item costs $9.69. Find the cost of the item before the sales ta was added. Eample 4: You are choosing between two car rental agencies. Avis charge $40/day plus $.0/mile to rent a car. Hertz charges $50/day plus $.08/mile. You plan to rent the car for three days. After how many miles of driving will the total cost for each agency be the same? Page (Section.)
Eample 5: The perimeter of a triangular lawn is 6 meters. The length of the first side is twice the length of the second side. The length of the third side is 6 meters shorter than three times the length of the second side. Find the dimensions of the triangle. Eample 6: (Simple Interest Problem) Tricia received an inheritance of $5500. She invested part of it at 8% simple interest and the remainder at % simple interest. At the end of the year she had earned $540. How much did Tricia invest at each interest amount? Eample 7: How many liters of a 9% solution of salt should be added to a 6% solution in order to obtain 50 liters of a % solution? Eample 8: A student scores 8%, 86% and 78% on her first three eams. What score is needed on the fourth eam for the student to have an average of 85% for all four eams? Eample 9: A student scores 85%, 7%, 96%, and 98% for his first four chapter eams. If the fifth eam, the final eam, counts twice as much as each of the chapter eams, is it possible for the student to get a high enough final eam score to get a 90% average for the course? Page (Section.)
Eample 0: A rectangular swimming pool measures 8 feet by 0 feet and is surrounded by a path of uniform width around all four sides. The perimeter of the rectangle formed by the pool and the surrounding path is feet. Determine the width of the path. Eample : Sam can plow a parking lot in 45 minutes. Eric can plow the same parking lot in 0 minutes. If Sam and Eric work together, how long will it take them to clear the lot? Eample : A plane leaving Lansing International Airport travels due east at a rate of 500 mph. A second plane takes off 5 minutes later traveling in the same direction at 650 mph. How long will it take for the second plane to catch up to the first plane? Page (Section.)
. Homework Problems. When a number is decreased by 0% of itself, the result is 56. What is the number?. 5 % of what number is.6?. 5 is what % of 80? 4. Find.5% of $40. 4 5. After a 0% price reduction, a cell phone sold for $77. Find the original price of the phone. 6. Find two consecutive even integers such that the sum of twice the smaller integer plus the larger is 44. 7. The length of a rectangular garden plot is 6 feet less than triple the width. If the perimeter of the field is 40 feet, what are its dimensions? 8. Trinity College currently has an enrollment of,00 students with a projected enrollment increase of 000 students a year. Brown college now has 6,800 students with a projected enrollment decline of 500 students per year. Based on these projections, when will the colleges have the same enrollment? 9. Sam invested $6,000 in two different stocks. The first stock showed a gain of % annual interest while the second stock suffered a 5% loss. If the total annual income from both investments was $40, how much was invested at each rate? 0. How many liters of a 7% acid solution should be added to 0 liters of a 5% solution in order to obtain a 0% solution?. How many liters of skim milk (0% fat) must be added to liters of milk containing.5% butterfat in order to dilute the milk to % butterfat?. Amy scored 78%, 64%, 98%, and 88% on her first four eams. What score does she need on her fifth eam in order to have an 85% average for all five eams?. Scott showed improvement on his five math eams throughout the semester improving by % on each successive eam. If the fifth eam (final eam) counted twice as much as the first four eams and his average was 79% for all five eams, what score did he receive on his first eam? 4. The length of a rectangular tennis court is 6 feet longer than twice the width. If the perimeter of the court is 8 feet, find the dimensions of the court. 5. During a road trip, Tony drove one-third the distance that Lana drove. Mark drove 4 more miles than Lana. The total distance they drove on the trip was 46 miles. How many miles did each person drive? 6. A garden hose can fill a swimming pool in 5 days. A larger hose can fill the pool in days. How long will it take to fill the pool using both hoses? 7. The Smith family drove to their vacation home in Michigan in 5 hours. The trip home took only hours since they averaged 6 mph more due to light traffic. How fast did they drive each way?. Homework Answers:. 80. 40..5% 4..6 5. $96.5 6. 4 and 6 7. 44 ft by 6 ft 8. 9 years 9. $,000 @ %; $4000 @ 5% 0. 50 liters..5 liters. 97%. 7% 4. 6 ft by 78 ft 5. Tony: 46 miles; Lana: 8 miles; Mark:6 miles 6..875 days (45 hrs) 7. 9 mph; 65 mph Page 4 (Section.)
. Quadratic Equations In this section you will learn to: solve quadratics equations by. factoring. square root property. quadratic formula 4. completing the square 5. graphing (used mainly for checking not considered an algebraic solution) use the discriminant to find the number and type of solutions (roots, -intercepts, zeros) A quadratic equation in is an equation that can be written in the general form a + b + c = 0, where a, b, and c are real numbers, with a 0. A quadratic equation in is also called a second-degree polynomial equation. The Zero-Product Principle: If the product of two algebraic epressions is zero, then at least one of the factors equal to zero. If AB = 0, then A = 0 or B = 0. Solving Quadratic Equations by Factoring: Eample : 9 = Eample : = + 0 Steps:. Rearrange equation so that one side is 0. (Look for GCF.). Factor. (Use sum/product idea when a =. If a, use grouping*.). Set each factor equal to 0. 4. Solve each equation. 5. Check in original equation or by graphing (observe -intercepts). *Go to Class Pages on math home page (www.math.msu.edu) for steps using Grouping Method. Page (Section.)
Eample : 4 = (Use guess & check or grouping method.) Solving Quadratic Equations by the Square Root Method: Square Root Property: If a > 0 then = a has two real roots: = a or = a Reminder: Any time you choose to take a square root when solving an equation, you must include ±. (Eample: If = 4, then = ±.) Eample 4: = 47 Eample 5: (8 ) = 5 Solving Quadratic Equations Using the Quadratic Formula: Quadratic Formula: If a + b + c = 0, where a 0, then b ± b 4ac a =. Eample 6: Solve and simplify: = 5 Page (Section.)
Eample 7: Solve and simplify: 4 + 6 = Eample 8 (optional): Solve for t and simplify: h = t 6t In the quadratic formula, Beware: The discriminant is NOT b ± b 4ac =, the value of b 4ac is called the discriminant. a b 4ac!!! b 4ac > 0 b 4ac < 0 b 4ac = 0 Eample 9: Determine the number and type of solutions for the equations below. (Do not solve.) (a) = (b) 5 + 49 = 70 Page (Section.)
Solving a Quadratic Equation by Completing the Square: Recall: Perfect Square Trinomials ( + ) = + 6 + 9 + + 6 = ( + 6) 8 + = ( ) + 8 + = ( ) + 00 = ( ) Eample 0: + 5 = 0 Steps:. Divide each term by the coefficient of.. Move the constant term to the right side.. Form a perfect square trinomial by adding (/ of the new coefficient of ) to both sides. 4. Factor the left side (perfect square trinomial). Add the terms on the right side. 5. Finish solving using the Square Root Method. Page 4 (Section.)
. Homework Problems Solve Problems -6 by factoring:. = 5. 5 =. 0 = + 5 4. a ( a ) 5 = 0 5. 4 = 0 6. m = 7m + 6 Solve Problems 7-9 by using the quadratic formula: 7. + 5 = 8 8. 4 8 + = 0 9. 4 = + 7 Solve Problems 0- using the square root method: 0. 5 =. ( + ) + 5 =. ( 4) = 8 49 y. Solve for t: h = 6t 4 4. Solve for : = b a For Problems 5-7, determine the number and type of solutions by eamining the discriminant. 5. = 6. 0 + 49 = 0 7. 9 + 49 = 4 Solve each of the following quadratic equations by completing the square: 8. + 8 + 5 = 0 9. = 0 0. 4 = 7 8 5. Homework Answers:. 0. {-, 5}. {-5} 4. {-, 5} 5. {-, 5} 6., ± ± 9 5 4 ± h + 4 7. {, 5} 8. 9. 0. ±. φ.. ± 4 7 4 4. + a b y ± 5. 0 real roots 6. real roots 7. real root 8. {-5, -} 9. {-, 5} b 0. ± Page 5 (Section.)
.4 Application of Quadratic Equations In this section you will learn to: solve rational equations by changing to quadratic form use quadratic equations to solve word problems Solving Rational Equations: Steps: Eample : 4 =. Multiply both sides by the LCD.. Simplify both sides.. Change to quadratic form: a + b + c = 0. 4. Solve using a quadratic technique. 5. Check answer in original equation. (Be aware of restrictions!) Eample : + = 4 5 4 Eample : The length of a rectangle eceeds its width by feet. If its area is 54 square feet, find its dimensions. Page (Section.4)
Eample 4: The MSU football stadium currently has the 4 th largest HD video screen of any college stadium. The rectangular screen s length is 7 feet more than its height. If the video screen has an area of 5760 square feet, find the dimensions of the screen. (MSU math fact: The area of the video screen is about 600 ft larger than Breslin s basketball floor.) Eample 5: When the sum of 8 and twice a positive number is subtracted from the square of the number, the result is 0. Find the number. Eample 6: Find at least two quadratic equations whose solution set is, 5. Eample 7: The height of an object thrown upward from the roof of a building 00 feet tall, with an initial velocity of 00 feet/second, is given by the equation h = 6t + 00t + 00, where h represents the height of the object after t seconds. How long will it take the object to hit the ground? (Round answer to nearest hundredth.) Page (Section.4)
Eample 8: In a round-robin tournament, each team is paired with every team once. The formula below models the number of games, N, that must be played in a tournament with teams. If 55 games were played in a round-robin tournament, how many teams were entered? N = Eample 9: John drove his moped from Lansing to Detroit, a distance of 0 km. He drove 0 km per hour faster on the return trip, cutting one hour off of his time. How fast did he drive each way? Eample 0 (optional): When tickets for a rock concert cost $5, the average attendance was 00 people. Projections showed that for each 50 decrease in ticket prices, 40 more people would attend. How many attended the concert if the total revenue was $7,80? Page (Section.4)
.4 Homework Problems. Solve: = +. Solve: = +. The base of a triangle eceeds its height by 7 inches. If its area is 55 square inches, find the base and height of the triangle. 4. A regulation tennis court for a doubles match is laid out so that its length is 6 feet more than two times its width. The area of the doubles court is 808 square feet. Find the length and width of a doubles court. 5. If 0 games were played in a round-robin tournament, how many teams were entered? (Refer to Eample 9 in class notes for formula.) 6. A quadratic equation has two roots: ¾ and -5. (a) Find a quadratic equation where the coefficient of the term is. (b) Find a second equation that has only integers as coefficients. 7. The height of a toy rocket launched from the ground with an initial velocity of 8 feet/second, is given by the equation h = 6t + 8t, where h represents the height of the rocket after t seconds. How long will it take the rocket to hit the ground? (Round answer to nearest hundredth.) 8. The height of an object thrown upward from the roof of a building 00 feet tall, with an initial velocity of 00 feet/second, is given by the equation h = 6t + 00t + 00, where h represents the height of the object after t seconds. At what time(s) will the object be 00 feet above the ground? (Round answer to nearest hundredth.) 9. Jack drove 600 miles to a convention in Washington D. C. On the return trip he was able to increase his speed by 0 mph and save hours of driving time. (a) Find his rate for each direction. (b) Find his time for each direction. 0. When tickets for a rock concert cost $, the average attendance was 500 people. Projections showed that for each $ increase in ticket prices, 50 less people would attend. At what ticket price would the receipts be $5600..4 Homework Answers:. { 0} ±. {0, }. base: in; height: 5 in 4. 78 ft by 6 ft 7 5 5. 6 teams 6. (a) + = 0 ; (b) 4 + 7 5 = 0 (answers vary) 7. 8 seconds 4 4 8..5 and 5 seconds 9. (a) 40 mph; 50 mph (b) 5 hours; hours 0. $4 ($ increase) or $8 ($4 decrease) Page 4 (Section.4)
.5 Comple Numbers In this section you will learn to: add, subtract, multiply and divide comple numbers simplify comple numbers find powers of i solve quadratic equations with comple roots The imaginary unit i is defined as i =, where i =. The set of all numbers in the form a + bi with real numbers a and b and i, the imaginary unit, is called the set of comple numbers. (Standard form for a comple number is a + bi.) Rational Numbers THE COMPLEX NUMBER SYSTEM Real Numbers Irrational Numbers Imaginary Numbers Eample : Simplify each imaginary number below: 9 = 6 = 8 = 7 = = 5 Equality of Comple Numbers: a + bi = c + di if and only if a = c and b = d. Addition of Comple Numbers: ( a + bi) + ( c + di) = ( a + c) + ( b + d) i Subtraction of Comple Numbers: ( a + bi) ( c + di) = ( a c) + ( b d) i Multiplication of Comple Numbers: ( a + bi)( c + di) = ( ac bd) + ( ad + bc) i NOTE: Add, subtract, or multiply comple numbers as if they were binomials. Page (Section.5)
Eample : Add: ( i) + (-4 + 5i) Eample : Subtract: ( i) (-4 + 5i) Eample 4: Multiply: ( i)(-4 + 5i) Recall: (a + b) and (a b) are conjugates. Therefore, (a + b)(a b) = a b. a + bi and a bi are called comple conjugates therefore: ( a + bi)( a bi) = Eample 5 (Division): Simplify i 4 + 5i (Hint: Multiply numerator and denominator by conjugate of the denominator.) Eample 6 (Division): Find the reciprocal of 4 + i. Page (Section.5)
Eample 7: Simplify each of the following powers of i. i = 4 i = i = 0 i = i = 6 i = Recall: = i. Therefore b = ( ) b = b = i b. Eample 8: Simplify and write each answer in standard form. (a) 7 + 75 (b) 4 9 (c) ( + 5) (d) ( + )( ) (e) + Eample 8: Solve: a + 4a + 8 = 0 Eample 9: Solve: 5 + = 5 Page (Section.5)
.5 Homework Problems. Simplify each epression: (a) 69 (b) 00 (c) 08 (d) (e) 7 6 (f) 4 5 i (g) 6 i (h) 7 i 50 49. Perform the indicated operation for each problem below. Simplify each problem and write the answer in standard form. (a) ( 7i) + ( 8 4i) (b) ( 6 i ) ( 9 + 5i) (c) ( 4i )( 8 + 5i) (d) ( 4 i 5 ) (e) 5 i (f) i + i (g) i + i (h) ( + i) (i) 00 + 8 (j) 6 5 (k) 7 (l) ( 5i) (m) ( 4) (n) 8 7 9. Simplify: 5 ( + i) 4. Simplify: ( + i)( i) + i 5. Find the reciprocal of 7 + i. 6. Solve each of the following: (a) = 7 (b) + 5 = 6 (c) 9 4 + 8 = 0 (d) + = 49 5.5 Homework Answers:. (a) i (b) 0i (c) 6i (d) i (e) 9i (f) i (g) - 7 (h) i. (a) -5 i (b) 5 7i (c) -4 + 47i (d) -9 40i (e) -5i (f) i 4 (g) i (h) + i (i) i (j) -0 (k) -9 (l) + 0i (m) + i (n) i 5 5 7 4 ± 4. -6 + 8i 4. 4 + 5. i 6. (a) ± 4i ; (b) ± i ; (c) or ± ; 58 58 (d) 5 ± i Page 4 (Section.5)
.6 Polynomial & Radical Equations In this section you will learn to: solve polynomial equations using factoring solve radical equations solve equations with rational eponents solve equations in quadratic form A polynomial equation of degree n is defined by a n n + a n n +..... + a + a + a0 = 0, where n is a nonnegative integer (whole number), an, an, an,... a, a, a0 are real numbers, a 0, n a is called the leading coefficient, n a 0 is called the constant term (Recall: 0 0 =, for 0, therefore a 0 = a0 ), The degree of the polynomial is n (the highest degree of any term in the polynomial). Solving Polynomials by Factoring: Eample : Solve: 4 4 = Eample : Solve: 9y + 8 = 4y + 8y Eample : Solve: 6 = 0 Page (Section.6)
Solving Radical Equations: Eample 4: Solve: + = Steps:. Isolate radical(s).. Square both sides.. Epand. 4. Solve for. 5. Check answer! Eample 5: Solve: + 5 = (Hint: Move one of the radicals to the other side and repeat Eample 4 Steps - after epanding.) Solving Equations in Quadratic Form (using substitution): Eample 6: Solve: 0 = 0 Eample 7: Solve: 7 8 = 0 Page (Section.6)
5 Eample 8: Solve: 5 4 = 0 Eample 9: Solve: 4 = 9 Eample 0: Try this alternative shortcut to substitution using this eample for the following problems: 0 = 0 ( 5)( + ) = 0 = 5 or = 4 0 = 0 0 = 0 0 = 0 0 = 0 ( + ) ( + ) 0 = 0 Page (Section.6)
Solving Equations with Rational Eponents: Recall: if 0 = = For eample: ( 5) = 5 = 5 if < 0 m n m =, =, =,...... = n 5 8 Eample : Solve: 4 = 0 Eample : Solve: ( + 5) = 4 Page 4 (Section.6)
.6 Homework Problems Solve each of the equations below using any appropriate method: 5. 5 5 = 0. + 6 5. = = 4. 4 = + 0 5. + = 6. + = 5 7. 5 = 8. + 6 = 0 9. + 5 + = 0 0. 5 + 5 + = 4. + = +. 4 = 50. = + 9 8 4. 4 4 = 0 5. ( 7) = 4 6. + = + 6 7. + = 7 + 6 Solve each of the following equations using substitution: 8. = 0 9. 5 = 0. 8 = 0 7.6 Homework Answers:. {, 5} 7. 9, 4 0 ±. {0,, }. {-, 4} 4. { } 8. {-6, } 9. φ 0. { ± 4}. {-}. {-5, 0, 5}. ± 5. {, -} 6. {},, 4. {6} 5. {-, 5} 6. {, ± } 7. 5, 8., 4 5 9. 9, 4 0. {4} Page 5 (Section.6)
.7 Inequalities In this section you will learn to: use interval notation understand properties of inequality solve linear (and compound) inequalities solve polynomial inequalities solve rational inequalities Interval Notation: Inequality Graph Interval Notation Set Builder Notation < 4 < or > 5 all real # s Intersection Union Page (Section.7)
If Properties of Inequalities +/- Property of Inequality Multiplication/Division Property of Inequality a < b, then a + c < b + c a c < b c (Adding or subtracting does not affect the > or < sign.) c > 0 (c is positive) If a < b, then ac < bc a < c b c (Multiplying or dividing by a positive number does not affect the > or < sign.) c < 0 (c is negative) If a < b, then ac > bc a b > c c (When multiplying or dividing by a negative number, reverse the > or < sign.) Eample : Solve and graph the inequality below. Write the answer using interval notation. Eample : Solve and graph the inequality below. Write the answer using interval notation. 5 5 (7 + 9) > ( 5) Eample : Solve the compound ( and ) inequality < + 5. Write answer using interval notation. (a) Solve by isolating the variable. (b) Solve by writing each inequality separately. Eample 4: Avis charges $40/day plus $.0/mile to rent a car. Hertz charges $50/day plus $.08/mile. When is Avis a better deal if you are renting a car for three days? Page (Section.7)
A polynomial inequality is any inequality of the form: f ( ) < 0 (graph is below the -ais) f ( ) 0 (graph is on or below the -ais) f ( ) > 0 (graph is above the -ais) f ( ) 0 (graph is on or above the -ais) where f is a polynomial function. Eample 5: Solve 6 > 5. Write the solution using interval notation. Steps for Solving Polynomial Inequalities:. Epress as f() > 0 or f() < 0. (Get 0 on right side.). Set f() = 0 and solve for to get Boundary Points.. Plot the boundary points on a number line to obtain Intervals. 4. Test Values within each interval and evaluate f () for each value. If f () > 0, then f () is + for interval. If f () < 0, then f () is for interval. 5. Write the solution using interval notation. Check the solution on your calculator. Eample 6: Solve + 9 9. Write the solution using interval notation. Page (Section.7)
Eample 7: Solve + 4 + 8. Write the solution using interval notation. A rational inequality is any inequality of the form: f ( ) < 0 (graph is below the -ais) f ( ) 0 (graph is on or below the -ais) f ( ) > 0 (graph is above the -ais) f ( ) 0 (graph is on or above the -ais) p( ) where f is a rational function. ( f ( ) =, where p and q are polynomials and q ( ) 0) q( ) Eample 8: Solve 0. Write the solution using interval notation. + 5 Steps for Solving Rational Inequalities:. Epress as f () > 0 or f () < 0. (Get 0 on right side.) *. Find values that make the numerator & the denominator = 0. These are the Boundary Points. (Note Restrictions!). Plot the boundary points on a number line to obtain Intervals. 4. Test Value within each interval and evaluate f () for each value. If f () > 0, then f () is + for interval. If f () < 0, then f () is for interval. 5. Write the solution using interval notation. Check the solution on your calculator. Page 4 (Section.7)
Eample 9: Solve +. Write the solution using interval notation. Eample 0: A ball is thrown vertically from a rooftop 40 feet high with an initial velocity of 64 feet per second. During which time period will the ball s height eceed that of the rooftop? (Use h ( t) = 6t + v0t + s0 where v 0 = initial velocity, s 0 = initial height/position, and t = time. (You may also want to graph this function on your calculator using the viewing rectangle [0, 0, ] by [-00, 500, 00]). Page 5 (Section.7)
.7 Homework Problems Solve each of the inequalities below and write the answer using interval notation.. + 5 ( + ). ( + ) 4( + 5). ( + ) ( + 7) < 4. ( 5) 5. + > + ( + ) 6. + < 5 + 4 + 7. 0 < 4 8. 8 > 0 9. + 0 0. + 4 + 4. 9. 4 + 8 0. 9 6 + < 0 4. < 4 4 5. > 0 + 4 6. 0 + 4 + 7. 0 4 + 8. + 9. 5 0. 0 45 4., 4., 5 5., 7 7 ( 7. [-, 5) 8. (, ) (4, ) 9., 0. (, ] [, ]. {0} [ 9, ) 4 (, ]. φ 4. (, 0), 5. ( ) ( 4, ),,.7 Homework Answers:. [, ). [, ) 6., ). { }, 6. [ ) 7. [,4) 8. (, 5] (, ) 9., 0. 5, Page 6 (Section.7)
.8 Absolute Value In this section you will learn to: solve absolute value equations solve absolute value inequalities apply absolute inequalities The absolute value of a real number is denoted by, and is defined as follows: If 0 (non-negative number), then =. If < 0 (negative number), then = -. Absolute Value Equations: If k 0 and = k, then = k or = -k. Solving Absolute Value Equations: Eample : Solve: + = 7 Eample : Solve: = 5 Eample : Solve: + + 6 = Eample 4: Solve: 6 = 7 + 4 + Eample 5: Solve: = 4 Page (Section.8)
Eample 6: Solve (using guess and check) and graph each of the following: = > < Which solutions involve and? Which use the concept of or? Absolute Value Inequalities: If k > 0, then k is equivalent to k or k (union). Solving Absolute Value Inequalities: If k > 0, then k is equivalent to k and k (intersection). Eample 7: Solve and graph the inequality below. Write answer in interval notation. Eample 8: Solve and graph the inequality below. Write answer in interval notation. 5 4 < 5 + Eample 9: Solve: + 8 Eample 0 (optional): Solve: 0 < + 5 Page (Section.8)
Eample : The temperatures on a summer day satisfy the inequality t 74 0, where t is the temperature in degrees Fahrenheit. Epress this range without using absolute value symbols. Eample : A weight attached to a spring hangs at rest a distance of inches off of the ground. If the weight is pulled down (stretched) a distance of L inches and released, the weight begins to bounce and its distance d off of the ground at any time satisfies the inequality d L. If equals 4 inches and the spring is stretched inches and released, solve the inequality to find the range of distances from the ground the weight will oscillate. Page (Section.8)
.8 Homework Problems: Solve each of the absolute value equations below:. 5 + 8 =. + 4 + =. 4. 5 m 7 + = 5. 5 = 0 + 5 6 7 = Solve each inequality. Write the solution using interval notation. + 6. + 4 + 5 < 8 7. 5 > 8. 4 9. 5 0. 4 + < 7. >. A Steinway piano should be placed in room where the relative humidity h is between 8% and 7%. Epress this range with an inequality containing an absolute value.. The optimal depths d (in feet) for catching a certain type of fish satisfy the inequality 8 d 50 400 < 0. Find the range of depths that offer the best fishing..8 Homework Answers:. {, 9}. {-6, -}. {-.5, } 4. {, 5} 5. {-5, -, 5, 7} 6. (-5, -) 7. (, 9) 8. (, ], 9. [0, 0] 0. φ. (, 0) (, ). h 55 < 7. (00,400) Page 4 (Section.8)