# Graphing and Solving Nonlinear Inequalities

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1 APPENDIX LESSON 1 Graphing and Solving Nonlinear Inequalities New Concepts A quadratic inequality in two variables can be written in four different forms y < a + b + c y a + b + c y > a + b + c y a + b + c Using a procedure similar to graphing linear equalities a quadratic inequality can be graphed. Eample 1 Graphing a Quadratic inequality a. Graph y > Step 1: Graph y = as a boundary. Use a dashed curve because the inequality symbol is >. Step : Shade inside the parabola since the solution consists of y-values greater than the y-values on the parabola for the corresponding -values O y 8 Check Test a point in the solution region. Substitute (1, 3) into the inequality y > (1) + (1) > 0 b. Graph y Step 1: Graph y as a boundary. Use a solid curve because the inequality symbol is. Step : Shade below the parabola since the solution consists of y-values less than the y-values on the parabola for the corresponding -values O y 8 Check To verify the solution region test a point. Substitute (3, ) into the inequality. y (3) + (3) Saon Algebra 1

2 A quadratic inequality in one variable can be written in four different forms a + b + c < 0 a + b + c 0 a + b + c > 0 a + b + c 0 Quadratic inequalities can be solved using tables, graphs, or algebraic methods. Eample Solving with a Table APPENDIX LESSONS Solve - 3 using a table. Step 1: Write the inequality as Step : Make a table of values The inequality is true for values of between -1 and 3 inclusively. The solution of the inequality is Eample 3 Solving with a Graphing Calculator Table Solve - - using a graphing calculator. Step 1: Use a graphing calculator to graph each side of the inequality. Set Y1 equal to - - and set Y equal to. Step : View the table comparing the two equations. Step 3: Identify the values of where Y1 = - - are less than or equal to the values of Y =. The solution set is - 3. Appendi Lesson 1 831

3 Eample Solving with a Graphing Calculator Graph Solve < using a graphing calculator. Step 1: Use a graphing calculator to graph each side of the inequality. Set Y1 equal to and set Y equal to. Step : Calculate the points of intersection. Step 3: Identify the values of where Y1 Y. The solution set is < <. Lesson Practice (E 1) (E 1) (E ) (E 3) (E ) a. Graph y > b. Graph y c. Solve - 3 using a table. d. Solve using a graphing calculator. e. Solve < using a graphing calculator. 83 Saon Algebra 1

4 APPENDIX LESSON New Concepts Graphing Piecewise and Step Functions APPENDIX LESSONS When a function has a different rule for different pieces of its domain, it is called a piecewise function. This kind of function is a combination of two or more functions. It assigns a different value to each domain interval. A piecewise function that is constant for each part of the domain is called a step function. Eample 1 Evaluating a Step Function Evaluate the function for =, = -, and = 6. f() = 10 if - 8 if > - When =, then f() = 10 because -. When = -, then f(-) = 10 because - -. When = 6, then f(6) = 8 because 6 > -. Eample Evaluating a Piecewise Function Evaluate the function for =, = -, and = 6. f() = - 1 if < 6 8 if 6 When =, then < 6. Use the piece of the function, f() = - 1. f() = () - 1 Substitute for into f(). = -8-1 Multiply and. = -9 Simplify. When = -, then < 6. Use the piece of the function, f() = - 1. f(-) = (-) - 1 Substitute - for into f(). = - 1 Multiply and -. = -5 Simplify. When = 6, then 6. Use the piece of the function, f() = 8. f(6) = 8 6 Substitute 6 for into f(). = 8 36 Simplify the eponent. = 88 Multiply. Appendi Lesson 833

5 Eample 3 Graphing a Step Function Graph the function. f() = -1 if 3 if > Graphing a step function is a lot like graphing inequalities. You will use open circles to indicate > or < and closed circles to show or. Begin by considering the function at =. This is where the steps separate. Because f() = -1, graph the point (, -1) with a closed circle. f() = -1 for. Draw a ray from the point etending to the left, along the line y = -1. This is one horizontal step. Net consider the other piece, f() = 3 for >. 6 O y 6 At (, 3), draw an open circle because f() 3. Draw a ray going to the right. This is another horizontal step. Eample Graphing a Piecewise Function Graph the function if - 1 f() = -5 if -1 < - 10 if > The function is made of two linear pieces and a quadratic piece with a domain divided at = -1 and =. Find the value of the two surrounding functions for these values to see if the graph is continuous. Use a table to find points and graph each piece. The shaded regions are coordinates that will not be included in the graph of f(). f() = f() = -5 f() = y Graph each value. There will be an open circle at (, -6) and a closed circle at (, -10) to clearly show the value of the function at =. No open circle is needed at = -1 because the function is connected at that point by the two pieces of the function. 83 Saon Algebra 1

6 Eample 5 Application: Ticket Prices At an amusement park, children under three years of age are free. Ages 3 to 1 pay \$0. Everyone older than 1 pays \$30. Write the function that represents this information, and graph the function. First, identify the intervals for the independent variables. Let represent age in years. under three < 3 ages 3 to older than 1 > 1 Then, write the function rule. f() is the price of the ticket. 0 if < 3 f() = 0 if if > 1 Graph the function. APPENDIX LESSONS y O Lesson Practice Evaluate each step function for the values given. (E 1) a. f() = - if 1 for = -3 and = 10. if > 1 b. f() = 6 if < 9 for = 8 and = if 9 Evaluate each piecewise function for the values given. (E ) c. f() = 3 if < 0 for = and = if 0 d. f() = 3 if - 1 for =-5 and = if > -1 Graph each step function. (E 3) e. f() = 7 if < 5 if 5 3 if < -3 f. f() = 0 if -3 < 3-3 if 3 Appendi Lesson 835

7 Graph each piecewise function. (E ) g. f() = if < - + if - 3 if 1 h. f() = 6-3 if 1 < < - if (E 5) (E 5) i. Allowance A child less than 5 years old does not get an allowance. Starting at 5 years old, he gets 3 times his age per month. At 10 years, the rate increases to times his age per month. Write the function that represents this information, and graph the function. j. Rides At an amusement park, there are 15 rides that have no height requirement. If a person is at least feet tall, there are a total of 0 available rides. To be granted access to all rides in the park, a person must be at least.5 feet tall. Write a function that represents the number of available rides based on a person s height. Sketch a graph of that function. 836 Saon Algebra 1

8 APPENDIX LESSON 3 New Concepts Understanding Vectors APPENDIX LESSONS To say that you biked 3 miles tells how far you went, but to say that you biked 3 miles north tells how far you went and in what direction. A vector is a quantity with both magnitude (size) and direction. 3 miles north can be represented by a vector. A vector is represented by a line segment with a half-arrow that indicates direction, not a continuation of the segment infinitely as in a ray. This vector can be named MN or ν. v N Terminal Point M Initial Point Component form is also used to name a vector. It identifies the horizontal change () and vertical change (y) from the initial point to the terminal point in the form,, y. The horizontal change is positive to the right and negative to the left. The vertical change is positive up and negative down. Eample 1 Writing Vectors in Component Form Write each vector in component form. a. AB The horizontal change from A to B is 5. The vertical change from A to B is -. The component form of AB is 5, -. b. RS with R(-1, ) and S(6, 3). A B RS = - 1, y - y 1 RS = 6 - (-1), 3 - RS = 7, - 1 Horizontal change is - 1 and vertical change is y - y 1. Substitute the coordinates of the given points. Subtract the initial point s coordinates from the terminal point s coordinates. Simplify. The length of the vector is called its magnitude. It is written EF or ν. Derived from the distance formula, the formula for the length of a vector is a, b = a + b. Appendi Lesson 3 837

9 Eample Finding the Magnitude of a Vector Find the magnitude of the vector to the nearest tenth. -3, 5 a, b = a + b -3, 5 = (-3) + 5 = = The direction of a vector is the angle formed by it and a horizontal line. Begin at the positive -ais and measure counterclockwise to the vector. Then, use inverse trigonometric functions to find the angle. Eample 3 Finding the Direction of a Vector Find the direction of the vector to the nearest degree. A boat s velocity is given by the vector, 8. First, draw the vector on a coordinate plane. Use the origin as the initial point. 8 6 y F The horizontal change and the vertical change make right triangle FGH. G is the angle formed by the vector and the -ais. tan G = 8 _. So m G = tan -1 ( 8 _ ) 63. O G 8 H 6 8 Equal vectors are two vectors that have the same magnitude and direction. They do not have to have the same initial and terminal points. Parallel vectors may have different magnitudes, but have the same or opposite direction. Equal vectors are always parallel vectors. Eample Identifying Equal and Parallel Vectors a. Identify equal vectors. A B D F Equal vectors have the same magnitude and direction. AB = GH b. Identify parallel vectors. C E G H Parallel vectors have the same or opposite directions. AB GH and CD EF 838 Saon Algebra 1

10 Lesson Practice Write each vector in component form. (E 1) a. Write the vector in component form. B APPENDIX LESSONS b. Write the vector in component form. A C D Write each vector in component form. (E 1) c. PQ with P(, -6) and Q(1, -1). d. JK with J(3, 7) and K(8, -). Find the magnitude of each vector to the nearest tenth. e., -9 f. 6, 1 g. Water Current The river s current is given by the vector 3, 1. Find the direction of the vector to the nearest degree. i. Identify the equal vectors. j. Identify the parallel vectors. (E 3) (E ) (E ) M N E F D G K L Appendi Lesson 3 839

11 APPENDIX LESSON Using Variation and Standard Deviation to Analyze Data New Concepts {1,, 3,, 5, 6, 7, 8, 9} The mean of the data set is 5. Standard deviation measures how the data is spread from the mean. It is a measure of variation. The variance, represented by the symbol σ, is the average of the squared differences from the mean. To calculate the variance. Find the mean of the data. Subtract each value from the mean and square the result. Find the average of the squared results. The standard deviation, represented by the symbol σ, is the square root of the variance. Eample 1 Finding the Standard Deviation Ten students are asked how many CDs they own. Their responses are recorded in the data set. {10, 15, 13, 0, 8, 11, 10, 9, 1, 16} Find the standard deviation of the data. First, find the mean of the data by adding the data and dividing by = _ = 1.6 Net, subtract each value in the data set from the mean and square the result. Value () Difference (1.6 - ) Difference Squared (1.6 - ) Now, find the average of the differences squared = _ = 1.. Finally, take the square root to get the standard deviation Saon Algebra 1

12 The standard deviation describes the spread of the data. When the standard deviation is low, the data tends to be close to the measure of central tendency, or mean. When the standard deviation is high, the data is more spread out. An outlier is a number that is much greater or much less than the other values in the data set. Outliers have a great impact on the mean and standard deviation and can cause them to misrepresent the data set. One way to determine whether a value is an outlier is to see if it is more than 3 standard deviations from the mean. APPENDIX LESSONS Eample Eamining Outliers The population of southern states is shown. Find the mean and standard deviation of the data. Identify any outliers, and if one is found, eplain how it affects the mean. State TX OK AK LA MS AL FL GA NC SC VA WV MD DE KY TN Population in millions First, find the mean of the state populations Net, subtract each value in the data set from the mean and square the result. Now, find the average of the difference squared, _ , and take the square root to get the 16 standard deviation An outlier would be outside the 3 standard deviations from the mean, 6.7 ± 3(5.69). Negative population would not make sense, so check to see if any state has a greater population than (5.69) = 3.77 million. There are no outliers in this data because there are no populations larger than 3.77 million. All data is within 3 standard deviations of the mean. Population Difference (6.7 - ) Difference Squared (6.7 - ) Appendi Lesson 81

13 Some data is said to be normally distributed. The shape of the data looks like a bell, so it is often called a bell-shaped curve. The mean is at the center. 68% 95% 99.7% -3SD -SD -1SD mean +1SD +SD +3SD As the graph indicates, 68% of the data falls within one standard deviation of the mean. 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean. Eample 3 Using the Normal Distribution The ages of people at a park are normally distributed. The mean is 18 years and the standard deviation is 6 years. Between what two ages do 95% of the ages fall? SOULTION Because it is a normal distribution, 95% of the data falls within standard deviations of the mean. 18 ± (6) = 18 ± 1 95% of the ages fall between 6 and 30. Lesson Practice Find the standard deviation of the data. (E 1) a. An ATM machine records the values of the withdrawals made in one day. {0, 100, 0, 00, 0, 0, 100, 0, 80, 0, 0, 0, 100, 0, 100} b. A group of students is asked how many movies they watched in the last month. Their responses are recorded in the data set. {, 10, 6, 8,, 5, 30,,, 3, 1} Find the mean and standard deviation of the data. Identify any outliers, and if one is found, eplain how it affects the mean. c. Twelve students are asked how many books they read last year. Their (E ) responses are recorded in the data set. {1, 15, 30, 1, 13, 9, 10, 10, 11, 1, 1, 8} d. A teacher records the scores on a test. {90, 95, 90, 85, 80, 80, 90, 0, 95, 90, 85, 90, 95, 80, 100} (E 1) e. Test Results The results on a test are normally distributed with a mean of 85 and a standard deviation of 5. Between what two scores are 68% of the scores? f. Salaries The salaries of educators are normally distributed with a mean of \$35,000 and a standard deviation of \$10,000. Between what two scores are 99.7% of the salaries? (E 3) (E 3) 8 Saon Algebra 1

14 APPENDIX LESSON 5 New Concepts Evaluating Epressions with Technology APPENDIX LESSONS A graphing calculator can help you evaluate epressions for several values of the variable. Eample 1 Using a Graphing Calculator to Evaluate Epressions a. Use a graphing calculator to evaluate for = 50, 150, 50, 350 and 50. Press. Enter for Y 1. Press to set the table values. Enter the first value of, 50, for TblStart. For ΔTbl, enter the difference in the -values, 100. Press. In the first column, you will see the values of. The second column shows the value of the epression for each value of. b. Use the table to find the value of the epression when = 550. Find 550 in the first column, and look across from it. The value is 908,599. c. Use the table to find the value of if the epression is equal to 368,199. Find 368,199 in the second column. It is net to = 350. Appendi Lesson 5 83

15 A spreadsheet can also be used to evaluate epressions. Eample Using a Spreadsheet to Evaluate an Epression Evaluate for = 11, 13, 15, 17, and 19. Enter 11, 13, 15, 17, and 19 in the first column, A1 to A5. Enter the epression in cell B1, using A1 instead of a variable. The epression should be typed as = 5 A1^ - 1 A1-16 After pressing enter, the value of the epression appears in the cell. Copy the epression by clicking on the bottom right corner of B1. Hold the mouse while you drag to highlight the cells B through B5. 8 Saon Algebra 1

16 Column B will be filled with the values of the epression. APPENDIX LESSONS The spreadsheet will automatically evaluate the epression using the corresponding value of in column A. b. Use the spreadsheet to find the value of the epression when =. Enter in the first column, and copy the epression into the corresponding row of column B. The value is 10. Lesson Practice Use a graphing calculator to evaluate for the given values. (E 1) a. = b. = c. = 6 Use a graphing calculator to evaluate for the given values. (E 1) d. = 8 e. = 78 f. = 108 Use a spreadsheet to evaluate for the given values. (E ) g. = 6 h. = 1 i. = 18 Use a spreadsheet to evaluate for the given values. (E ) j. = k. = 9 l. = 1 Appendi Lesson 5 85

17 Skills Bank Compare and Order Rational Numbers Skills Bank Lesson 1 A rational number is a number that can be written as a ratio of two integers. Eample 1 Comparing Rational Numbers Compare 7_ 10 and 5_ 1. Write <, >, or =. Method 1: Multiply to find a common denominator = 10 Multiply the denominators _ 1 _ _ 10 > _ 50 10, so _ 7 10 > _ 5 1 Write fractions with a common denominator. Method : Find the least common denominator (LCD). 7_ 10 _ 6 5_ 6 1 _ 5 Write fractions using the LCD of _ 60 > _ 5 60, so 7_ 10 > _ 5 1 Eample Ordering Rational Numbers Order the numbers - 5_,.75, -3, 1_, - 1_ from least to greatest. 5 Write each fraction as a decimal. Graph the numbers on a number line. - 5 _ = -1.5, 1 _ =.5, - 1_ 1_ = Read the numbers from left to right: -3, - 1_ 5, - 5_, 1_,.75. The numbers are in order from least to greatest Skills Bank Practice Compare. Use >, <, or =. a. _ 5 7_ 8 1 b. 3_ 11 3_ 10 c. - 3_ 7 - _ 5 Order from least to greatest. d. -, _ 7 8, 0.8,.1, 1 _ 1 3 e. 0.7, -1, -_ 5, _ 3, -.3, - 9_ 86 Saon Algebra 1

18 Decimal Operations Skills Bank Lesson To add or subtract decimals, align the numbers at their decimal points. Then perform the operation the same way as adding or subtracting whole numbers. Eample 1 Adding and Subtracting Decimals a. Find the sum of.5 and b. Find the difference of and Write the problem vertically Align the decimal points To multiply decimals, multiply first. Then place the decimal so that the product has the same number of decimal places as the total number of decimal places in the two factors. To divide decimals, multiply the divisor and the dividend by a power of 10 in order to make the divisor a natural number. Then divide as with whole numbers. SKILLS BANK Eample Multiplying and Dividing Decimals a. Find the product of 1.5 and Write the problem vertically Since the factors have a total of 3 decimal places, there should be 3 decimal places in the product. b. Find the quotient of 3.7 and Multiply the divisor and dividend by 10 so the divisor is a natural number. Skills Bank Practice a. Find the sum of 19.3 and.5. b. Find the difference of and c. Find the product of.8 and d. Find the quotient of and Simplify. e f g h Skills Bank 87

19 Fraction Operations Skills Bank Lesson 3 To add or subtract fractions with unlike denominators, first find a common denominator. Eample 1 Adding and Subtracting Fractions a. Add 5_ 6 and 3_ 8. Method 1: Multiply to find a common denominator. 6 8 = 8 5 6( 8 8) + _ 3 _ 8( 6 Multiply by fractions 6) equal to 1. = _ _ 18 Add. 8 _ = 58 8 = _ 9 or 1 _ 5 Simplify. Method : Find the lowest common denominator (LCD). Multiples of 6: 6, 1, 18,, Multiples of 8: 8, 16,, The LCD is. 5_ 6( _ ) + 3 _ 8( 3 _ = _ 0 + _ 9 = _ 9 or 1 _ 5 Multiply by fractions 3) equal to 1. Add. b. Subtract 1_ from 7_ 8. 7_ 8-1 _ ( _ = _ _ 8 = _ 3 8 ) Write equivalent fractions using a denominator of 8. Eample a. Multiply _ 3 5_ 6. Multiplying and Dividing Fractions Multiply the numerators and denominators. Then simplify if possible. _ 3 _ 5 6 = _ = _ 5 9 b. Divide 5_ 3_ 5. Write the reciprocal of 3_ 5 and then multiply. 5_ 5 _ 3 = 5 _ 1 = _ 5 1 or _ 1 1 Multiply by 5_ 3. Skills Bank Practice Add, subtract, multiply, or divide. Simplify if possible. a. 7_ 1 + _ 3 8 b. 9_ 10 - _ 5 c. _ 5 9 _ 3 d. _ 16 _ 9 8 e. _ _ _ f. _ _ Saon Algebra 1

20 Divisibility Skills Bank Lesson A number is divisible by another number if the quotient is a whole number without a remainder. Eample 1 Divisibility Rules A number is divisible by if its last digit is even (0,,, 6, or 8). 3 if the sum of its digits is divisible by 3. if its last two digits are divisible by. 5 if its last digit is 0 or 5. 6 if it is divisible by both and 3. 9 if the sum of its digits is divisible by if its last digit is 0. Determining the Divisibility of Numbers SKILLS BANK a. Determine whether is divisible by, 3,, 5, and 6. The last digit is even. divisible 3 The sum of the digits is divisible by 3. + = 6 divisible The last two digits are divisible by. divisible 5 The last digit is not 0 or 5. not divisible 6 The number is divisible by both and 3. divisible is divisible by, 3,, and 6. b. Determine whether both the numerator and denominator in the fraction _ are divisible by, 3,, and 5. The last digit is even both divisible 3 The sum of the digits in 16 is not divisible by = = 6 not both divisible The last two digits are divisible by both divisible 5 The last digit in 16 is not 0 or not both divisible Both the numerator and denominator in 16_ 60 are divisible by and. Skills Bank Practice Determine whether each number is divisible by, 3,, 5, 6, 9, and 10. a. 90 b. 830 c. 10 d. Determine whether both the numerator and denominator in the fraction _ 1 5 are divisible by, 3,, 5, and 6. Skills Bank 89

21 Equivalent Decimals, Fractions, and Percents Skills Bank Lesson 5 Numbers can be written as decimals, fractions, and percents. The table shows common fractions and their equivalent decimals and percents. Eample 1 Fraction Decimal Percent 1_ 0.5 5% 1_ 3_ 1_ 5 1_ % % 0. 0% % Writing Fractions As Decimals and Percents Find the equivalent decimal and percent for each fraction. a. 7_ Find the equivalent decimal. Divide the numerator by the denominator = 70% Find the equivalent percent. Move the decimal two places to the right. 7_ 10 is equivalent to 0.7 and 70%. _ b. 9 9 = 0. Divide the numerator by the denominator. 0. =. % Move the decimal two places to the right. _ 9 is equivalent to 0. and. %. Skills Bank Practice Write the equivalent decimal and percent for each fraction. a. _ 3 b. _ 5 10 c. 3 _ 8 e. 7 _ 9 d. 5_ 11 f. 3 _ 850 Saon Algebra 1

22 Repeating Decimals and Equivalent Fractions Skills Bank Lesson 6 A terminating decimal, such as 0.75, has a finite number of decimal places. A repeating decimal, such as and , has one or more digits after the decimal point repeating indefinitely. A repeating decimal can be written with three dots or a bar over the digit or digits that repeat, such as 0. 3 and Eample 1 Writing an Equivalent Fraction for a Terminating Decimal Write each decimal as a fraction in simplest form. a = _ _ 100 = _ 7 0 The decimal is in the hundredths place, so use 100 as the denominator. Simplify. SKILLS BANK b = 1 9 _ 10 The decimal is in the tenths place, so use 10 as the denominator. Eample Writing an Equivalent Fraction for a Repeating Decimal Write as a fraction. To eliminate the repeating decimal, subtract the same repeating decimal. n = Let n represent the fraction equivalent to n = Since digits repeat, multiply both sides of the equation by 10 or n = Subtract the original equation. 99n = 7 Combine like terms. n = _ 7 99 = _ is equivalent to 3_ 11. Skills Bank Practice Divide both sides by 99 and simplify. Write an equivalent fraction in simplest form for each decimal. a b c d e. 0.8 f. 1.5 g h. 0. Skills Bank 851

23 Equivalent Fractions Skills Bank Lesson 7 Fractions that represent the same amount or part are called equivalent fractions. _ 1_ Eample 1 Finding Equivalent Fractions For each fraction, write two equivalent fractions. a. _ 3 _ b Choose any whole number. Multiply the numerator and the denominator by that number. 3_ = _ = _ 9 1 3_ = _ = _ _ is equivalent to 9_ 1 and 15_ 0. Eample Simplify. _ 8 Find a number that is a factor of the numerator and the denominator. Divide both by that number. 36_ 0 = _ 36 0 = _ _ 0 = _ 36 0 = _ _ 0 is equivalent to 9_ 10 and 18_ 0. Writing Fractions in Simplest Form Using the GCF Find the greatest common factor (GCF) of and 8. The GCF is. _ 8 = _ 8 = _ 1 Divide the numerator and denominator by. Skills Bank Practice For each fraction, write two equivalent fractions. a. _ 3 b. _ 1 c. _ Simplify. e. _ 1 h. _ 8 60 f. _ i. 90_ 360 g. d. _ _ Saon Algebra 1

25 Greatest Common Factor (GCF) Skills Bank Lesson 9 The greatest common factor, or GCF, is the largest factor two or more given numbers have in common. For eample, and 5 are common factors of 10 and 0, but 5 is the greatest common factor. One way to find the GCF is to make a list of factors and choose the greatest factor that appears in each list. Another way is to divide by prime factors. Eample 1 Finding the GCF a. Find the GCF of and 60. : 1,, 3,, 6, 8, 1, List the factors of each number. 60: 1,, 3,, 5, 6, 10, 1, 15, 0, 30, 60 Find the greatest common factor., 3,, 6, and 1 are common factors. The GCF of and 60 is 1. b. Find the GCF of 5 and Divide both numbers by the same prime factor Keep dividing until there is no prime factor that divides into both numbers without a remainder or 3 = 18 The GCF of 5 and 7 is 18. Eample Using the GCF to Simplify Fractions a. Write 1_ 8 in simplest form. Divide 1 and 8 by the GCF, 7. 1_ 8 = _ = _ 3 b. Write 1 9_ 1 in simplest form. Divide 9 and 1 by the GCF, 3. 9_ 1 = _ = _ 3 1 _ 9 1 = 1 _ 3 Skills Bank Practice Find the GCF. a. 7 and 60 b. 5 and 89 c. 1 and 56 d. 10 and 960 e. 3, 6, and 1 f. 7, 1, and 9 g.,, and 0 h. 0, 5, and 80 Write each fraction in simplest form. i. 8_ j. _ 15 k. _ l. _ m. 5 _ Saon Algebra 1

26 Least Common Multiple (LCM) and Least Common Denominator (LCD) Skills Bank Lesson 10 The least common multiple, or LCM, is the smallest whole number, other than zero, that is a multiple of two or more given numbers. Eample 1 Finding the LCM a. Find the LCM of 6 and 10. List the multiples of each number. Multiples of 6: 6, 1, 18,, 30, 36,, 8, 5, 60, Multiples of 10: 10, 0, 30, 0, 50, 60, 30 and 60 are common multiples. Find the common multiples that are in both lists. The LCM of 6 and 10 is 30. Find the least common multiple. SKILLS BANK b. Find the LCM of 1 and Divide both numbers by the same prime factor. Keep dividing until there is no prime factor that divides into both numbers without a remainder. 3 3 or 3 = 36. The LCM of 1 and 18 is 36. The least common denominator, or LCD, is the least common multiple of two or more denominators. Eample Finding the LCD and Writing Equivalent Fractions Find the LCD of 3_ 8 and 5_ 1. Use the LCD to write equivalent fractions. The LCM of 8 and 1 is, so is the LCD. 3_ 8 = _ = _ 9 5_ 1 = _ 5 1 = _ 10 3_ 8 and 5_ 1 are equivalent to 9_ and 10_. Write an equivalent fraction using a denominator of. Write an equivalent fraction using a denominator of. Skills Bank Practice Find the LCM. a. 9 and 15 b. 0 and 5 c. and 8 d. 1 and 1 e. 5, 50, and 100 f. 8, 16, and 8 g., 3, and 0 h. Use the LCD to write equivalent fractions for 1_ and 7_ 15. Skills Bank 855

27 Mental Math Skills Bank Lesson 11 Mental math means to find an eact answer quickly in your head. Mental math strategies use number properties. Eample 1 Using Properties to Add or Multiply Whole Numbers a. Find the sum of Look for sums that are multiples of 10. = Use the Commutative Property. = (3 + 57) + (3 + 8) Use the Associative Property. = Add. = 10 b. Find the product of 5. 5 Look for products that are multiples of 10. = 5 Use the Commutative Property. = ( 5) Use the Associative Property. = 10 Multiply. = 0 c. Find the product of = 8 (0 + 7) Break apart 7 into = (8 0) + (8 7) Use the Distributive Property. = Multiply. = 376 Add. Skills Bank Practice Find each sum or product. a b c. 58 d e. 3 7 f. 6 5 g h Saon Algebra 1

28 Prime and Composite Numbers and Prime Factorization Skills Bank Lesson 1 A prime number is a number that has eactly two factors, 1 and itself. For eample, 5 is a prime number because its only factors are 1 and 5. A composite number has more than two factors. For eample, 8 is a composite number because its factors are 1,,, and 8. The number 1 is neither prime nor composite. Eample 1 Determining Whether a Number is Prime or Composite Determine whether each number is prime or composite. a. 18 b. 13 1,, 3, 6, 9, 18 List the factors. 1, 13 List the factors. 18 is a composite number. 13 is a prime number. SKILLS BANK Every composite number can be written as the product of two or more prime numbers. This product is called the prime factorization of a number. Eample 36 Using a Factor Tree to Find the Prime Factorization Choose any two factors of 36. Continue to factor until each branch ends in a prime number. 3 The prime factorization of 36 is 3 3 or 3. Skills Bank Practice Determine whether each number is prime or composite. a. 17 b. 15 c. 3 d. 9 Find the prime factorization of each number. e. 7 f. 8 g. 3 h. i. 76 j. 3 k. 5 l. 5 Skills Bank 857

29 Classify Angles and Triangles Skills Bank Lesson 13 You can classify an angle by its measure. Classification of Angles An acute angle measures less than 90. A right angle measures eactly 90. An obtuse angle measures more than 90 and less than 180. A straight angle measures eactly 180. You can classify a triangle by its angle measures. Classification of Triangles by Angle Measures An acute triangle has three acute angles. An equiangular triangle has three congruent acute angles. A right triangle has one right angle. An obtuse triangle has one obtuse angle. You can also classify a triangle by its side lengths. Classification of Triangles by Side Lengths An equilateral triangle has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle has no congruent sides. 858 Saon Algebra 1

30 Eample 1 Classifying Angles Classify each angle according to its measure. a. b. This is a straight angle, because the figure is a line and the angle measures 180. c. This is an acute angle because the angle measure is less than 90. This is an obtuse angle, because the angle measure is greater than 90 but less than 180. d. This is a right angle because the angle measure is equal to 90. SKILLS BANK Eample Classifying Triangles Classify each triangle according to its angle measures and side lengths. a. > 90 b. The figure has one obtuse angle and at least congruent sides. So, this is an obtuse isosceles triangle. The figure has one right angle and no congruent sides. So, this is a right scalene triangle. Skills Bank Practice Classify each angle according to its measure. a. b. c. Classify each triangle according to its angle measures and side lengths. d. e. f. Skills Bank 859

31 Classify Quadrilaterals Skills Bank Lesson 1 A quadrilateral is a two-dimensional figure with four sides and four angles. The table shows five special quadrilaterals and their properties. Parallelogram Opposite sides are parallel and congruent. Opposite angles are congruent. Rectangle Parallelogram with four right angles Rhombus Parallelogram with four congruent sides Square Rectangle with four congruent sides Trapezoid Quadrilateral with eactly two parallel sides May have two right angles Eample 1 Classifying Quadrilaterals a. Identify which statement is always true. A trapezoid is also a parallelogram. A square is also a rhombus. A parallelogram is also a rectangle. A rectangle is also a square. A square is also a rhombus is true, because a square is a parallelogram with four congruent sides. b. Identify which statement is not always true. A quadrilateral has sides. A quadrilateral has angles. A quadrilateral has straight sides. A quadrilateral has right angles. A quadrilateral does not always have right angles. Skills Bank Practice Complete each statement. a. A square is also a. b. A rhombus is sometimes a. c. All trapezoids are also. d. A is any two-dimensional figure with four straight sides and four angles. 860 Saon Algebra 1

32 Complementary and Supplementary Angles Skills Bank Lesson 15 Two angles with measures that have a sum of 90 are complementary angles. Two angles with measures that have a sum of 180 are supplementary angles. Eample 1 Identifying Complementary and Supplementary Angles a. Are A and B complementary or supplementary angles? b. Are K and L complementary or supplementary angles? 3 A B 56 m A + m B = = 90 A and B are complementary. 35 K L15 m K + m L = = 160 K and L are neither complementary nor supplementary. SKILLS BANK Eample Finding Missing Angle Measures a. M and N are supplementary angles. Find m N. b. E and F are complementary angles. Find m F. M m N = 180 m N = m N = 1 N 67 E 67 + m F = 90 m F = m F = 3 F Skills Bank Practice Classify each pair of angles as complementary or supplementary. Then find the missing angle measure. a. b. c d. D and E are complementary angles. If the measure of D is 50, what is the measure of E? e. W and T are supplementary angles. If the measure of W is 50, what is the measure of T? Skills Bank 861

33 Congruence Skills Bank Lesson 16 Congruent segments are segments that have the same length. Congruent angles are angles that have the same measure. Figures are congruent if all of their corresponding angles and sides are congruent. Hint The symbol for congruent is. A B Statement: ΔABC ΔDEF C D E Congruent Triangles F Corresponding Angles A D B E C F Corresponding Sides AB DE BC EF AC DF _ AB DE = _ BC EF = _ AC DF In a congruence statement, the order of the letters shows which angles and sides are congruent. Eample 1 Identifying the Corresponding Angles and Sides Find the congruent angles and sides. Then write a congruence statement. D I E H F G DE IH EF HG DF IG D corresponds to I. E corresponds to H. F corresponds to G. DE corresponds to IH. EF corresponds to HG. DF corresponds to IG. E D G 13 F DEF IHG 13 I H Skills Bank Practice Write a congruence statement for each pair of figures. a. b. L 0 K 10 J P T Q 10 0 B T J L D 8 7 K P 8 Y 86 Saon Algebra 1

34 Estimate the Perimeter and Area of Figures Skills Bank Lesson 17 Perimeter is the distance around a figure. The perimeter of a polygon is the sum of its side lengths. The area of a figure is the amount of surface it covers. Perimeter and Circumference Formulas Rectangle P = l + w or P = (l + w) Circle C = πr or C = πd Rectangle A = lw Circle A = πr Area Formulas Eample 1 Estimating Perimeter a. Estimate the perimeter of the figure. 8 feet b. Estimate the perimeter of the trapezoid. SKILLS BANK 8 feet Estimate the length of the top, sides, and bottom of the figure. right and left: 8 feet bottom: 8 feet top: 8 feet P (8) The perimeter is about 3 feet. Find the length of the top, side, and bottom of the trapezoid. top: units left: units bottom: 9 units Estimate the length of the diagonal line. diagonal line: 5 units P The perimeter is about units. Eample Estimating Area Estimate the area of the circle. Estimate the area by counting the squares. 1 full squares almost full squares 8 quarter full squares: 8 corners: 1 The area of the circle is about 19 units. Skills Bank Practice a. Estimate the perimeter of the figure. b. Estimate the area of the figure. Skills Bank 863

35 Nets Skills Bank Lesson 18 A net is a two-dimensional representation of a solid that can be folded to form a three-dimensional figure. A polygon is a closed plane figure formed by three or more line segments. Eample 1 Identifying a Net of a Three-Dimensional Figure Draw the net that represents the pizza bo. Eample Drawing a Three-Dimensional Figure from a Net Draw the three-dimensional figure that the net represents. Skills Bank Practice a. Draw the net that represents the can. b. Draw the three-dimensional figure that the net represents. 86 Saon Algebra 1

36 Parts of a Circle Skills Bank Lesson 19 A circle is the set of points in a plane that are a fied distance from a given point, the center. A chord is a line segment that connects points on the circle. JK and GH are chords. R A diameter is a chord that passes through the center of the circle. PR is a diameter. K J A A circle is named by its center. This is circle O. P O G B H A radius is a line segment that connects a point on the circle with the center of the circle. AO, BO, PO, and RO are radii. SKILLS BANK Eample 1 Naming Parts of a Circle Name the center, radii, diameters, and chords. C B D A E F G Center A Radii AB, AD, AE, AG Diameters DE Chords CF, CB, DE The plural of radius is radii. A diameter is also a chord. Skills Bank Practice Name the center, radii, diameters, and chords of each circle. a. T U b. W Z Y Z V X T Y X W Skills Bank 865

37 Perspective Drawing Skills Bank Lesson 0 You can see up to three sides of a figure when drawing a three-dimensional object. This means you have to visualize how a figure looks from other angles. Orthogonal views show how a figure looks from different perspectives. For figures constructed with cubes, the orthogonal views will be groups of squares. Eample 1 Drawing a Figure from Different Perspectives Draw the front, top, and side views of the figure. From the front and all side views, there appears to be 3 stacked cubes, with cubes on each side. The top view shows that cubes are on the sides of the bottom cube. Front Side Top Skills Bank Practice a. Draw the front, top, and side views of the figure. 866 Saon Algebra 1

38 Surface Area of Prisms and Pyramids Skills Bank Lesson 1 The surface area, S, is the total area of the two-dimensional surfaces that make up the figure. Prism Pyramid Eample 1 Formulas for Surface Area of Prisms and Pyramids B: area of base S = B + Ph P: perimeter of base h: height S = B + _ 1 B: area of base Pl P: perimeter of base l: slant height Finding the Surface Area of Prisms and Pyramids Find the surface area of each figure. a. b. 8 cm SKILLS BANK 9.6 m 7 cm 6 cm 6.8 m. m S = B + Ph = (. 6.8) + () (9.6) = (8.56) = = 68.3 m S = B + 1 _ Pl = (7 6) + _ 1 (6)(8) = + 10 = 16 cm Skills Bank Practice Find the surface area of each figure. a ft b. 9 m 9.6 m 8. m 15.3 ft 1. ft Skills Bank 867

39 Tessellations Skills Bank Lesson A tessellation is a pattern of plane figures that completely covers a plane with no gaps or overlays. Eample 1 Creating Tessellations Determine whether each figure can be used to create a tessellation. a. b. The rhombus can create a tessellation. There are no gaps or overlays. A pentagon cannot create a tessellation. There will be gaps and overlays. Gap Skills Bank Practice Determine whether each figure can be used to create a tessellation. If not, eplain why not. a. b. c. 868 Saon Algebra 1

40 Three-Dimensional Figures Skills Bank Lesson 3 A polyhedron is a three-dimensional figure that is made up of polygons which are called faces. A polyhedron has flat faces and straight edges. The faces intersect at edges. A verte is any point in which three or more edges intersect. Verte Face Some three dimensional figures are not polyhedra because they are not made up of polygons. Edge SKILLS BANK Eample 1 Determining Whether a Three-Dimensional Shape Is a Polyhedron Determine whether the three-dimensional shape is a polyhedron. If yes, tell how many faces, edges, and vertices the shape has. a. b. This shape is not a polyhedron. This shape is a polyhedron. There are 6 faces, 1 edges, and 8 vertices. Skills Bank Practice Determine whether the three-dimensional shape is a polyhedron. If yes, tell how many faces, edges, and vertices the shape has. a. b. Skills Bank 869

41 Transformations in the Coordinate Plane Skills Bank Lesson A transformation is a change in the size or position of a figure. If you transform the preimage, or original figure ABC, then the transformed figure, or image, is named A B C. Transformations include translations or slides, reflections or flips, and rotations or turns. Preimages and images are congruent for all transformations. Eample 1 Finding Transformations a. Reflect ABC across the y-ais. y O - A B - C b. Translate ABC 3 units left and units down. - y A O - C B The y-ais is a line of symmetry. Move each verte 3 units left and units down. y y A B B - O A A B A O B C C C C Skills Bank Practice a. Reflect ABC across the y-ais. y O - - A B C b. Give the coordinates for the points that describe the translation 5 units left. - O - y B A C D 870 Saon Algebra 1

42 Vertical Angles Skills Bank Lesson 5 When two lines intersect, the nonadjacent angles are called vertical angles. Vertical angles always have the same measure, so they are congruent angles. Eample 1 Finding the Measure of Vertical Angles Find m WVY, m YVZ, and m ZVX, where m XVW = 70. a. W Y V X Z m XVW + m WVY = 180 XVW and WVY are supplementary m WVY = 180 Substitute. m WVY = 110 m YVZ = m XVW Vertical angles have the same measure. m YVZ = 70 m ZVX = m WVY Vertical angles have the same measure. m ZVX = 110 SKILLS BANK Skills Bank Practice a. Name the two pairs of vertical angles. b. Find m ABQ, m ABC, and m CBR. B C A C E B A D 100 Q R c. Find m EFG, m GFH, and m HFI, where m EFI = 0. E F G d. Find m BAC, m DAE, and m EAB, where m CAD = 10. B C I H A E D Skills Bank 871

43 Volume of Prisms and Cylinders Skills Bank Lesson 6 The volume is the amount of space a solid occupies. Volume is measured in cubic units. To estimate volume, imagine unit cubes filling a figure. Prism Cylinder Eample 1 Formulas for the Volume of Prisms and Cylinders V = Bh B: area of base h: height of prism V = πr h r: radius h: height Finding the Volume of Prisms and Cylinders Find the volume of each figure. Use 3.1 for π. Round to the nearest hundredth. a. b. 3 m 6 m 3 m m m V = Bh = ( ) 3 = 8 3 = m 3 V = πr h 3.1 (3 ) 6 = 3.1 (9) 6 = m 3 Skills Bank Practice Find the volume of each figure. Use 3.1 for π. Round to the nearest hundredth. a. m 11 m m b. 7 m 8 m 87 Saon Algebra 1

44 Making Bar and Line Graphs Skills Bank Lesson 7 In a bar graph, bars are used to represent and compare data. The bars can be horizontal or vertical. In a line graph, points that represent data values are connected using segments. Line graphs often show a change in data over time. Eample 1 Making a Bar or Line Graph a. Use the data to make a bar graph. Favorite Activities Activity Golf Movie Amusement Park Number of People Find the appropriate scale. Use the data to determine the length of the bars. Title the graph and label the aes. Number of People Favorite Activities Golf Movie Amusement Park Activities b. Use the data to make a line graph. U.S. Households with a Computer Year Percent 8% 15% % 36% Find the appropriate scale. Make a point for each data value. Connect the points with line segments. Title the graph and label the aes. U.S. Households with a Computer Percent of People Year SKILLS BANK Skills Bank Practice a. Use the data to make a bar graph. Favorite Subject in School Subject Art PE English Math Science Number of Students b. Use the data to make a line graph. Average High Temperature in Palm Beach, Florida Month March April May June Temperature Skills Bank 873

45 Making Circle Graphs Skills Bank Lesson 8 A circle graph compares part of the data set to the whole set of data. In a circle graph, data is displayed as sections of a circle. Each section has an angle at the center. The total measure of the angles at the center of the circle is 360. The entire circle represents all of the data. Eample 1 Making a Circle Graph Use the data in the table to make a circle graph. Step 1: Find the angle measures by multiplying each percent by 360. Cheese: 0% 360 = = 1 Supreme: 10% 360 = = 36 Pepperoni: 50% 360 = = 180 Step : Use a compass to draw a circle. Favorite Pizza Toppings Toppings Students in Class Cheese 0% Supreme 10% Pepperoni 50% Step 3: Draw a circle and radius with a compass and straightedge. Then use a protractor to draw the first angle, 1. Then draw the second and third angles, 36 and 180. Favorite Pizza Toppings Supreme 10% Cheese 0% 180 Pepperoni 50% Step : Label the graph and give it a title. Skills Bank Practice a. In a survey, people were asked what kind of pet they owned. The table shows the results of the survey. Use the table to make a circle graph. Pet Owners Dog 36% Cat 5% Fish 15% No pets % % No Pet 15% Fish 5% Cat 87 Saon Algebra 1

46 Making Line Plots Skills Bank Lesson 9 How often a data value occurs in a data set is called its frequency. A line plot is a graph made up of a number line and columns of s. Other markers can be used to show a frequency. A cluster is a group of data values that are grouped together. Eample 1 Making a Line Plot a. In a survey, 8 people waiting at a bus stop were asked their age. Their ages are shown in the frequency table below. Make a line plot. Identify any gaps or clusters in the data set. Age Frequency Age Frequency SKILLS BANK Draw a number line that includes the minimum and maimum age. Use an to represent each person. Title the graph and the ais. There is a gap between 16 and 19. There is a cluster between 19 and 5. Ages of Bus Riders X X X X X X X X X X X X X X X X X X X X X X X X X X X X Age Skills Bank Practice a. Make a line plot of the lowest temperatures for the last two weeks. 55 F, 60 F, 65 F, 65 F, 65 F, 60 F, 60 F, 70 F, 65 F, 65 F, 70 F, 65 F, 65 F, 60 F b. What are the minimum and maimum temperatures that were recorded? c. What was the most common temperature? Skills Bank 875

47 Venn Diagrams Skills Bank Lesson 30 A Venn diagram shows the relationship between sets. Eample 1 Making a Venn Diagram 167 people taste tested two new brands of cereal. 7 people did not like either brand, 100 people liked Brand A, and 110 people liked Brand B. How many people only liked Brand A? Make a Venn diagram to represent the data. Draw and label two intersecting circles to show the set of people that liked Brand A and Brand B. 7 Cereal Taste Testing Brand A 50 Brand B There must be people that liked both brands of cereal, because = 17, and only 167 people taste tested the cereal. The overlap is = 50. This means 50 people were counted twice because 50 people liked both Brand A and Brand B. Out of 100 people who liked Brand A, 50 of them also liked Brand B. So, 50 people liked only Brand A. Skills Bank Practice Out of a group of 133 people, 55 people carpool to work, 67 take the bus to work, and 30 do not carpool or take the bus to work. Make a Venn diagram. Then use the Venn diagram to find how many people use both a carpool and a bus. 876 Saon Algebra 1

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