What is the ratio of white blood cells to red blood cells? If the expression is a polynomial, identify it as a monomial, a binomial, or a trinomial.
MONEY Write a polynomial to represent the value of t ten-dollar bills, f fifty-dollar bills, and h one-hundred-dollar bills. GRAVITY The height above the ground of a ball thrown up with a velocity of 96 feet per second from a height of 6 feet is 6 96t 16t2 feet, where t is the time in seconds.
According to this model, how high is the ball after 7 seconds? A second item sold at the same price brings in a profit of s3 30s2 s Write a polynomial that expresses the total profit from the sale of both items. If P is the perimeter, and P 10x 5y, find the measure of the third side.
Practice Multiplying a Polynomial by a Monomial Find each product. What is the product of twice the integer added to three times the next consecutive integer? Write an expression that represents the amount of money invested in the traditional account. Write a polynomial model in simplest form for the total amount of money T Kent has invested after one year.
What is the product of the next two consecutive even integers? Find an expression for the volume of a rectangular pyramid whose base has an area of 3x2 12x 9 square feet and whose height is x 3 feet.
What trinomial represents the area of the enlarged graph? In a guinea pig, pure black hair coloring B is dominant over pure white coloring b. Suppose two hybrid Bb guinea pigs, with black hair coloring, are bred. Find an expression for the genetic make-up of the guinea pig offspring.
What is the probability that two hybrid guinea pigs with black hair coloring will produce a guinea pig with white hair coloring?
Then classify each number as prime or composite. The area of a rectangle is 84 square inches. Its length and width are both whole numbers.
What is the minimum perimeter of the rectangle? What is the maximum perimeter of the rectangle? Baxter wants to tile a wall to serve as a splashguard above a basin in the basement. She plans to use equal-sized tiles to cover an area that measures 48 inches by 36 inches. What is the maximum-size square tile Ms. Baxter can use and not have to cut any of the tiles? How many tiles of this size will she need? Chapter 8 9 Glencoe Algebra 1 Lesson Find the prime factorization of each integer.
Check your solutions. A landscaping company has been commissioned to design a triangular flower bed for a mall entrance. The final dimensions of the flower bed have not been determined, but the company knows that the height will be two feet less than the base. The area of the flower bed can be 1 2 represented by the equation A b b. Write this equation in factored form.
Suppose the base of the flower bed is 16 feet. What will be its area? The height h of the rocket in feet above the ground after t seconds is modeled by the equation h 60t 16t2. How long was the rocket in the air before it returned to the ground? Find all values of k so that the trinomial x2 kx 35 can be factored using integers. A construction company is planning to pour concrete for a driveway. The length of the driveway is 16 feet longer than its width w.
Write an expression for the area of the driveway. Find the dimensions of the driveway if it has an area of square feet.
Janeel has a inch by inch photograph. She wants to scan the photograph, then reduce the result by the same amount in each dimension to post on her Web site. Janeel wants the area of the image to be one eighth that of the original photograph. Write an equation to represent the area of the reduced image. Find the dimensions of the reduced image.
If the trinomial cannot be factored using integers, write prime. DIVING Lauren dove into a swimming pool from a foot-high diving board with an initial upward velocity of 8 feet per second. Find the time t in seconds it took Lauren to enter the water. Use the model for vertical motion given by the equation h 16t2 vt s, where h is height in feet, t is time in seconds, v is the initial upward velocity in feet per second, and s is the initial height in feet.
Hint: Let h 0 represent the surface of the pool. Enrique caught the ball on its way down at a point 4 feet above the ground. How long was the ball in the air before Enrique caught it? Use the model of vertical motion from Exercise If the polynomial cannot be factored, write prime.
The distance d that the rock falls in t seconds is given by the equation d 16t2. How long does it take the rock to hit the ground? Cooper contested a speeding ticket given to him after he applied his brakes and skidded to a halt to avoid hitting another car. In traffic court, he argued that the length of the skid marks on the pavement, feet, proved that he was driving under the posted speed limit of 65 miles per hour.
The ticket cited his speed at 70 miles per hour. Use the formula 1 2 s 24 d, where s is the speed of the car and d is the length of the skid marks, to determine Mr. Was Mr. Cooper correct in claiming that he was not speeding when he applied the brakes? If so, factor it. If increasing the radius of a circle by 1 inch gives the resulting circle an area of square inches, what is the radius of the original circle?
The area of just the frame itself is 69 square inches. What is the width of the frame? Determine the domain and range. Identify the vertex as a maximum or minimum. Then graph the function. Miranda throws a set of keys up to her brother, who is standing on a balcony 38 feet above the ground. The equation h 16t2 40t 5 gives the height h of the keys after t seconds.
How long does it take the keys to reach their highest point? How high do the keys reach? Will her brother be able to catch the keys? What is a reasonable domain and range for this problem? A player hits a baseball into the outfield. The equation h 0. What is the equation of the axis of symmetry? What is the maximum height reached by the baseball? An outfielder catches the ball three feet above the ground.
How far has the ball traveled horizontally when the outfielder catches it? If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.
Two numbers have a sum of 2 and a product of 8. The quadratic equation n2 2n 8 0 can be used to determine the two numbers. Graph the related function f n determine its x-intercepts.
What are the two numbers? A footbridge is suspended from a parabolic support. The function h x 1 2 x 25 12 6 12 6 O 6 12 h x 9 represents the height in feet of the support above 0 represents the midpoint of the bridge. Graph the function and determine its x-intercepts. What is the length of the walkway between the two supports? Round to the nearest tenth if necessary.
Jaime owns a business making decorative boxes to store jewelry, mementos, and other valuables. The function y x2 50x models the profit y that Jaime has made in month x for the first two years of his business. The height H of the rock t seconds after Mikaela throws it is represented by the equation H 16t2 32t To the nearest tenth of a second, how long does it take the rock to reach the lake below? Hint: Replace H with 0. Then determine the number of real roots of the equation.
A roofer tosses a piece of roofing tile from a roof onto the ground 30 feet below. He tosses the tile with an initial downward velocity of 10 feet per second.
Write an equation to find how long it takes the tile to hit the ground. Use the model for vertical motion, H 16t2 vt h, where H is the height of an object after t seconds, v is the initial velocity, and h is the initial height.
Hint: Since the object is thrown down, the initial velocity is negative. How long does it take the tile to hit the ground?
She releases the ball from a height of 6 feet. The equation h 16t2 30t 6 represents the height h of the ball after t seconds. If the ball must reach a height of 25 feet for Quyen to catch it, does the ball reach Quyen? Hint: Substitute 25 for h and use the discriminant.
State the y-intercept. Then use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value. Explain why or why not. Klemperer told her English class that each week students tend to forget one sixth of the vocabulary words they learned the previous week. Suppose a student learns 60 words.
The number of words remembered can be described by the function W x 60 student remember after 3 weeks? If a lab researcher begins with 50 cells, how many cells will there be after one day, two days, and three days? Hint: Use the exponential function y 50 2x. Chapter 9 37 Glencoe Algebra 1 Lesson 5 x , where x is the number of weeks that pass.
Sports radio stations numbered in Write an equation for the number of sports radio stations for t years after If the trend continues, predict the number of sports radio stations in this format for the year During the s, the forested area of Guatemala decreased at an average rate of 1.
If the forested area in Guatemala in was about 34, square kilometers, write an equation for the forested area for t years after If this trend continues, predict the forested area in What will the value of the piece of machinery be after 7 years?
Find the value of the car in 8 years. If the population of Osaka was 11,, in and it continues to decline at the same rate, predict the population in SKY DIVING When a skydiver jumps from an airplane, the time t it takes to free fall a given distance can be estimated by the formula t 2s , where t is in seconds and s is in 9.
If Julie jumps from an airplane, how long will it take her to free fall meters? To estimate how long a thunderstorm will last, meteorologists can use the formula t d3 , where t is the time in hours and d is the diameter of the storm in miles.
A thunderstorm is 8 miles in diameter. Estimate how long the storm will last. Give your answer in simplified form and as a decimal. Will a thunderstorm twice this diameter last twice as long? A rectangle is 5 7 2 3 centimeters long and 6 7 3 3 centimeters wide.
Find the perimeter of the rectangle in simplest form. Find the area of the rectangle in simplest form. Practice Radical Equations Solve each equation. Check your solution. The voltage V in a circuit is given by V resistance in ohms. PR, where P is the power in watts and R is the If the voltage in a circuit is volts and the circuit produces watts of power, what is the resistance in the circuit?
Suppose an electrician designs a circuit with volts and a resistance of 10 ohms. How much power will the circuit produce? Assuming no air resistance, the time t in seconds that it takes an object to fall h feet can be determined by the equation t h. If a skydiver jumps from an airplane and free falls for 10 seconds before opening the parachute, how many feet does the skydiver fall? Suppose a second skydiver jumps and free falls for 6 seconds.
How many feet does the second skydiver fall? Justify your answer. If a television screen measures 24 inches high and 18 inches wide, what size television is it?
Darla told Tri that she has a inch television. The height of the screen is 21 inches. What is its width? Tri told Darla that he has a 5-inch handheld television and that the screen measures 2 inches by 3 inches. Is this a reasonable measure for the screen size? Chapter 10 31 Glencoe Algebra 1 Lesson Stephan would like to store a square theater prop that is 7 feet on a side.
Will it fit through the door diagonally? Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. Three players are warming up for a baseball game. Player B stands 9 feet to the right and 18 feet in front of Player A. Player C stands 8 feet to the left and 13 feet in front of Player A. Draw a model of the situation on the coordinate grid. Assume that Player A is located at 0, 0. What is the distance between Players B and C?
If they superimpose a coordinate grid on the map of their neighborhoods, Maria lives at 9, 1 and Jackson lives at 5, 4. When the discriminant is positive the quadratic equation has two solutions. When the discriminant is zero the quadratic equation has one solution. When the discriminant is negative the quadratic equation has no real solutions.
For a quadratic equation of the form , ,. To determine the number of solutions of each quadratic equation, we will look at its discriminant. You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use. What about the method of completing the square? Most people find that method cumbersome and prefer not to use it.
We needed to include it in this chapter because we completed the square in general to derive the Quadratic Formula. You will also use the process of completing the square in other areas of algebra.
Identify the most appropriate method to use to solve each quadratic equation:. Since the equation is in the , the most appropriate method is to use the Square Root Property.
We recognize that the left side of the equation is a perfect square trinomial, and so Factoring will be the most appropriate method. Put the equation in standard form. While our first thought may be to try Factoring, thinking about all the possibilities for trial and error leads us to choose the Quadratic Formula as the most appropriate method.
Access these online resources for additional instruction and practice with using the Quadratic Formula:. Use each clue to graph points that show the locations of his ships.
Three or more points in a row indicate the location of a single ship. State the domain and range of this relation. State the inverse of the relation. Plot the points in your graph. Clue 4: Determine whether each of the following relations is a function. If the relation is a function, graph the given points. If it is not a function, do not graph it. This is the number of ships that you should have found in the fleet. It is defined as the ratio of the rise, or vertical change in y, to the run, or horizontal change in x, as you move from one point to the other.
Given the coordinates of two points, x 1 , y 1 and x 2 , y 2 , on a line, the slope m of the line can be found as follows. What is the slope of the line that passes through 4, 6 and 2, 3? The height of the ramp needs to be 21 cm.
What will be the length of the ramp? Standardized Test Practice What is the slope of the line that passes through 1, 3 and 2, 6? In direct variation we say that y varies directly with x or y varies directly as x. In the direct variation equation, y kx, k is the constant of variation. The constant of variation in a direct variation equation has the same value as the slope of the graph. For example, y 5x is a direct variation because it is in the form of y kx. The constant of variation of y 5x is 5.
The slope of the linear graph of y 5x is 5. All direct variation graphs pass through the origin. For the equation y 2x, which passes through points 2, 4 and 5, 10 , show that the slope and the constant of the variation are equal. Write and solve an equation if y varies directly with x and y 40 when x 5. Therefore, y 8x. Name the constant of variation for each equation. Then determine the slope of the line that passes through the given pair of points.
Assume that y varies directly with x. If y 32 when x 4, find x when y If y 15 when x 6, find x when y Standardized Test Practice Which equation is not an example of a direct variation? Finding Intercepts To find the x-intercept, substitute 0 for y in the equation and solve for x. To find the y-intercept, substitute 0 for x in the equation and solve for y. Slope-Intercept Form If a line has a slope of m and a y-intercept of b, then the slope-intercept form of a Linear Equation of an equation of the line is y mx b.
Example Find the x- and y-intercepts of the graph of 2x 3y 5. Then, write the equation in slope-intercept form. Practice x 5 2 The x-intercept is 5 2. Slope-Intercept Form: 2x 3y 5 3y 2x 5 Subtract 2x from each side.
Find the x- and y-intercepts of the graph of each equation. Then write the equation in standard form. Standardized Test Practice What is the slope-intercept form of an equation for the line that passes through 0, 1 and 3, 37? It is also possible to write an equation for any line with a given slope and any point on the line. In addition, since you know the slope formula, m y 2 y1 , you can also write an equation of any line given two x2 x1 points.
Replace m with the given slope and the coordinates given the slope of the given point for x and y. Rewrite the and one point. Chose any of the two given points to use in place given two points. Rewrite the equation with the slope for m and the y-intercept for b. Write an equation in slope-intercept form from the given Examples information. The slope is 3 and the line passes b. The line passes through the points through the point 5, Use the slope formula. Practice Write an equation in slope-intercept form from the given information.
Standardized Test Practice Which is the correct slope-intercept equation for a line that passes through the points 15, 47 and 19, 59? The linear equation of a vertical line, which has an undefined slope, through a point x 1 , y 1 is x x 1. Write the equation, first in pointslope form and then in standard form, of the line that passes through 2, 3 and has a slope of 5. Point-Slope Form y y 1 m x x 1 y 3 5 x 2 y 3 5x 10 Distribute.
Standard Form 5x y 7, where A 5, B 1, and C 7. Write the point-slope form of an equation of the line that passes through 0, 3 and 4, 0. Write the point-slope form of an equation of the line that passes through the point 1, 4 and has a slope of 2 5. Write the standard form of an equation of the line that passes through the given point and has the given slope. Standardized Test Practice What is the standard form of an equation of the line that passes through 3, 3 and 1, 1?
If two nonvertical lines have the same slope, then they are parallel. All vertical lines are parallel. If the product of the slopes of two lines is 1, then the lines are perpendicular. The slopes of two perpendicular lines are negative reciprocals of each other.
Determine whether the graphs of b. Write an equation in slope-intercept Since 3 2 2 3 1, these lines are perpendicular. Rewrite each line in slope-intercept form to form of the line that is parallel to the graph of x 6y 12 and has an x-intercept of 9.
Find the slope of the line given. An x-intercept of 9 means the new line passes 6 y 0 1 6 x 9 m 1 6 , x 1 , y 1 9, 0 y 1 6 x 3 2 Slope-intercept form Practice Determine whether the graphs of each pair of equations are parallel, perpendicular, or neither.
Standardized Test Practice What is the slope of a line perpendicular to y 3x 2? In a scatter plot, the two sets of data are plotted as ordered pairs in the coordinate plane. As a negative correlation. As correlation. In this case; x and x increases, y also x increases, y decreases. Try These Together Explain whether a scatter plot for each pair of variables would probably show a positive, negative, or no correlation between the variables.
Practice Explain whether a scatter plot for each pair of variables would probably show a positive, negative, or no correlation between the variables. Standardized Test Practice What type of correlation is there between the number of hours spent talking long distance on the telephone and the amount of the telephone bill?
A positive correlation B no correlation C negative correlation D need more information Answers: 1. Then use the clues below to graph a group of segments and one line. The segments will not be connected in order, but when you finish they will form a recognizable figure. CLUE 1 Plot 4, 5 and 5, 3. Connect them with a line segment. What is the slope of this segment? Write an equation for the line that contains this segment. Use the slope m 2 to rise and run once. Connect the two points with a line segment.
Write an equation in point-slope form for the line that contains this segment. Plot the point indicated by the y-intercept. Use the slope to rise and run once. Connect the two points. CLUE 8 Connect 5, 1 to 0, 6. Use the slope to rise and run twice. CLUE 10 Connect 2, 5 and 0, 3 with a line segment.
Write an equation in slopeintercept form for the line that contains this segment. Also true for 2. Also true for The solutions of an inequality can be graphed on a number line or written using set-builder notation. Example Solve 3m 7 4m 1. Check your solution, and graph it on a number line.
Graph the point 8 using an open circle, since 8 is not part of the solution. Solve and graph z 16 5. Solve and graph j 1 2 9. Practice Solve each inequality. Then check your solution, and graph it on a number line. A number decreased by 3 is at least Twice a number is more than the difference of that number and 4. Standardized Test Practice Which number is a solution of 2x x 8?
A 12 B 11 C 9 D 6 Answers: 1—6. Multiplication and Division Properties for Inequalities For all numbers a, b, and c, the following are true. If c is positive and a b, then ac bc and c c , and if c is positive and a b, a b then ac bc and c c.
If c is negative and a b, then ac bc and c c , and if c is negative and a b, a b then ac bc and c c. These properties also hold true for inequalities involving and. Example Solve 5y 12 and check your solution. Check: Let y be 2. Solve 3a 27 and check. Solve 5 7 s 5 and check. One fifth of a number is at most Standardized Test Practice Solve 1 2 x 1 2.
The important exception is that multiplying or dividing an inequality by a negative number reverses the sign of the inequality. Example Solve 3f 7 f 9. Try These Together Solve each inequality. Define a variable, write an inequality, and then solve. Standardized Test Practice Solve 1 x 3 0. The graph of a compound inequality containing and is the intersection of the graphs of the two inequalities that make up the compound inequality.
To find the intersection, determine where the two graphs overlap. Compound inequalities that contain the word or are true if one or more of the inequalities is true. The graph is the union of the graphs of the two inequalities that make up the compound inequality.
Solve each compound inequality. Then graph the Examples solution set. Graph the solution set of a 9 and 2. Graph the solution set of d 6 or a 9.
Graph the solution set of n 7 and n 4. Then graph the solution set. The sum of three times a number and two lies between 8 and Eight less than 4 times a number is at most 24 and at least Standardized Test Practice If the replacement set is all integers, find the solution set for 1 x 1 3.
Also true for x n Examples a. Rewrite as a compound inequality. Solve a 4 7 and graph the 2. Solve 6s 4 8 and graph the solution set. HINT: The solution will be two points. HINT: The solution will be a line segment. The weight w in a bicycle trailer is allowed to vary from 60 pounds by no more than 40 pounds. The height h of a person allowed on a roller coaster can vary from 65 inches by no more than 13 inches.
Standardized Test Practice Solve x 5 7. An equation defines the boundary or edge of the half-plane. Graphing Inequalities in Two Variables 1.
Find the boundary by graphing the equation related to the inequality. If the inequality symbol is or , draw the boundary as a dashed line. If the inequality symbol is or , draw the boundary as a solid line to show that the points on the boundary are included in the solution set. Determine which of the two half-planes contains the solutions by choosing a point in each half-plane and testing its coordinates in the inequality.
If the coordinates make the inequality true, shade that half-plane. Example Graph y 2x 1. Solve the equality for y: y 2x 1. Then, graph the related equation y 2x 1. Draw the line as a solid line since the inequality symbol is less than or equal to.
Select a point in each of the half-planes and test it in the inequality. Test 0, 0 Test 1, 1 y 2x 1 y 2x 1 0 2 0 1 1 2 1 1 0 1 True 3 1 False Therefore, the half-plane that contains the point 0, 0 should be shaded. Practice Find which ordered pairs from the given set are part of the solution set for each inequality.
Jobs It takes a librarian 1 minute to renew an old library card and 3 minutes to make a new card. Together, she can spend no more than 30 minutes renewing and making cards. Write an inequality to represent this situation, where x is the number of old cards she renews and y is the number of new cards she makes. Standardized Test Practice Which ordered pair is a solution of x 2y 7? The student who raises the most money wins the grand prize. To find out what the grand prize is, match each inequality to its graph on the right.
Use the letter to the right of the graph to fill in the blanks that correspond with the number of the problem. A solution to a system of two equations with two variables is an ordered pair of numbers that satisfies both equations. One way to solve a system of equations is to carefully graph the equations on the same coordinate plane.
The coordinates of the point at which the graphs intersect is the solution to the system. If the graphs of the two equations coincide, meaning they are the same line, then there are infinitely many solutions to the system. A system of equations with at least one ordered pair that satisfies both equations is consistent.
It is possible for the graphs of the two equations to be parallel. In this case, the system is inconsistent because there are no solutions that satisfy the two equations. Chapter 7 Example Graph the system of equations to find the solution. Check this estimate by replacing x with 2 and y with 1 in each equation. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, name it. Practice Graph each system of equations.
Standardized Test Practice A canoe can be paddled 10 miles upstream, against the river current, in 5 hours. Paddling downstream the same distance takes 1 hour. Write and then graph a system of equations to solve for the speed c of the canoe in still water and the speed r of the river current.
Express the solution to the system as an ordered pair c, r. In general, if you solve a system of equations and the result is a true statement, such as 5 5, the system has infinitely many solutions; if the result is a false statement, such as 5 7, the system has no solution. Example Use substitution to solve the system of equations x y 1 and 2x y 1. Step 1: Solve one of the equations for x or y. Step 2: Substitute this value into the other equation.
The solution to the system is 2, 3. Check: Substitute 2 for x and 3 for y in each of the original equations and check for true statements. Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
Packs of AA batteries are on sale. What was the price of a single CD? If the coefficients of those terms are the same, the elimination method can be applied by subtracting the equations. Examples Practice Solve each system of equations using elimination. The solution of the system is 1, 2. Try These Together State whether addition, subtraction, or substitution would be most convenient to solve each system of equations.
Then solve the system. How much does a single can of juice cost? Standardized Test Practice Solve the system. Examples Solve each system of equations using elimination. Then, equation by 2 so that the coefficients of the x-terms add the equations and solve for x. Then, add the 2 x y 2 5 2x 2y 10 equations and solve for y. Try These Together 2x 45 43 Substitute 5 for y. Use elimination to solve each system of equations. Algebra Solve using elimination: 0 and 0. Standardized Test Practice By which number could you multiply the first equation of the following system to solve the system by elimination?
Recall that the graph of an inequality is a half-plane. The intersection of the two half-planes graphed in a system of inequalities represents the solution to the system.
Example Graph the system of inequalities to find the solution. Then, graph each inequality. This region is shaded dark gray. Notice that the boundary lines y x 3 and y x 3 are included in the solution, since the inequalities contained and symbols. Practice Solve each system of inequalities by graphing. Algebra Solve by graphing.
Standardized Test Practice A dieter limits a snack to 90 Calories. Which is a possible snack combination of Calorie apricots and 3-Calorie celery stalks? A 4 apricots B 3 apricots C 2 apricots D all of these 3 celery stalks 10 celery stalks 8 celery stalks Answers: 1—9.
The treasure hunt is taking place on a giant coordinate grid that is laid out on the floor of your school gym. You are competing with other parents and students for a grand prize. The treasures for which you are searching are numbered stickers on the floor of this giant coordinate grid. Specifically, you are given a list of four items and a starting point.
To locate the treasures, you must plot the intersection of the two graphs listed for each treasure. Your starting point is the intersection of the graphs of y 2 x 5 and 3 2y x Find the coordinates of your starting point by graphing the two equations. Determine the coordinates for the location of each treasure by graphing each pair of equations given in the figure below.
Then determine which treasure is closest to your starting point. A monomial is a number, a variable, or a product of a number and one or more variables. Monomials that are real numbers are called constants. To simplify a product involving monomials, write an equivalent expression in which: 1 there are no powers of powers, 2 each base appears exactly once, and 3 all fractions are in simplest form.
Product You can multiply powers with the same base by adding exponents. For any number a, of Powers and all integers m and n, a m a n a m n. For any number a, and all a Power integers m and n, a m n a mn. Chapter 8 Power of A power of a product is the product of the powers. For all numbers a and b, and any a Product integer m, ab m a m b m.
Power of a Monomial The power of a power property and the power of a product property can be combined into the power of a monomial property. For all numbers a and b, and all integers m, n, and p, a m b n p a mp b np. Examples Practice Simplify each expression. Standardized Test Practice Simplify a 2 b ab 2 3. For all integers m and n and any nonzero number a, a m an a m n. For any nonzero number a, a 0 1.
For any nonzero number a and any integer n, a n a 1 n and a 1 n a n. Examples Simplify each expression. Assume that no denominator is equal to zero.
Money Matters You can use the formula P A 1 1 i i n to find the monthly payment on a loan of A dollars that is paid back in equal monthly payments over n months. The variable i represents annual interest rate Standardized Test Practice Simplify x y x2. D 7 Answers: 1. For this reason, it is not always desirable to express numbers in standard notation. Large numbers such as these may be expressed in scientific notation.
Scientific A number is expressed in scientific notation when it is in the form a 10 n , Notation where 1 a 10 and n is an integer. If there is a positive or negative correlation, describe its meaning in the situation. Determine what relationship, if any, exists in the data. Use the points , Predict the price of a ticket in Draw a scatter plot and determine what relationship, if any, exists in the data.
Draw a line of fit for the scatter plot. Write the slope-intercept form of an equation for the line of fit. Cases 40 30 20 0 4. ZOOS The table shows the average and maximum longevity of various animals in captivity.
Sample 10 Year Longevity years Avg. Predict the maximum longevity for an animal with an average longevity of 33 years. Then find the correlation coefficient. Year 42 48 46 50 52 48 Goals Scored 2. Year Percentage TAXES The table shows the estimated sales tax revenues, in billions of dollars, for Massachusetts each year since Year Tax Revenue 3. Source: Beacon Hill Institute 4.
Year Diapers Sold 60, 65, 66, 65, 70, a. Find an equation for the median-fit line. How many diapers should SureSave anticipate selling in ? To determine the ideal level of acidity, a farmer measured how many bushels of barley he harvests in different fields with varying acidity levels.
Soil Acidity pH Bushels Harvested 5. Find an equation for the regression line. According to the equation, how many bushels would the farmer harvest if the soil had a pH of 10? Is this a reasonable prediction?
Year 21 17 16 16 14 Turtles Hatched 2. Predict the number of students who were learning English in California in Predict the number of students who will be learning English in California in Below is a table showing the population of Detroit each decade. Year Population millions 1. Census Bureau a. Find the correlation coefficient and explain the meaning of its sign.
Estimate the population of Detroit in State the domain and range. Determine the domain and range of each function. Every fraction of a minute is rounded up to the next minute.
Draw a graph to represent the cost of using the cell phone. Check your solution, and then graph it on a number line.
A number decreased by 10 is greater than A number increased by 1 is less than 9. Seven more than a number is less than or equal to Twenty less than a number is at least A number plus 2 is at most 1. The sum of a number and 17 is no less than Twice a number minus 4 is less than three times the number. Twelve is at most a number decreased by 7. Eight plus four times a number is greater than five times the number. If a plane is flying at an altitude of 5.
Jamal is planting a bush that needs a hole 18 centimeters deep for the roots. The instructions suggest an additional 8 centimeters depth for a cushion. If Jamal wants to add even more cushion, and the topsoil in his yard is 30 centimeters deep, how much more cushion can he add and still remain in the topsoil layer?
Define a variable, write an inequality, and solve each problem. Three times a number is at most nine. One third of a number is no more than nine.
Negative three times a number is more than nine. Three times a number is less than nine. Negative three times a number is at least nine. One third of a number is greater than or equal to nine.
Four times a number is greater than One eighth of a number is less than or equal to 3. Negative twelve times a number is no more than Negative one sixth of a number is less than Eight times a number is at least Negative four times a number is less than five.
Four fifths of a number is no more than five. Four times a number is fewer than five. Negative four times a number is no less than five. Four times a number is at most five. Four fifths of a number is more than five. Negative three times a number is at least Two thirds of a number is no more than Negative three fifths of a number is less than If the river rises more than 2 feet, it will exceed flood stage.
How long can the river rise at this rate without exceeding flood stage? Add 3 to each side. Distributive Property b. Subtract 33 from each side. Divide each side by 5. Four more than the quotient of a number and three is at least nine.
The sum of a number and fourteen is less than or equal to three times the number. Negative three times a number increased by seven is less than negative eleven. Five times a number decreased by eight is at most ten more than twice the number. Seven more than five sixths of a number is more than negative three. Four times the sum of a number and two increased by three is at least twenty-seven. A number is less than one fourth the sum of three times the number and four.
Two times the sum of a number and four is no more than three times the sum of the number and seven decreased by four. The base of the triangle is 16 feet. What is the height of the triangle? She practices for three fourths of an hour each session. If Nabuko has already practiced 3 hours in one week, how many sessions remain to meet or exceed her weekly practice goal? A number plus one is greater than negative five and less than three.
A number decreased by two is at most four or at least nine. The sum of a number and three is no more than eight or is more than twelve. Two times a number plus one is greater than five and less than seven. A number minus one is at most nine, or two times the number is at least twenty-four. Write an inequality to represent the latitude of the prevailing westerlies. Write an inequality to represent the latitudes where the prevailing westerlies are not located. If you eat no fewer than 4 and no more than 7 cookies, how many grams of fat will you consume?
Serena will make the B team if she scores within 8 points of the team average of Solve each inequality. The height of the plant must be within 2 inches of the standard inch show size. Based on the size of the chicken, the actual number of Calories in the entree can vary by as many as 40 Calories from this amount.
Write an absolute value inequality to represent the situation. What is the range of the number of Calories in the chicken entree? You have boxes in 12 inch and 15 inch heights, and want to stack them as high as possible to fit. Write an inequality that represents this situation.
Does Satchi have enough money to buy 2 videos and 3 CDs? If it has one solution, name it. How many treats does Nick need to sell per week to break even? Write a system of equations to represent the situation. Graph the system of equations. How many CDs and videos did the store sell in the first week?
What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pay scale? Which is the better offer? How many adult tickets and student tickets were purchased? Use elimination to solve each system of equations. The sum of two numbers is 28 and their difference is 4.
What are the numbers? Find the two numbers whose sum is 29 and whose difference is The sum of two numbers is 24 and their difference is 2. Find the two numbers whose sum is 54 and whose difference is 4. Two times a number added to another number is Three times the first number minus the other number is Find the numbers. The sum of two numbers is 41 and their difference is 5.
Four times one number added to another number is One number added to three times another number is Five times the first number added to three times the other number is Together, English and Farsi are spoken as a first language in countries. In how many countries is English spoken as the first language? In how many countries is Farsi spoken as the first language?
What were the prices for the gloves and hats? Two times a number plus three times another number equals The sum of the two numbers is 7. Four times a number minus twice another number is The sum of the two numbers is Find the amount they charged to wash a car and vacuum a car. Eight times a number plus five times another number is The sum of the two numbers is 1.
Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. The return trip took three hours. Find the rate at which Laura and Brent paddled the canoe in still water. If the digits are reversed, the new number is 45 more than the original number. Determine the best method to solve each system of equations. Then solve the system.
Write and solve a system of equations to find how many pumpkins and squash they sold? He worked 5 times as many hours during the week as he did on the weekend. Write and solve a system of equations to determine how many hours of overtime Ramiro worked on the weekend.
Some of these baskets were worth 2-points and others were worth 3-points. In total, she scored 30 points. Write and solve a system of equations to find how 2-points baskets she made.
There are 3 times as many non-fiction books as fiction books. Write and solve a system of equations to determine the number of nonfiction and fiction books. Tia and Ken each sold snack bars and magazine subscriptions for a school fund-raiser, as shown in the table. Number Sold Tia Ken Item snack bars 16 20 magazine subscriptions 4 6 a. Define variable and formulate a system of linear equation from this situation.
What was the price per snack bar? Determine the reasonableness of your solution. The film club has 4 members and gains 4 new members every month. Write and solve a system of equations to find when the number of members in both clubs will be equal. Then identify the element in each matrix. If the matrix does not exist, write impossible. Write a matrix to organize the temperatures. What are the dimensions of the matrix?
Which day and location had the highest temperature? Then identify the position of the circled element in each matrix. Dulles a. Write a matrix to organize the sales data. In which city does Burger Hut sell more food than its competitors? Use an augmented matrix to solve each system of equations. Write a system of linear equations to represent the situations. Write the augmented matrix. What is the cost of each type of ticket? O Chapter 6 Make a graph to show the number of hours Diego works out at the gym and the number of miles he walks per week.
List three possible solutions. Explain your reasoning. A panel of five light switches can set in twice this many ways. In how many ways can five light switches be set? HOBBIES Tawa wants to increase her rock collection by a power of three this year and then increase it again by a power of two next year. If she has 2 rocks now, how many rocks will she have after the second year? Assume that no denominator equals zero. A cubic millimeter of the blood contains white blood cells and red blood cells.
What is the ratio of white blood cells to red blood cells? Evaluate each product. Express the results in both scientific notation and standard form. G is a constant equal to 6. The mass of the earth m1 is equal to 5. Express the distance r in scientific notation. Compute the amount of gravitational force between the earth and the moon. Express your answer in scientific notation. If so, identify the polynomial as a monomial, binomial, or trinomial. Write each polynomial in standard form.
Identify the leading coefficient. MONEY Write a polynomial to represent the value of t ten-dollar bills, f fifty-dollar bills, and h one-hundred-dollar bills. According to this model, how high is the ball after 7 seconds? Write a polynomial that expresses the total profit from the sale of both items. What is the product of twice the integer added to three times the next consecutive integer? Write an expression that represents the amount of money invested in the traditional account.
Write a polynomial model in simplest form for the total amount of money T Kent has invested after one year. Find each product.
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