Chapter Chapter Maintaining Mathematical Proficiency (p. ). 6 8 Chapter Mathematical Practices (p. ). x + < x. x > x +.. = 6. =. The solution is x <. The solution is x >.. x + x +. + = + =. = = 6. + = + = 7. 9 < 9 6 8 The solution is x.. Explorations (p. ). a. Statement: The temperature t in Sweden is at least C. Inequality: t C Graph: b. Statement: The elevation e of Alabama is at most 7 feet. Inequality: e 7 ft Graph: 7 8. < 6 9. <. 7.. 7 > = 8 6 8 8 8 6 <. Because a < b and a is the opposite of a and b is the opposite of b, b < a.. a. x ; Sample answer: The inequality is true for all values of x that are greater than or equal to. b. x > ; Sample answer: The inequality is true for all values of x that are greater than. c. x ; Sample answer: The inequality is true for all values of x that are less than or equal to. d. x < ; Sample answer: The inequality is true for all values of x that are less than.. Sample answer: An inequality can be used to describe the relationship between two expressions or quantities. More specifically, it tells which quantity is greater and which is less and whether the quantities may or may not be equal.. a. Sample answer: A tube has to pass through a hole with a diameter of. millimeters. So, the diameter of the tube must be less than. millimeters. b. Sample answer: Your mom gives you $6 for dinner out with your friends. So, you have to order something that will cost at most $6 with tax. c. Sample answer: The temperature must stay above C, or the school will cancel the ski bus. d. Sample answer: In some school districts, each student must do at least hours of community service each semester in order to graduate. Copyright Big Ideas Learning, LLC Algebra 6
Chapter. Monitoring Progress (pp. 7). A number b is fewer than.. b <. An inequality is b <... 7 is at least twice a number k minus. 7 k An inequality is 7 k.. c + < + <? < So, is a solution of the inequality.. m? ()? + 6 9 So, is not a solution of the inequality.. x. ()?... So, is a solution of the inequality. 6. x > () >? >? 9 > So, is not a solution of the inequality. 7. b > 8.. g 9. r <. v 6...8..6 6 8. An inequality that represents the graph is x.. Exercises (pp. 8 6) Vocabulary and Core Concept Check. A mathematical sentence using the symbols <, >,, or is called an inequality.. x + > 8 + >? 8 8 > 8 no; Because 8 is not greater than itself, is not a value of x that makes the inequality true. So, is not in the solution set.. Draw an open circle when a number is not part of the solution. Draw a closed circle when a number is part of the solution. Draw an arrow to the left or right to show that the graph continues in that direction.. The sentence, w is no more than 7 is different, because an inequality that represents this sentence is w 7, but the other three sentences can be represented by the inequality w 7. Monitoring Progress and Modeling with Mathematics. A number x is greater than. x > An inequality is x >. 6. A number n plus 7 is less than or equal to 9. n + 7 9 An inequality is n + 7 9. 7. Fifteen is no more than a number t divided by. An inequality is t. t 8. Three times a number w is less than 8. w < 8 An inequality is w < 8. 9. One-half of a number y is more than. y > An inequality is y >.. Three is less than the sum of a number s and. < s + An inequality is < s +.. Thirteen is at least the difference of a number v and. v An inequality is v. 6 Algebra Copyright Big Ideas Learning, LLC
Chapter. Four is no less than the quotient of a number x and.. An inequality is x.. x.. Let w be the weight of the second fish. The second fish weighs at least. lb more than the first fish. w. +. An inequality is w. +., or w.7.. Let x be the additional people that can enter the pool. Additional people are at most maximum capacity minus people in pool. x 6 An inequality is x 6, or x 7.. r + > 8 + >? 8 6 > 8 So, r = is not a solution of the inequality. 6. x < 8 ( ) <? 8 + <? 8 8 < 8 So, x = is not a solution of the inequality. 7. s 9 ()? 9 9 So, s = is a solution of the inequality. 8. 7 y 7? (7) 7 So, y = 7 is a solution of the inequality. 9. > x >? > So, x = is a solution of the inequality.. z? So, z = is not a solution of the inequality.. n +? ( ) +? + So, n = is a solution of the inequality.. (s) < ( ) <? <?. < So, s = is not a solution of the inequality.. z +.? () +? +? + So, z = is a solution of the inequality. m 6 > (8) 6 >? 6 >? >? > So, m = 8 is not a solution of the inequality.. a. 8 feet inches = 8() inches + inches = 7 inches Let h be the heights (in inches) of every other person that has ever lived. An inequality is h < 7. b. 9 feet = 9() inches = 8 inches h < 7 8 < 7 Because 8 < 7 is not true, 9 feet is not a solution of the inequality. Copyright Big Ideas Learning, LLC Algebra 6
Chapter 6. a. Words: Other competitor bench press weights are at most Winner s bench press weight Least amount bench press weight Variable: Let x be the weights that the other competitors bench pressed. Inequality: x An inequality is x, or x 77. b. x 79? 79 77 Because x = 79 pounds does not make the inequality true, it is not a solution of the inequality. 7. Because is to the right of on the number line, it is greater than. So, y = 8 does not make the inequality true and therefore is not a solution of the inequality. y + 7 < + 7 <? < So, 8 is not in the solution set. 8. The statement 6 6 is true because 6 is less than or equal to itself. So, x = 8 is a solution of the inequality. x + 6 (8) +? 6 +? 6 6 6 So, 8 is in the solution set. 9. x. z. > t. < w. v. s < 6. < p 6. r 7. x < 7 6 8 8. n 9. z.. w <..6.8.. 7.....6. An inequality is x.. An inequality is x.. An inequality is x >.. An inequality is x <.. C; The temperature must increase by no less than 76 F 7 F = F in order for the temperature to be no less than 76 F. So, the increase must be greater than or equal to F and is represented by x. 6. For a truck with axles, an inequality that represents the total possible weights w (in pounds) of the vehicle and its contents is w,.,, 6, 8, For a truck with axles, an inequality is w 6,.,, 6, 8, For a truck with axles, an inequality is w 8,.,, 6, 8, 7. If the longest natural arch is () = 8 inches, an inequality that represents the lengths (in inches) of all other natural arches is < 8. 6 8 8. Sample answer: If a student works no more than hours each week and the hours are divided evenly over a -day work week, then an inequality that represents how many hours h the student can work each day is h, or h. 9. Sample answer: If today is the rd day of the month, and you let x represent the number of days left in the month, then an inequality to represent this situation is + x.. a. Let T be the known melting points (in degrees Celsius) of all metallic elements. An inequality is T.87. b. T.87.87.87 Because.87 C is greater than or equal to itself, it is possible for a metallic element to have a melting point of.87 C. 6 Algebra Copyright Big Ideas Learning, LLC
Chapter. Words: Variable: Price per one-way ride Number of one-way rides is less than or equal to Cost of a monthly pass Let x be how many one-way rides you buy. Inequality: $.9 x..9x.9()?. Because x = makes the inequality true, it is cheaper to pay the one-way fare for rides than to buy a monthly pass.. Your cousin is correct because is less than or equal to itself. So, x = is a solution of the inequality.. Sample answer: Some highways have a minimum speed of miles per hour and a maximum speed of 7 miles per hour. Let s be the speed you can legally travel on the highway. Then the situation can be represented by the inequalities s and s 7.. Let b be the amounts of all the losing bids. An inequality is b < 7,. 7,,, 6, 8,. The area of the triangle is bh = (x)(6) = x, which equals when x =. So, the area is less than square meters when x < meters. 6. The area of the triangle is bh = (x)() = x, which equals 8 when x =.6. So, the area is greater than or equal to 8 square feet when x.6 feet. 7. The area of the trapezoid is ( b + b ) h = (8 + )(x) = 6x, which equals 8 when x =. So, the area is less than 8 square centimeters when x < centimeters. 8. The area of the rectangle is w = x, which equals when x = 6. So, the area is greater than square inches when x > 6 inches. meters 9. a. seconds =, or about.7 meters per second 7 So, an inequality is r > or about.7. 7.7.68.7.7.7 b. no; The human body is only able to run so fast. Eventually the graph will reach speeds at which it is impossible for the human body to run. Maintaining Mathematical Proficiency 6. x + = Check: x + = + =? x = = The solution is x =. 6. y 9 = Check: y 9 = + 9 + 9 9 =? y = The solution is y =. = 6. 6 = + y Check: 6 = + y 6 =? + = y 6 = 6 The solution is y =. 6. = y Check: = y + + =? = y The solution is y =. 6. v = x y z = v yz = xyz yz v yz = x The rewritten literal equation is x = v yz. 6. s = r + x s r = r r + x s r = x s r = x s r = x The rewritten literal equation is x = s r. 66. w = + (x ) w = + (x) () w = + x w = x + w = x + w = x w = x The rewritten literal equation is x = w. Copyright Big Ideas Learning, LLC Algebra 6
Chapter 67. n = x + n = x + n = x + n = x + n = x n = x n = x The rewritten literal equation is x = n.. Explorations (p. 6). a. no; It is possible, for T and C to be equal if every one of a quarterback s completed passes results in a touchdown. b. yes; The sum of the number of completed passes and the number of intercepted passes is always less than or equal to the number of attempted passes because a quarterback must attempt a pass in order for it to be either completed or intercepted. c. no; It is possible, for N and A to be equal if every one of the quarterback s attempted passes are intercepted. d. yes; The difference of the number of attempted passes and the number of completed passes is always greater than or equal to the number of incomplete passes, because all attempted passes must be completed, intercepted, or incomplete.. a. Sample answer: Attempts Completions Yards Touchdowns Interceptions 8.Y + C + T N P = A 8.() + () + () () = = = Whenever a quarterback earns a negative ranking, the inequality P < will be true. Because < is a true statement, P = is a solution of the inequality. One way a quarterback can earn a negative ranking after only attempted passes is by completing only half of them for only yards and no touchdowns while having the other half intercepted. b. Sample answer: Attempts Completions Yards Touchdowns Interceptions 7 6 6 8.Y + C + T N P = A 8.(6) + (7) + () (6) = =,. P + Check: P + +? P All values of P that are greater than or equal to will make the inequality true. So, P = is a solution of the inequality. One way a quarterback can earn a ranking of is by completing 7 out of attempted passes for 6 yards and touchdowns while only throwing 6 interceptions. c. Sample answer: Attempts Completions Yards Touchdowns Interceptions 8 7 8.Y + C + T N P = A 8.(7) + (8) + () () = = 7,7.8 76 P > Check: P > + + 76 >? P > 7 7 > All values of P that are greater than or equal to 7 will make the inequality true. So, P = 76 is a solution of the inequality. One way a quarterback can earn a ranking of 76 is by completing 8 out of attempted passes for 7 yards and touchdowns while only throwing interceptions.. In order to isolate a variable, you can undo addition by subtracting the same value from each side of an inequality, or you can undo subtraction by adding the same value to each side of an inequality. 66 Algebra Copyright Big Ideas Learning, LLC
Chapter. a. x + < b. x + + x < x 8 The solution is x <. The solution is x 8. c. > x d. x + + + 6 > x x The solution is x < 6. The solution is x.. Monitoring Progress (pp. 6 6). b > 9 + + b > 7 The solution is b > 7. 7. m + + m 8 The solution is m 8. 6 8. > y + + ( + = = ) > y The solution is y <.. k + k The solution is k.. 6 z + 6 6 6 ( 6 6 = 6 = ) z The solution is z. 6. p +.7 >..7.7 p > The solution is p >. 7. Words: Watts used by microwave oven + Watts used by the toaster <? Overload wattage + 8 <? 8 8 < 8 no; If both the microwave oven and the toaster are plugged in at the same time, then 8 watts are being used by the circuit, which is the point at which the circuit overloads. So, you cannot have both the microwave oven and the toaster plugged into the circuit at the same time.. Exercises (pp. 6 66) Vocabulary and Core Concept Check. Because is subtracted from each side of the inequality, then by the Subtraction Property of Inequality, the inequalities are equivalent.. Sample answer: When you are solving both equations and inequalities with addition, in order to isolate the variable on one side, you undo subtraction by adding the same number to each side, which produces an equivalent equation or inequality, respectively. Monitoring Progress and Modeling with Mathematics. k + < You would subtract from each side.. v > You would add to each side.. b 9 You would add 9 to each side. 6. 7 + p You would subtract 7 from each side. 7. x < + + x < The solution is x <. Copyright Big Ideas Learning, LLC Algebra 67
Chapter 8. s 8 + 8 + 8 9 s The solution is s 9. 9 6 8 9. 6 m + + 7 m The solution is m 7. 7 6 8. c > + + c > 8 The solution is c > 8. 6 8. r + < r < The solution is r <.. 8 + y 8 8 y The solution is y.. 9 + w > 7 9 9 w > The solution is w >.. q + q The solution is q.. h () h + h 8 The solution is h 8. 8 7 9 6. > t ( ) > t + 9 > t The solution is t < 9. 9 7 7. j + 9 < 8 j + 6 < 8 6 6 j < The solution is j <. 8. + y + y + + y 6 The solution is y 6. 6 8 9. p p 7 p 7 + 7 + 7 7 p The solution is p 7. 7 9. 8 z + 6z > + 6 8 + z > 9 8 8 z > 9 The solution is z > 9. 9 7 9 7 68 Algebra Copyright Big Ideas Learning, LLC
Chapter. A number plus 8 is greater than. n + 8 > n + 8 > 8 8 n > An inequality is n + 8 >, and the solution is n >.. A number minus is at least. n n + + n An inequality is n, and the solution is n.. The difference of a number and 9 is fewer than. n 9 < + 9 + 9 n 9 < n < An inequality is n 9 <, and the solution is n <.. Six is less than or equal to the sum of a number and. 6 n + 6 n + 9 n An inequality is 6 n +, and the solution is n 9.. a. Words: Current weight of your bag + Additional weight Maximum weight Variable: Let w be how much weight (in pounds) you can add to your bag. Inequality: 8 + w 8 + w 8 8 w So, you can add no more than pounds to your bag. b. w 6. Words: Price of book Variable: + Additional money spent Amount for free shipping Let x be how much more you must spend. Equation: 9.76 + x 9.76 + x 9.76 9.76 x. In addition to the book, you must spend at least $. in order to get free shipping. 7. The values that are to the right of should have been shaded, not the values to the left of. 7 < x + + < x So, the solution is x >. 8. A should have been added to each side of the inequality, not just the left. + x 9 + + x 9 + x So, the solution is x. 9. Words: Number of goals so far + Additional goals needed Record number of goals Variable: Let g be the possible numbers of additional goals the player can score. Inequality: 9 + g 9 9 + g 9 9 9 g In order to match or break the NHL record, the player must score or more goals. 9 +? no; Because 9 + = is not less than or equal to, you cannot add both a laptop and a pair of boots to your bag without going over the weight limit. Copyright Big Ideas Learning, LLC Algebra 69
Chapter. a. Words: Your st jump score + Your nd jump score > Competitor s st jump score + Competitor s nd jump score Variable: Let s be your score on your second jump. Inequality: 9. + s > 7. + 9.8 9. + s > 7. + 9.8 9. + s > 6.9 9. 9. s > 7. So, you must earn a score greater than 7. points on your second jump. b. Both are correct because 8. > 7. and 7. > 7... A; By the Subtraction Property of Equality you can subtract from each side of the inequality. x b <, and the result when simplified is the equivalent inequality x b <. D; By the Addition Property of Equality, you can add (b x ) to each side of the inequality, and the result when simplified is < b x.. Perimeter <. x +. +. <. x + 9.7 <. 9.7 9.7 x <.6 So, the side labeled x is less than.6 inches in length.. Perimeter 8.7 x + 6. +.9 +. 8.7 x +. 8.7.. x. So, the side labeled x is at most. feet long.. Sample answer: You received $ for your birthday, and you have already decided to buy a shirt that costs $. What is the most you can spend on some new songs and application downloads? An inequality to represent this situation is + x, and the solution is x 6, which matches the graph.. no; You cannot check all of the numbers in the solution set of an inequality, because the solution set contains infinitely many numbers. x + + x 8 For the equation x, the solution set includes all numbers greater than or equal to 8. So, you can check numbers such as 8, 9.,, or,76. All will make the inequality true as well as all the numbers in between and greater. 6. a. no; Sample answer: The set of students who have brown hair and the set of students who have brown eyes are separate sets with some overlap. You cannot determine from the diagram whether there are more students with brown hair or more with brown eyes. b. no; Sample answer: Because you cannot determine whether there are more students with brown hair or more with brown eyes, adding to the number of students with brown hair does not guarantee that the sum will be at least as big as the number of students with brown eyes. c. yes; Sample answer: The number of students with brown hair is at least as large as the number of students with both brown hair and brown eyes because the set of students with both brown hair and brown eyes is contained completely within the set of students with brown hair. d. yes; Sample answer: Because the inequality H X from part (c) must be true, if you add to the number of students with brown hair, the sum is still at least as large as the number of students with both brown hair and brown eyes. e. no; Sample answer: It is possible for all of the students with brown hair also to have brown eyes, in which case H would be equal to X, and the inequality H > X would not be true. f. yes; Sample answer: Because the inequality H X from part (c) must be true if you add to the number of students with brown hair, the sum is always going to be greater than the number of students with brown eyes. 7. a. x + 8 8 8 x 6 The numbers that are not solutions of x + 8 < are x 6. 6 8 b. x <.7 + + x < 7.7 The numbers that are not solutions of x.7 are x < 7.7. 7.7 7. 7. 7. 7.6 7.8 8. 7 Algebra Copyright Big Ideas Learning, LLC
Chapter 8. c d b + < a + a d 7 + + + 7 + 7 c d + b + < a a + d Because b + < a, it must be true that b < a. Similarly, because a + d, you can conclude that a < d. Finally, because c d +, you know that c > d, or d < c. So, b < a < d < c. Maintaining Mathematical Proficiency 9. 7 ( 9) =. ( ) =. 7 ( ) = 9. ( ) =. 6x = Check: 6x = 6x 6 = 6 x = The solution is x =. 6() =? =. y = Check: y = y = (6) =? y = 6 = The solution is y = 6.. s = Check: s = s = () =? s = = The solution is s =. 6. n = 7. Check: n = 7. n = ( 7.) 9. =? 7. n = 9. 7. = 7. The solution is n = 9... Explorations (p. 67). a. x x 6 9 6 <? x No No No No Yes Yes Yes The graph on the right represents the solution set of the inequality 6 < x. So, the solution is x >. b. i. x < x x 6 8 x <? Yes Yes Yes No No No No The solution is x <. ii. x x x 6 9? x Yes Yes Yes No No No No The solution is x. iii. x < 8 x x 6 8 x <? 8 Yes Yes Yes Yes Yes No No The solution is x <. iv. 6 x x x 6 9 6? x Yes Yes Yes Yes No No No The solution is x. Dividing each side of an inequality by the same positive number produces an equivalent inequality. Copyright Big Ideas Learning, LLC Algebra 7
Chapter. a. x x 9 6 6 <? x Yes Yes Yes No No No No The graph on the left represents the solution set of the inequality 6 < x. So, the solution is x <. b. i. x < x x 8 6 x <? No No No No Yes Yes Yes The solution is x >. ii. x x x 9 6. a. 7x < b. x 7x x 7 < 7 x < x The solution is x <. The solution is x. c. < x d. x x > x > x x The solution is x. The solution is x <.. Monitoring Progress (pp. 68 7). n 7.. w 7 n 7 7 ( ) n 7 The solution is n 7. (.) w w The solution is w.? x No No No No Yes Yes Yes 9 7 The solution is x. iii. x < 8 x x 8 6. b 6. >.q b. >.q 6. b 9 > q The solution is b 9. The solution is q <. x <? 8 No No Yes Yes Yes Yes Yes 7 9 The solution is x >. iv. 6 x x x 9 6 6? x No No No Yes Yes Yes Yes The solution is x. When dividing each side of an inequality by the same negative number, the direction of the inequality symbol must be reversed to produce an equivalent inequality.. Divide each side of the inequality by the same number. If the number is positive, this produces an equivalent inequality. If the number is negative, the inequality must be reversed to be equivalent.. p < 7 6. x p > 7 p > 8 The solution is p > 8. 8 6 7. z ( z ) z The solution is z. 8. 9m > 6 9m 9 < 6 9 m < 7 The solution is m < 7. x ( ) x The solution is x. 7 9 7 7 Algebra Copyright Big Ideas Learning, LLC
Chapter 9. r..y r.y.. r y The solution is r. The solution is y. 7 9. Words: Cost per copy Variable: Number of copies 6 8.6 Let c be the numbers of copies you can make. Inequality:. c.6.c.6.c..6. c.6 Because you cannot make a partial copy, you can make at most copies.. Consider the formula for distance d = rt. Words: Variable: Distance Maximum speed limit Time Let t be the time (in hours) it takes the school bus to travel 6 miles. Inequality: 6 t 6 t 6 t t If the maximum speed limit for a school bus is miles per hour, it will take the school bus at least hours to travel 6 miles.. Exercises (pp. 7 7) Vocabulary and Core Concept Check. Sample answer: When you solve x <, you divide each side by positive in order to undo multiplication, which produces an equivalent inequality. When you solve x < 8, you divide each side by negative. So, you must reverse the direction of the inequality symbol in order to produce an equivalent inequality.. Sample answer: x Monitoring Progress and Modeling with Mathematics. x < 8. y 9 x y 9 < 8 x < y The solution is x <. The solution is y. 6. n 6. < 7t 7 < 7t n 7 n < t The solution is n. The solution is t >. 7. 6 8 x > 8. a <. x > () x > The solution is x >. a <. a <.8 The solution is a <.8...6.8.. 9. w. 8 w w The solution is w. 6 8 t 8 () 8 8 t t The solution is t.. t <. 9y > 9 t 9y 9 < 9 > 9 t > y < The solution is t >. The solution is y <.. z. c c z z c The solution is z. The solution is c. 6 8 6 8 Copyright Big Ideas Learning, LLC Algebra 7
Chapter. n 6. w 6 n n The solution is n. 7. < m () > ( m ) > m The solution is m <. 8 6 8. > y () < ( y ) 9 < y The solution is y > 9. 7 9 9. Words: Cost per fish w 6 w The solution is w. 78 76 Number of fish Amount you have to spend Variable: Let p be the prices (in dollars) you can pay per fish. Inequality: p p p p. You can spend no more than $. per fish.. Words: Change in temperature per hour Number of hours Total change in temperature Variable: Let t be the time (in hours) the temperature is dropping. Inequality: t t t t. It will take at least. hours for the temperature to drop at least 6 F.. 6 < y 6 < y < y The solution is y >.. 7v 7v 7 7 v The solution is v.. 9 x 9 9 ( 9 x ) 9 x The solution is x 9.. > n < n < n The solution is n >.. x > x > x > 8 The solution is x > 8. 6..y <..y. <.. y < The solution is y <. 6 7. The direction of the inequality symbol should be reversed only if each side is multiplied or divided by a negative number. > x () > x 9 > x The solution is x < 9. 8. Because each side of the inequality is divided by negative, the direction of the inequality symbol must be reversed to produce an equivalent inequality. y y y 8 The solution is y 8... 6 6 6 6 7 Algebra Copyright Big Ideas Learning, LLC
Chapter 9. Words: Cost Area of per square carpet foot Total amount you can spend Variable: Let c be the costs per square foot that you can pay. Inequality: c $ ft ( ft)( ft) $7 $c 96 ft $7 ft $c 96 $7 $c 96 $7 96 96 $c $ 7 $.7 So, you can spend no more than $.7 per square foot on carpet for your bedroom.. a. C; Multiplying each side by m results in x < m. b. A; Multiplying each side by m results in x > m. c. B; Multiplying each side by m results in x < m. d. D; Multiplying each side by m and reversing the inequality symbol results in x > m.. a. Consider the formula for distance d = rt. Words: Variable: Distance run Maximum speed Number of hours Let d be the distance (in miles) you run. Inequality: d 6. d 6. d.6 You run at most.6 miles. b. d 6. d. Your friend is correct. In hours, the farthest you could run is. miles.. x x x Sample answer: x. Words: Production cost per penny Variable: Number of pennies produced > Total spent in production costs Let n be how many pennies are produced. Inequality: $. n > 6,,.n > 6,,.n. > 6,,. n >,, Over million pennies are produced when the U.S. Mint pays more than $6 million in production costs.. x x x no; By the Division Property of Inequality for c <, x is equivalent to x as shown. So, x is not equivalent to x.. a. A > B or B < A b. A < B or B > A c. As numbers move farther from zero on a number line, their absolute values become larger; A > B and A > B ; A < B and A > B. 6. Sample answer: If x >, then when you multiply each side by x, x would be equivalent to. However, if x x <, then when you multiply each side by x, the direction of the inequality symbol must be reversed, and x is equivalent to x. 7. r > C π > π C π > π C > π So the circumference of the circle is greater than π units in length. Copyright Big Ideas Learning, LLC Algebra 7
Chapter 8. Consider the formula for distance d = rt. Words: Distance Time < Speed Variable: Let d be the possible distances a beginner can travel. Inequality: d.7 < 8 ( Note: min =.7 h ) d.7 < 8.7 d.7 <.7 8 d <. A beginner will travel less than. miles in minutes of practice time. 9. Words: Total number of employees Fraction of the employees who work full-time Maximum number of employees who work part-time Variable: Let p be the fraction of the employees who work part-time. Inequality: 6 p 6p 6p 6 6 p At least of the employees work part-time. Maintaining Mathematical Proficiency. x + = Check: x + = () + =? x = + =? x = = x = The solution is x =.. n + = n Check: n + = n + n + n () + =? (). = n + =? + + = = n = n = n The solution is n =. z + = z 8 z z = z 8 ( = ) = + 8 + 8 = z = z 6 = z The solution is z = 6. Check: z + = z 8 (6) + =? (6) 8 + =? 8 7 = 7. 8% =.8 and.8 =.8 So, 8% >.8.. % = = So, 6 >. = 6. % =. So, % >.. 7. 6% = 6 = = 9 and So, > 6%. =. y 8 = Check: y 8 = + 8 + 8 () 8 =? y = 8 =? y = () = y = The solution is y =. 76 Algebra Copyright Big Ideas Learning, LLC
Chapter. Explorations (p. 7). a. x + x + Write the inequality. x x Subtract x from each side. x + Simplify. Subtract from each side. x Simplify. The solution is x, which is represented by graph B. b. x + > x + 9 Write the inequality. + x + x Add x to each side. > x + 9 Simplify. 9 9 Subtract 9 from each side. > x Simplify. > x Divide each side by. > x Simplify. The solution is x <, which is represented by graph A. c. 7 x + x Write the inequality. 7 9x Simplify. 7 9 9x 9 Divide each side by 9. x Simplify. The solution is x, which is represented by graph E. d. x + x 6 < x + 7x Write the inequality. x 6 < x Simplify. + 6x + 6x Add 6x to each side. < 8x Simplify. 8 < 8x 8 Divide each side by 8. < x Simplify. The solution is x >, which is represented by graph C. e. (x ) x > x 6 Write the inequality. (x) () x > x 6 Distributive Property x 9 x > x 6 Simplify. x 9 > x 6 Simplify. + x + x Add x to each side. x 9 > Simplify. + 9 + 9 Add 9 to each side. x > Simplify. The solution is x >, which is represented by graph D. f. x 6x 8 8x x Write the inequality. x 8 9x Simplify. + 9x + 9x Add 9x to each side. x 8 Simplify. x 8 Divide each side by and reverse the direction of the inequality symbol. x Simplify. The solution is x, which is represented by graph F.. To solve a multi-step inequality, simplify each side of the inequality, if possible. Then use inverse operations to isolate the variable. Be sure to reverse the inequality symbol when multiplying or dividing by a negative number.. Sample answer: x 7 x and (x + ) x + 9 Check: x 7 x (x + ) x + 9 x x (x) + () x + 9 7 7x x + x + 9 7 x x 7 7x 7 x + 9 x x x x The solution of both inequalities is x.. Monitoring Progress (pp. 7 76). b < 7 + + b < 8 b < 8 b < The solution is b <. Copyright Big Ideas Learning, LLC Algebra 77. 8 9c 8 8 8 9c 9c 9 9 c The solution is c. 6 8 n. + > n > n < n < The solution is n <.
Chapter. 6 v v ) ( v v The solution is v.. x x x x x + + x 8 x 8 x The solution is x. 6. (k ) < k + (k) () < k + k < k + k k < The inequality < is true. So, all real numbers are solutions. 7. (n ) > n +. (n) ( ) > n +. n + > n +. + n + n >. The inequality >. is false. So, there is no solution. 8. (a ) a (a) () a 6a a 6a 6a a + + 8 a 8 a a The solution is a. 9 + 9 + 77 + 89 + x 9. 8 + x 8 + x 8 + x x 7 A score of at least 7 points will allow you to advance.. Exercises (pp. 77 78) Vocabulary and Core Concept Check. Sample answer: For solving both multi-step inequalities and multi-step equations, you begin by simplifying each side, if necessary. Then, you use inverse operations to isolate the variable. The processes are different, however, because when solving multi-step inequalities only, you have to remember to reverse the inequality symbol when multiplying or dividing by a negative number.. Sample answer: Because the terms with the variable are the same, they will cancel. Then the resulting inequality, 8 is false. Monitoring Progress and Modeling with Mathematics. 7b + + 7b 7b 7 7 b The solution is b, which is represented by graph B.. p + p 8 p 8 p The solution is p, which is represented by graph A.. g + g 8 g 8 g The solution is g, which is represented by graph C. 78 Algebra Copyright Big Ideas Learning, LLC
Chapter 6. ( f ) ( f ) f f ( )( f) ( )() The solution is f, which is represented by graph D. 7. x > 7 + + x > x > x > The solution is x >. 6 8 8. y + 9 9 9 y y y The solution is y. 9. 9 7 8v 7 7 v v v The solution is v.. > t + + > t < t < t The solution is t >.. w + > w > w > w > The solution is w >.. + m 6 m. m m The solution is m. 8 6 p + 9 > 9 9 p > p < p < The solution is p <. 8 r. + 6 r r r The solution is r.. 6 (a + ) 6 (a + ) a + a The solution is a. Copyright Big Ideas Learning, LLC Algebra 79
Chapter 6. 8 (b ) 8 (b ) 6 b + + b The solution is b. 8 6 7. m > 7 m + m + m + m > 7 m > The solution is m >. 8. 8n + 8n 9 8n 8n 9 The inequality 9 is false. So, there is no solution. 9. d < d + 8 + d + d < d + 8 8 8 < d < d < d The solution is d >.. 8 + f > f + f + f 8 + f > 8 8 f > 6 f > 6 f > The solution is f >.. 8g g + g g + g g g The inequality is true. So, all real numbers are solutions.. w > w + w 7 w > w 7 w w > 7 The inequality > 7 is true. So, all real numbers are solutions.. 6( + ) < ( + 6) 6( ) + 6() < ( ) + (6) 6 + 8 < 6 + 8 6 6 8 < 8 The inequality 8 < 8 is false. So, there is no solution.. (c 7) (c ) (c) (7) (c) () c c c c The inequality is true. So, all real numbers are solutions.. ( t ) > (t ) ( t ) () > (t) () t 8 > t 6 t t > The inequality > is false. So, there is no solution. 6. ( b + ) 6(b + 9) ( b ) + () 6(b) + 6(9) b b + 6b + b b + 9 b The solution is b 9. 7. 9j 6 + 6j (j ) j 6 (j) () j 6 j 6 j j The inequality is true So, all real numbers are solutions. 8 Algebra Copyright Big Ideas Learning, LLC
Chapter 8. 6h 6 + h < (h ) 8h 6 < (h) () 8h 6 < 8h 6 8h 8h < The inequality < is false. So, there is no solution. 9. In the first step, you need to use the Distributive Property on the left side. x + 6 ( x + 6 ) + (6) ) ( x x + x The solution is x.. The inequality 7 is false because is farther to the right on the number line than 7. So, the original inequality has no solution. ( x) x 7 () ( x) x 7 + x x 7 x x 7 Because 7 is false, there is no solution.. Words: Value per bill Number of bills + Minimum balance Current balance Variable: Let n be the number of $ bills you can withdraw. Inequality: n + n + n n You can withdraw no more than $ bills.. Words: Minimum earnings per hour Number of hours + Cost of materials Selling price Variable: Let h be how long (in hours) the woodworker can spend building the cabinet. Inequality: h + h + h 7 h In order to earn at least $ an hour, the woodworker can spend no more than hours building the cabinet.. Area > 6 ft ft (x )ft > 6 ft (x ) > 6 (x) () > 6 x 6 > 6 + 6 + 6 x > 96 x > 96 x > So, the value of x must be greater than.. Words: Forest Park membership fee Woodland membership fee + Forest Park nightly fee + Woodland nightly fee Number of nights Number of nights Variable: Let n be the number of nights you plan to camp. Inequality: + n + n + n + n n n + n 8 n 8 n n no; If you plan to camp for less than nights, Woodland Campgrounds would charge less and therefore be the better choice. If you plan to camp for more than nights, Forest Park Campgrounds charges less and would therefore be the better choice. Copyright Big Ideas Learning, LLC Algebra 8
Chapter. 8 ft 7 ft ft h 8 ft Use the Pythagorean Theorem to find the height the ladder will reach. h + = 7 h = 7 h = 9 h = 7 So, the ladder and the truck can reach a height of 7 + 8 = 78 feet. Solve the inequality n 78 to find how many stories the ladder can reach. n 78 n 78 n 7.8 The ladder can reach 7 stories. 6. a. no more than $ b. Gasoline is $. per gallon, and a car wash is $8. c..x + 8 d. The graph of y =.x + 8 intersects y = when x = 9. So, you can buy no more than 9 gallons of gas. 7. Area 9(π ) π r ( r r ) 9(π ) π r r 9(π ) r (π ) 9(π ) r (π ) 9(π ) (π ) (π ) r 9 r 9 r So, when r is at least units long, the area of the shaded region is greater than or equal to 9(π ). 8. Sample answer: Let x be the runner s time (in minutes) in his last race.. +. +.8 +. + x < 98. + x < 98. + x < 98. + x < 98. 98. x <.9 In order to achieve his goal, the runner must finish his last race in under.9 minutes. 9. a(x + ) < x + x a(x) + a() < x + ax + a < x + If a =, then an equivalent inequality is x + < x +, and when you subtract x from each side, you get <, which is always true. So, all real numbers are solutions of the original inequality when a =.. x + 8 + ax ax a x + ax + 8 ax a x( + a) + 8 ax a Let + a = a. a a = a Then x[ + ()] + 8 ()x () x( + 6) + 8 9x 9x + 8 9x 9x 9x 8 If a =, then the inequality 8 is equivalent to the original inequality. Because 8 is always true, all real numbers are a solution of the original inequality when a =. Maintaining Mathematical Proficiency. Six times a number y is less than or equal to. 6y An inequality is 6y.. A number p plus 7 is greater than. p + 7 > An inequality is p + 7 >.. The quotient of a number r and 7 is no more than 8. r 7 An inequality is r 7 8. 8 8 Algebra Copyright Big Ideas Learning, LLC
Chapter.. What Did You Learn? (p. 79). Sample answer: Because the Xianren Bridge is the world s longest natural arch, no other natural arch is the same length or longer. So, the length of every other arch is less than feet, the length of the Xianren Bridge.. Sample answer: If you make a mistake and write your inequality incorrectly for part (a), then you may also get the wrong answer when you answer part (b). So, checking its reasonableness is important to be sure that the correct result is used.. Sample answer: Because carpet is sold by the square foot, you need to know the area of the room in square feet. To find the area of the room, you multiply the length and width of the room, and because both were measured in feet, the units of the product are square feet... Quiz (p. 8). A number z minus 6 is greater than or equal to. z 6 An inequality is z 6.. Twelve is no more than the sum of. times a number w and..w + An inequality is.w +.. x <. x 8. 9 + q 9 9 q 6 The solution is q 6. 6. z ( 7) < z + 7 < 7 7 6 8 z < The solution is z <. 7. < y + + < y The solution is y >. 8. p 8 p 8 p 6 The solution is p 6. 6 8 9. 6 > w 6 < w < w The solution is w >.. x > x < x < The solution is x <.. y 7 7 + 7 + 7 y y 8 The solution is y 8.. 8(g ) (g + ) 8(g) 8() (g) + () g 6 < g + g g < The inequality < is true. So, all real numbers are solutions.. 6(x ) (x + ) 6(x) 6() (x) + () x 6 x + x x The inequality is false. There is no solution. Copyright Big Ideas Learning, LLC Algebra 8
Chapter. a. Let s be the distance (in yards) you can swim. s Let t be how long (in minutes) you can tread water. t 6 8 Let u be the length (in yards) you can swim under water without taking a breath. u b. ft yd yd = 8. yd ft ft yd yd =.7 yd ft no; Your swimming distance, feet, is approximately equal to 8. yards, which is less than the required minimum of yards. So, even though the other two satisfy the respective requirements, this one does not.. Words: Current volume of water + Additional volume Maximum volume Variable: Let V be the additional volumes the pelican s bill can contain. Inequality: + V 7 + V 7 V 6 The pelican s bill can contain at most an additional 6 cubic inches. 6. a. Words: Amount saved per week Number of weeks Minimum price Variable: Let w be how long (in weeks) you need to save to purchase a bike. Inequality: w w w w 8 You need to save for at least 8 weeks in order to have enough money to purchase a bike. b. It will take less time to save up enough money to buy a bike. The inequality would now be 6 + w. 6 + w 6 6 w w w, or So, it will take at least weeks to save up the rest of the money needed to buy a bike.. Explorations (p. 8). a. x and x < b. x > and x c. x and x d. x > and x < 6 e. Sample answer: You use and because the numbers in the shaded interval make both inequalities true.. a. x or x > b. x < or x c. x or x d. x < or x > 6 e. Sample answer: You use or because the numbers in the shaded interval make either inequality true, but not both.. Sample answer: You can join two inequalities with the word and or the word or. If you are describing all the numbers between two endpoints, you use the word and because the numbers in the interval make both inequalities true. If you are describing the numbers either below or above two endpoints, you use the word or, because the numbers in the shaded interval make either inequality true, but not both.. Monitoring Progress (pp. 8 8). A number d is more than and less than. d > and d < An inequality is < d <. 8. A number a is fewer than or no less than. a < or a An inequality is a < or a. 8 Algebra Copyright Big Ideas Learning, LLC
Chapter. m + < m < 6 The solution is m < 6. 6 8. < k < 7 + + + < k < < k < < k < 6 The solution is < k < 6. 6 8. c + or c 8 > + 8 + 8 c c > 7 c c or c > 7 The solution is c or c > 7. 7 8 6. p + < 7 or p p < p p < p p < or p The solution is p < or p. 8 7. C 9 (F ) 9 9 (F ) 9 7 F 7 () 9 + + + F 9 The solution is F 9. So, the temperature rating of the winter boots is F to 9 F.. Exercises (pp. 8 86) Vocabulary and Core Concept Check. Sample answer: The graph of x and the graph of x or x both have closed dots as endpoints on and. The first one, however, is shaded between and, and the second one is shaded below and above.. Sample answer: The compound inequality that does not belong is a < 6 or a > 9. When you graph this compound inequality, the whole number line will be shaded, because the shaded regions of the individual inequalities overlap. So, all real numbers are a solution of one part or the other of the compound inequality. Monitoring Progress and Modeling with Mathematics. < x (x > and x ). 7 < x < (x > 7 and x < ). x 7 or x 6. x or x > 6 7. A number p is less than 6 and greater than. p < 6 and p > An inequality is < p < 6. 6 8 8. A number n is less than or equal to 7 or greater than. n 7 or n > An inequality is n 7 or n >. 7 8 6 9. A number m is more than 7 or at most. m > 7 or m An inequality is m or m > 7 7 9 7. A number r is no less than. and fewer than 9.. r. and r < 9. An inequality is. r < 9... 9. 6 8. Let e be the depths (in feet) of slitsnails. e. Copyright Big Ideas Learning, LLC Algebra 8
Chapter. a. Let h be the elevations of trees in the low-elevation forest zone. 7 < h b. Let h be the elevations of flowers in the subalpine and alpine zones. < h,. 6 < x + < x 6 The solution is < x 6. 6 8. > r 9 < r 9 < r The solution is < r.. v + 8 < or v < v 8 8 > v < or v > The solution is v < or v >. 8 6. > w + or w 7 w 7 7 > w or w 9 The solution is w < 7 or w 9. 7 9 7. r + < 7 or r + 9 9 9 r < r 7 r < r 7 r < or r 7 The solution is r < or r 7. 7 8 8. < n + 9 < 9 9 9 < n < < n < < n < The solution is < n <. 8 9. < (x + 6) < 8 ( ) < (x + 6) < 8 < x + 6 < 6 6 6 6 < x < < x < < x < The solution is < x <. 6. < 7( b) or < 7() 7(b) < 7b < 7b 7 > 7b 7 > b or The solution is b < or b. 8 (b ) (b) () b + + b b b. In the second step, should have been subtracted from the as well as the and the 9. < x + < 9 < x < 6 > x > 6 > x > The solution is < x <. 86 Algebra Copyright Big Ideas Learning, LLC
Chapter. The graph should be shaded to the left of and to the right of, but not between them. x > or x + 8 < + + 8 8 x > or x < The solution is x < or x >.. C (F ) 9 9 () 9 9 (F ) 9 ( ) F 7 + + + F The solution is F. So, the possible temperatures of the interior of the iceberg are F to F.. Let h be the heights of skiers the shop does not provide skis for..6h < or.6h >.6h.6 <.6 h <.6.6h.6 >.6 or h >.6 The shop does not provide skis for skiers of heights less than or about 9. centimeters tall or greater than.6.6 or about 89.6 centimeters tall.. < c + < 8 < c < 8 > c > > c > Because there are no numbers that are both greater than and less than, the inequality has no solution. Note that the original inequality states that <, so there is no solution. 6. m or m > + + m > m 6 m > m 6 m or m > Because all numbers greater than or equal to are also greater than, every number greater than makes at least one of the inequalities true. So, the solution of the compound inequality is m >. 9 7 7. y + 8 and y + > 9 y y > 7 y y and y > 7 All numbers greater than 7 are also greater than, and make both inequalities true. So, the solution of the compound inequality is y > 7. 7 8. x 8 or x + > 9 + 8 + 8 x x > 6 x or x > 6 x > The graphs of these two inequalities overlap to cover the whole number line. So, all real numbers are a solution of one inequality or the other, and therefore all real numbers are a solution of the compound inequality. 8 6 9. n + 9 + n or n + < n + n n + n + n n + 9 < n + 9 9 n 9 or < n The graphs of these two inequalities overlap to cover the whole number line. So, all real numbers are a solution of one inequality or the other, and therefore all real numbers are a solution of the compound inequality. Copyright Big Ideas Learning, LLC Algebra 87
Chapter. x 8 < x and x 6 < x x < x + 6 + 6 x < + + < x and x < Because there is no overlap between the numbers greater than and less than, there are no numbers that make both inequalities true. So, the compound inequality has no solution.. (x 6)? (x ) (x) (6)? (x) () and (x + ) (x + 8) (x) + () (x) + (8) x? x x + x + 6 x x x x x? x + 6 + + x? x? x? and x 6 x 6 x In order for the solution of the compound inequality to be only one value, the first inequality must be (x 6) (x ). Then the solutions of the inequalities are x and x, and is the only number that satisfies both inequalities. So, for the compound inequality to have a solution of one value, x =, the inequality symbol to use is.. Sample answer: A lifeguard training class will be offered as long as at least 6 people sign up to take it, and there is only enough equipment for a maximum of people. So, the number of people p in the class must be 6 p, which is modeled by the graph.. x + 7 > x + > 7 + 7 > x 7 7 > x x > x > no; A value of does not make the inequality x > true.. a. Let P be the annual profits (in millions of dollars) from 6 to. P 9 b. P = R C P 9 R C 9 R 9 + + + 7 R no; The solution is 7 R. So, the company s annual revenue from 6 to was at least $7 million and no more than $ million. Because $6 million is too small to be in this range, it is not possible that the company had this annual revenue from 6 to. Maintaining Mathematical Proficiency. d 9 = 6 d 9 = 6 or d 9 = 9 d 9 = 9 6 9 d 9 = 9 () d = d = The solutions are d = and d =. 8 6. 7 p 7 = 7 p 7 = 7 7 p 7 = Because absolute value expressions must be greater than or equal to, the expression p 7 cannot equal. So, the equation has no solution. 88 Algebra Copyright Big Ideas Learning, LLC
Chapter 7. r + = r r + = r or r + = (r ) r r r + = r + = r + r + r + + r + = 6 = r 6 = r r = = r r = Check: r + = r + =? () =? 9 =? = The solutions are r = and r =. 8. w 6 = w + 7 r = r + = r + =? ( ) + =? 8 =? = w 6 = w + 7 or w 6 = (w + 7) w w = w + 7 7 7 = w ( ) = w 6 = w Check: w 6 = w 7 + w + w w 6 = 7 + 6 + 6 w = w = ( ) w = w 6 = w + 7 w 6 = w + 7 (6) 6 =? 6 + 7 6 =? 9 9 =? 9 9 = 9 The solutions are w = 6 and w =. 6 ( ) 6 =? + 7 6 =? + 8 =? 9 9 =? 9 9 = 9 9. Mean: + + + + 6 + 8 + + + + = 7 = 7 Absolute deviations: 7 = 6, 7 = 6, 7 =, 7 =, 7 6 =, 8 7 =, 7 =, 7 =, 7 =, 7 = 6 Mean absolute deviation: 6 + 6 + + + + + + + + 6 = = So, the data values deviate from the mean by an average of and are clustered close together.. Mean: + 6 + 8 + 8 + + + + + + 6 = = Absolute deviations: = 6, 6 =, 8 =, 8 =, =, =, =, =, =, 6 = 6 Mean absolute deviation: 6 + + + + + + + + + 6 = 8 =.8 So, the data values deviate from the mean by an average of.8 and are clustered close together..6 Explorations (p. 87). a. Sample answer: The inequality x + is true when the expression x + has a value less than or equal to and greater than or equal to. x + and x + b. x + and x +. a. x and x The solution is x and x ( x ). c. Sample answer: Write a compound inequality representing the distance between the absolute value expression and. x + = + =? = b. 6 8 6 8 All of the values between and are solutions of the absolute value inequality x + and the compound inequality from Exploration. c. Plot the distances to determine the endpoints of the solution. Copyright Big Ideas Learning, LLC Algebra 89
Chapter. a. A x -6 - - - 6-7 - 8 9 B x + The solutions given by the spreadsheet are and and the numbers in between, because they are the values of x that make x + less than or equal to. b. The spreadsheet method yields the same solutions as the other two methods. c. Sample answer: You can use a spreadsheet to quickly calculate the values of an absolute value expression for many values of the variable, and find the ones that give the expected solution.. Sample answer: You can solve an absolute value inequality algebraically, graphically on a number line, or numerically with a spreadsheet.. Sample answer: The algebraic method allows you to calculate exact values for the boundaries of the solution sets for all types of absolute value inequalities, even when the boundary points are not whole numbers. The graphical method provides a visual representation of how the endpoint of the solution set are related to each other and to the other numbers in the inequality, but it can be tedious to create and work with the graph especially if you do not know approximate values of the boundary points and/or they are not whole numbers. The numerical method helps you see how the values of an absolute value expression change as the value of the variable changes, but it can be time-consuming especially if you do not know approximate values of the boundary points, and it may not provide an exact answer if the boundary points are not whole numbers..6 Monitoring Progress (pp. 89 9). x. x. and x. The solution is. x..... k < By definition, the absolute value of an expression must be greater than or equal to. The expression k cannot be less than. So, the inequality has no solution.. w < w > and w < + + + + w > w < w > w < w > and w < 6 The solution is < w < 6. 6 8. x + > 8 x + < or x + > 8 x < or x > The solution is x < or x >. 6. n + + + n + By definition, the absolute value of an expression must be greater than or equal to. The expression n + will always be greater than. So, all real numbers are solutions. 6. d + 7 + 7 + 7 d + 6 d + 6 d + d + or d + d or d The solution is d or d. 9 Algebra Copyright Big Ideas Learning, LLC
Chapter 7. The mean price is 66 = $66. Let x represent a price you are willing to pay. x 66 7 7 x 66 7 + 66 + 66 + 66 89 x 79 The solution is 89 x 79. So, the prices you will consider must be at least $89 and at most $79. Five prices meet your condition: $6, $66, $67, $6, and $7..6 Exercises (pp. 9 9) Vocabulary and Core Concept Check. yes; By definition, the absolute value of an expression must be greater than or equal to. The expression x will always be greater than. So, all real numbers are solutions.. Solving the inequality w 9 requires a compound inequality joined by and. Solving the inequality w 9 requires a compound inequality joined by or. Monitoring Progress and Modeling with Mathematics. x < x > and x < The solution is < x <.. y. y. or y. The solution is y. or y.... 8. d + 9 > d + 9 < or d + 9 > 9 9 9 9 d < or d > The solution is d < or d >. 6. h h + + + h The solution is h. 7. s 7 By definition, the absolute value of an expression must be greater than or equal to. The expression s 7 will always be greater than. So, all real numbers are solutions. 8. c + > 7 c + < 7 or c + > 7 c < c > c < c > c < or c > The solution is c < or c >. 9. p + < By definition, the absolute value of an expression must be greater than or equal to. The expression p + cannot be less than. So, the inequality has no solution.. 9 n < < 9 n < 9 9 9 < n < > n >. > n > The solution is < n <.... 6t 7 8 + 8 + 8 6t 7 6t 7 or 6t 7 + 7 + 7 + 7 + 7 6t 6t 8 6t 6 6t 6 6 8 6 t or t The solution is t or t. Copyright Big Ideas Learning, LLC Algebra 9
Chapter. j + 6 > 6 6 j > By definition, the absolute value of an expression must be greater than or equal to. The expression j will always be greater than. So, all real numbers are solutions.. m > 8 m > 8 m > 6 m < or m > 6 m < m > m > m < m > or m < 8 The solution is m < 8 or m >. 8 6. 6b 8 6b 8 6b 8 By definition, the absolute value of an expression must be greater than or equal to. The expression 6b 8 will always be greater than. So, all real numbers are solutions of the original inequality.. w + 8 + + w + 8 8 w + 8 8 w + 8 w + 8 8 8 8 w w 6. u + < u < u u > 6 u < or u > 6 u < u > u > u < u > or u < The solution is u < or u >. 7. 6 f + + 7 > 7 7 7 6 f + > 6 f + > 6 6 f + > f + > or f + < f > f < f < f > f < or f > So, the solution is f < or f >. Or, in other words, all real numbers except are solutions of the inequality. 6 w The solution is w. 9 Algebra Copyright Big Ideas Learning, LLC