Advanced Algebra Honors. Most Problems Are Non

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Algebra 2 Honors Summer PacketCongratulations! You are going to be in Algebra 2 Honors!Here is a packet of pre-algebra/algebra topics that you are expected to know before you startAdvanced Algebra Honors. Most problems are non-calculator. To prepare yourself for the yearahead, do not use a calculator while doing these problems. If a calculator may be used to help withthe calculation process, it will be clearly marked: CALC.This packet contains notes, worked out examples, and practice problems. Be sure to read theexamples in each section before starting. You can show your work directly in this packet.This packet should be completed before the first day of school. We will go over the answers andquestions from this packet during first few days of school, so if you have a few questions, don’tworry. Remember to bring your completed packet with you on the first day of school!We look forward to meeting you in August! Number ersNaturalNumbers1

Definitions of Numbers:Complex: A number a bi where a and b are real numbers and i is the imaginary number. Allnumbers are complex.Real Numbers: Numbers that can be graphed on a number line.Imaginary Numbers: Numbers that contain the 1 iRational Numbers: Numbers that can be written as the ratio of two integers.Irrational Numbers: Numbers that cannot be written as the ratio of two integers.Integers: Positive and negative whole numbers.Whole Numbers: The set of the natural numbers and zero.Natural Numbers: The set of the counting numbers {1,2,3,4,5 }Complex NumbersCounting NumbersNatural NumbersReal NumbersImaginary NumbersIrrational NumbersRational NumbersNegative NumbersPositive NumbersOdd NumbersEven NumbersIntegers1) Check the box if the value belongs to that given set.a) 5b) 2/3c) -7d)3e)16f) 16g) 0h) 44i) j) 1.765k) -10000l) 1.5m) 62

Conversion of unitsExample 1: Convert 11 yards to inches.11 yards 3 feet 12 inches 396 inches1 yard1 footExample 2: Convert 45mph to feet per second.45 miles1 hour1 minute 5280 feet 66 feet per second1 hour60 minutes 60 seconds1 mileCALC: Evaluate by showing all steps!1. Convert 350 feet to yards.2. Convert 20 mi/hour to feet per second.3. Convert 6 feet per second to mph.3

Solving Linear Equations:Example 1:Example 2:3x 8 2953x 2155x (21)3x 357 p 10 9 p 610 2 p 64 2pp 2Solve the equation. Check your solution.1. 5t 302. 4q 5 353. 6 y 15 104. 4(2n 5) 8(n 2)Solve the equation. Check your solution.5. 48 j 25 12 j 116. 4( x 2) 4 x 84

Solving Linear Equations by Clearing Fractions:Example 1:11x x 102311( x x 10) *6233 x 2 x 605 x 60x 12Example 2:13 25( x ) 45 352x 3 4352( x 3 ) *124315 x 36 815 x 2828x 15Solve the following equations and check your answers.1211. 2 m 3 m 42. w 4 w 23235Solve the following equations and check your answers.2113. 6( x ) x 31285

Rewriting FormulasA formula is an equation that relates two or more variables.To rewrite a formula is to solve for another variable and write the equation as an equivalentequation.Example 1:Solve the distance formula for the rate.d r td rtExample 2:Recall the formula for the sum of all interior angles of a polygon, S 180(n-2)Solve this formula for n.Method #1:S 180(n-2)Method #2:S 180(n-2)S 180n – 360S n 2180S 2 n180S 360 180n( S 360) n180*Note: Both the formulas above are correct. They are mathematically equivalent.Formulas you should be familiar with from previous courses:QuantityFormulaMeaning of variablesDistanced rtd distance, r rate, t timeTemperature9F C 325F degrees in FahrenheitC degrees in CelciusArea of a triangle1A bh2A Area, b base,h heightPerimeter of a rectangleP 2l 2wArea of a trapezoid1A (b1 b2 )h2P perimeter, l length,w widthA Area, b1 base 1, b2 base 2,Area of a circleCircumference of a circleA r2C 2 r or C dh heightA Area, r radiusA Area, r radius,d diameter6

Rewrite the following formulas as requested.1. Solve the temperature formula for degrees Celsius.2. Solve the area of the trapezoid formula for one base.3. Solve the slope-intercept equation y mx b for the slope, m.Example 3:Example 4:Solve 7x – 4y 5 for x.7x 5 4y(5 4 y )x 7Solve 7x – 4xy 5 for xx(7 – 4y) 55x 7 4ySolve the following equations for x.1.y 6x 22.8y – x 94.2xy – x 3Solve the following equations for x.3.4x – 2xy 77

Graphing Linear InequalitiesThe graph of an inequality in one variable consists of all points on a number line that representthe solutions. Recall that an “open” circle means the value is not included in the solution and a“closed” circle means the value is included in the solution.Examples:x 4x 2 ---------------------------------------- ---------------------------------------- -2 -1 0 1 2 3 4 5-3 -2 -1 0 1 2 3 4 3 x 2(x -3 and x 2) --------------------------------------------- -4 -3 -2 -1 0 1 2 3 4x 2 or x 1 --------------------------------------------- -4 -3 -2 -1 0 1 2 3 4Important note: “and” means that all values in the solution make BOTH conditions true. This isthe “overlap” from the two given condition. “or” means that all values in the solution make atleast ONE condition true. It is possible to have a “no real numbers” or an “all real numbers”case.x 2 and x 1has no real solution (no “overlap”)x 2 or x 1all real numbers will make this condition true!Graph each inequality on the number line & simplify inequality if possible.1. x 3 and x 2 --------------------------------------------- -4 -3 -2 -1 0 1 2 3 42. x 1 or x 2 --------------------------------------------- -4 -3 -2 -1 0 1 2 3 43. x 2 or x 1 --------------------------------------------- -4 -3 -2 -1 0 1 2 3 48

Solving Linear InequalitiesA linear inequality in one variable can be written in one of the following forms, where a and bare real numbers and a 0 .ax b 0ax b 0ax b 0ax b 0Solving linear inequalities is the same as solving linear equations with one exception -- if youmultiply or divide BOTH sides of an inequality by a NEGATIVE number, then you mustreverse, or “flip,” the inequality sign.A solution of an inequality in one variable is a value that, when substituted for the variable,results in a true statement.Examples: Solve the inequality. Also, check your inequality by testing values for x that areinside and outside your solution set.Example 1:Example 2:x 2 4x 8 3x 2 8 3x 6x 24 x 5 9 x 10 5x 5 10 5x 15x 3Check x 2 4 x 8If x 2, 2-2 4(2) – 8 trueIf x 3, 3 – 2 4(3) – 8 trueIf x 0, 0 – 2 4(0) – 8 false, so x 0should not be in the solution (and it isn’t)Check 4 x 5 9 x 10If x 4, 4(4) 5 9(4)-10 falseIf x 3, 4(3) 5 9(3) – 10 falseIf x 0, 4(0) 5 9(0)-10 trueExample 3: 7 5x 2 8 5 5x 10 1 x 2Example 4:4 x 7 5 or 3x 2 234 x 12 or 3x 21x 3 or x 7(this means x -1 and x 2)Check 7 5x 2 8If x 0, -7 5(0) – 2 8 trueIf x -1, -7 5(-1) – 2 8 falseIf x 2, -7 5(-2)-2 8 trueCheck 4 x 7 5 or 3x 2 23Recall that each value for x just needs to makeone condition true (because of the “or”)If x 0, 4(0) – 7 5 trueIf x 10, 3(10) 2 23 true9

Solve the inequality. Also, do a “mental” check your inequality by testing values for x.1. 3x 5 8x 253. 5 3 2 x 72. 14 x 9 20 x 334. 3x 1 10 or 3 4 x 11Finding the Slope of a Line/Rate of changeFormula: m Example 1:y y2 y1 rise x x2 x1 runExample 2:Find the slope of the line connecting(3,5) and (7,2)y 2 5 3m x 7 3 4Oblique Line!(2,4) and (7,4)y 4 4 0m 0x 7 2 5Horizontal Line!Example 3:Example 4:Find the slope of the line connectingFind the value of k given the slope of the line(2,5) and (2,-8)y 8 5 13m undefinedx 2 20Vertical Line!Find the slope of the line connecting(2,4) and (3k, k 1) with slope 16y k 1 4 1 x3k 26k 3 1 3k 2 66(k 3) 1(3k 2)6k 18 3k 23k 16k 16310

For the following, find the slope and identify if the line is vertical, horizontal or oblique.1. (4, 4) and (4, 8)2. (2, -2) and (8, -2)3. (0, 8) and (9, 0)4. Find the value of k so that the line through the given points has the given slope.(2, -3) and (k, 7); m -2X and Y InterceptsExample 1:Example 2:The x-intercept(s) occur when y 0.To find the x-intercept, set the y-valueequal to zero and solve.The y-intercept occurs when x 0.To find the y-intercept, set the x-valueequal to zero and solve.y 2 x 240 2 x 2424 2 xx 12For #1-6,1. y y 3x 4y 3(0) 4y 4a) Find the x- intercept b) Identify the y- intercept and c) Identify the slope3x 323. 4 x 2 y 82. y 2 x 44. 5x 2 y 1011

Equations of Linear FunctionsPoint – Slope Form of a line is: y y1 m( x x1 ) where m is the slope of the line and( x1 , y1 ) is a point on the line.Slope – intercept form of a line is: y mx b where m and b are constant values. The slopeof the line is m and b is the y-intercept (0, b ).Standard Form of a line is: Ax By C where A, B,&C are integer values and A is positive.Example 1:Example 2:Write the equation of the line in all formsGiven:1m and (2,1)3Point-Slope Form:1y 1 x 2 3Slope-Intercept Form:11y x 33Standard Form:x 3 y 1connecting: (-2,5) and (3,1)y1 5 4 x 3 ( 2) 5Point-Slope Form: 4 4y 1 x 3 or y 5 x 2 55Slope-Intercept Form:m 417y x 55Standard Form:4 x 5 y 17#1-3. Write each line in all three forms, if possible. If not possible, write in correct form.1. Line connecting: (4, 7) & (0,9)2. Line connecting: ( 2,3) & (4,7)3. Line with slope: -3 and goes through (7,2)12

Graphing a Line in any formGiven a linear function, use the following methods to graph the line.Graphing a Line in any form:1) Construct a table of values.2) Plot enough points from the table to recognize a pattern.3) Connect the points with a line.Graphing a Line in Point-Slope Form:1) Determine the slope and point of the line.2) Graph the given point.3) Use the slope (m) and use it to plot at least a second point on the line.4) Draw a line through the two points.Graphing a Line in Slope-Intercept Form:1) Determine the slope and y-intercept of the line.2) Plot the y-intercept coordinate (0,b).3) Use the slope (m) and use it to plot at least a second point on the line.4) Draw a line through the two points.Graphing a line in Standard Form:1) Determine the x-intercept by plugging zero in for y and solving for x. Plot the point.2) Determine the y-intercept by plugging zero in for x and solving for y. Plot the point.3) Connect the points with a line.Graphing a horizontal line or vertical line:1) Determine two points on the line.2) Connect the points with a line.Example 1:Graphm y 2x 632(up 2, right 3)3b 6(0,6)13

Graph lines A-D on the same coordinate grid. Label each line.A: y 1x 33C: y 4B : 2 x 4 y 12D : x 3Graph lines E-H on the same coordinate grid. Label each line.E : y xF : 5 y 2 x 101G : y 3 ( x 2)2F : y 1 2( x 3)14

Systems of EquationsSystems of equations are classified as consistent or inconsistent. A consistent system has at least one solution. If a system of equations is classified asconsistent, it can be either independent or dependent. If the consistent system is independent, then it has one solution. If the consistent system is dependent, then it has infinitely many solutions. An inconsistent system has no solutions.Methods for Solving a System of Equations. Solve each system by using the indicated method.Then, classify the system.Examples:GraphingSubstitutionElimination 4 x 2 y 4 2 x 3 y 10 8 x 2 y 2 x 3 y 14 4 x 2 y 2 6 x y 5y 1. Solve equation (2) for y: x 14 3 y x The solution is 2, 2 and it is aconsistent, independent system.2. Substitute into equation(1):8 14 3 y 2 y 2112 24 y 2 y 2112 22 y 2 22 y 110y 53. Substitute value of y intoeither equation to solvefor x:x 14 3 5 x 1The solution is 1,5 and itis a consistent, independentsystem.1. Multiply equation (2) by2: 4 x 2 y 2 12 x 2 y 102. Add equations (1) and(2):16 x 83. Solve for x:x 124. Substitute value of x intoeither equation to solvefor y: 1 6 y 5 2 y 2 1 The solution is , 2 and 2 it is a consistent,independent system.15

1. Solve the system by graphing and then classify the system. 3x 4 y 8 a. 3 2 x 2 y 6 y 3x 2b. 5 x 2 y 2yy x x 2. Use any method that you would like to solve the system of equations. Then, classify thesystem. 3x 2 y 1a. 4 x 6 y 7 2 x 3 y 3b. 4 x 6 y 8 2 x 5 y 3c. 4 x 10 y 616

Use Problem Solving Strategies & ModelsWhen solving “real world” problems, it is helpful to write an equation in words before you writethe equation in mathematical symbols. This word equation is called a verbal model. Verbalmodels can be formulas, patterns, or diagrams. Here are some examples:Example 1: Use a formula.A bus travels at an average rate of 55 miles per hour. The distance between Chicago and SanFrancisco is 2130 miles. How long would it take for the bus to travel from Chicago to SanFrancisco?Verbal model& FormulaDistance Rate * TimeD rt2130 55t2130T hours or approximately 36.9 hours55Example 2: Look for a pattern.The table below shows the height h of a jet airplane t minutes after beginning its decent. Findthe height of the airplane after 15 minutes.Time (min), t02468Height (ft), h35,00032,00029,00026,00023,000Pattern Observed:The height decreases by 3,000 feet every two minutes.Verbal ModelHeight initial height – (rate of descent)*(time)& FormulaH 35000 – (3000/2)tH 35000 – 1500tH 35000 – 1500(15) 12,500 feet after 15 minutesExample 3: Draw a Diagram.You want to paint five 1-foot wide stripes on the wall. There should be an equal amount ofspace between the ends of the wall and the stripes and between each pair of stripes. The wall is14 feet long. How far apart should the stripes be?Diagram:1 ft1 ftEquation to solve:xxxx1 ftxx1 ft1 ftx 1 x 1 x 1 x 1 x 1 x 146x 5 14 6x 9 x 1.5 feet between each stripe17

CALC: Use one of the problem solving strategies to answer each question. Show all yourwork, as demonstrated by the examples on the previous page.1. If a jet airplane descends at the rate given in the table, what is its height after 27 minutes?Time (min), t0481216Height (ft), h46,00042,80039,60036,40033,2002. A car used 16 gallons of gasoline and traveled a total distance of 460 miles. The car’s fuelefficiency is 30 miles per gallon on the highway and 25 miles per gallon in the city. Howmany gallons were used on the highway? (The verbal model is provided for you this time Total distance(miles) HighwayGas usedfuel efficiency * on highway City fuelefficiency *Gas usedin city3. A moving company weighs 22 boxes you have packed that contain either books or clothes.The total weight of these boxes is 445 pounds. If each box of books weighs 35 pounds andeach box of clothes weighs 10 pounds, how many box of books did you pack?4. If the perimeter of a rectangle is 120 meters, and the length is 40 meters, find the width of therectangle.18

5. Your local cable company charges 30 per month for basic cable. Premium channels areavailable for a surcharge of 6 per channel. You have 70 a month budgeted for cable. Howmany premium channels can you purchase?6. You want to hang six 2-foot wide posters on the wall. There should be an equal space betweenthe posters and you also want the spaces to the far right and far left of the poster group to betwice the space between any two adjacent posters. The wall is 54 feet long. How far apartshould the posters be? (Express your answer in feet & inches.)7. You want to hang six 2-foot wide posters on a cylindrical kiosk that has a diameter of 10feet. There should be an equal space between the posters. How far apart should the postersbe? You may round to the nearest tenth of a foot.(Hint #1: Draw a diagram with an “unrolled” kiosk . Recall from geometry that the circumference of the kiosk is equal to the length of the unrolled rectangle.Hint #2: Because the posters are hanging on a cylinder-shaped wall, think about how many spaces there arebetween the posters.)19

8. On a track at an Air Force base in New Mexico, a rocket sled travels 3 miles in 6 seconds.What is the average speed in miles per hour.9. At a vegetable stand, you bought 3 pounds of peppers for 4.50. Green peppers cost 1 perpound and orange peppers cost 4 per pound. How many pounds of each kind of pepper didyou buy?10. You have two summer jobs. In the first job, you work 25 hours per week and earn 7.75 perhour. In the second job, you earn 6.25 per hour and can work as many hours as you want.You want to earn 250 per week. How many hours must you work at the second job?11. A quarter mile running track is shaped as shown. The formula for the inside perimeter is:P 2 r 2 x . Solve the perimeter for r.rrx12. Over a 30 day period, the amount of propane in a tank that stores propane for heating a homedecreases from 400 gallons to 214 gallons. What is the average rate of change in the amountof propane? (Include units)20

Algebra 2 Honors Summer Packet 1 Congratulations! You are going to be in Algebra 2 Honors! Here is a packet of pre-algebra/algebra topics that you are expected to know before you start Advanced Algebra Honors. Most problems are non-calculator. To prepare yourself for the year a

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