Math388 Lecture Notes PDF

Chapter 3 1 MULTIPLES ‫ﻣﺿﺎﻋﻔﺎﺕ‬ To find Multiples of a number: Example: To find Multiples of 8 We M...

Chapter 3 1 MULTIPLES ‫ﻣﺿﺎﻋﻔﺎﺕ‬ To find Multiples of a number: Example: To find Multiples of 8 We Multiply 8 by 1, 2, 3, 4, 5,.. We Multiply the number by 1, 2, 3, 4, 5,... 8x1 = 8 Example: Find six multiples of 8 8x2=16 Solution: 8, 16, 24, 32, 40, 48 8x3=24 8x4=32, Example: Find five multiples of 12 8x5=40, … Solution: 12, 24, 36, 48, 60 Practice: Find five multiples of 15 Practice: Find seven multiples of 9 Solution: ……………….………..……….. Solution: …………………….………………….. 2 1  DIVISIBILITY ‫ﻗﺎﺑﻠﻳﺔ ﺍﻟﻘﺳﻣﺔ‬ Divisibility: B is said to be divisible A number is divisible by by A if B 𝐴 is an integer.  2 if it is even (the one digit is 0, 2, 4, 6 or 8).  10 if the ones digit is 0. Example: Is 55 divisible by 5?  9 if the sum of its digits is divisible by 9. Solution: Yes since 55 5 11  5 if its one digit is 0 or 5. Example: Is 102 divisible by 5?  3 if the sum of all its digits is a multiple of 3. Solution: No since 102 5 20.4  6 if it is divisible by 2 and 3 at the same time. Example: Practice: Is 237 divisible by 12 Determine whether 655122 is divisible by 2,3,5,6,9 or10 Solution: ……………. Solution: 2 Yes, 3 Yes, 5 No, 6 Yes, 9 No, 10 No {2, 3, 6} Practice: Is 476 divisible by 17? Practice: Solution: ……………. Determine whether 43425 is divisible by 2, 3, 5, 6, 9 or 10 Solution: A) {2, 5, 6} B) {2, 3, 6} C) {3, 5, 9} D) {3, 3 6, 10} FACTORS Factors: A number b is a factor of a if a is divisible by b. Example: Find all Example: Find all Example: Find all Practice: Find all the factors of 10. the factors of 18. the factors of 24. the factors of 36. Solution: Solution: Solution: Solution: 10 = 1 10 18 = 1 18 24 = 1 24 10 = 2 5 18 = 2 9 24 = 2 12 Factors: 18 = 3 6 24 = 3 8 1, 2, 5, 10 Factors: 24 = 4 6 1, 2, 3, 6, 9, 18 Factors: 1, 2, 3, 4, 6, 8, 12, 24 4 2  Prime and Composite Numbers Prime Numbers: A number is said to be prime if its only factors are 1 and the number itself. Composite Numbers: A number that is not prime is composite. Note that The number 1 is not prime neither composite. Example: Determine whether each of the following numbers is Prime, composite, or neither: 23, 30, 31, 29, 27, 5, 42, 21, 37, 87 Solution: 23 Prime 30 Composite 31 Prime 29 Prime 27 Composite 5 ………….. 42 ………….. 21 ………….. 37 ………….. 87 ………….. 90 ………….. 19 ………….. 5 Prime Factorization Prime Factorization: We write the number as a product of prime numbers Example: Find the Example: Find the Practice: Find the prime factorization of 24. prime factorization of 36. prime factorization of 90 Solution: Solution: 24 Solution: 36 2 12 2 18 2 6 2 9 2 3 3 3 2x2x2x3 2x2x3x3 6 3  Fraction Notation 10 14 Special Cases: 0, 0, 0 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 0 0 Example: Find an equivalent expression. Practice: Find an equivalent expression. ? ? ? ? ? ? a) b) c) a) b) c) Solution: Solution: a) ? 6 ? a) ? = ………………………. b) ? 33 ? b) ? = ……………………… c) ? 20𝑥 ? c) ? = ……………………… 7 Determine whether two fractions are equivalent = is equivalent to 𝑎 𝑑 𝑏 𝑐 Example: Check whether and are equal. Example: Use = or ….. Solution: Solution: 2 6 12 and 3 4 12 5 8 40 and 6 7 42 Then = Then Practice: Check whether and are equal. Practice: Use = or Solution: a) …… b) …… 8 4  Simplify Fractions Example: Simplify Example: Simplify Example: Simplify Practice: Simplify Solution: Solution: Solution: Solution: = 5 4 8 6 = 9 Multiply and simplify fractions Example: Example: Practice: Multiply and simplify Multiply and simplify Multiply and simplify Solution: Solution: a) b) Solution: 3 7 3 7 5 18 5 18 14 6 14 6 3 10 3 10 a) = 1 5 2 3 3 2 2 3 2 5 b) = 3 10 5  Divide and simplify fractions We multiply the first fraction by the reciprocal of the second fraction Example: Example: Practice: Divide and simplify Divide and simplify Divide and simplify a) b) Solution: Solution: Solution: a) = = = = b) = 11 6  Chapter 4 1 LEAST COMMON MULTIPLE (LCM) LCM of two numbers is the smallest common multiple of both numbers. Example: Find the LCM of 12 and 16 Practice: Find the LCM of 15 and 10 Solution: Solution: Multiples of 12: 12, 24, 36, 48, 60, ….. Multiples of 16: 16, 32, 48, 64, 80, …. LCM = 48 Example: Find LCM of 6𝑥 𝑦 𝑎𝑛𝑑 9𝑥 𝑦 Practice: Find LCM of 4𝑥y 𝑎𝑛𝑑 12𝑥y Solution: Solution: Multiples of 6: 6, 12, 18, 24, …… Multiples of 9: 9, 18, 27, 36,.…. LCM = 18𝑥 𝑦 2 1  ADDITION AND Subtraction Example: Add and simplify a) b) c) d) e) f) 5 Solution: Practice: Add and simplify a) a) b) c) 3 b) Solution: c) a) = d) = = = b) = e) = = = f) 5 c) 3= 3 ADDITION AND Subtraction Example: Add and simplify Practice: Add and simplify a) b) c) d) e) a) 𝑥 𝑥 b) 𝑦 𝑦 c) Solution: Solution: a) a) 𝑥 𝑥= b) b) 𝑦 𝑦= c d c) = e) 4 2  ORDER Common Denominator: Compare the numerators. Different Denominators: Change them first into fractions with the same denominator, and compare the numerators. Example: Use < or > to form true statements: a) b) c) d) Solution: a) b) c) < d) Practice: Use < or > to form true statements: a) b) c) 5 MIXED NUMERALS Mixed Numeral: A number of the form a , a, b and c are integers and c 0 Example: Convert to a fraction Example: Convert to a mixed numeral a) 2 b) 1 c) 2 a) b) c) d) Solution: Solution: a 2 a) = 10 b) = 16 b) 1 c) 2 c) =-5 d) =7 6 3  ADDITION AND SUBTRACTION OF MIXED NUMERALS Example: Add the following mixed numerals Practice: 5 + 1 𝑎 2 +3 b 3 -4 c 2 4 Solution: Solution: a) 2 + 3 b) 3 - 4 c) 2 4 = + = - = = + = - = = + = = - = = 7 7 =5 = -1 2 MULTIPLICATION AND DIVISION OF MIXED NUMERALS Example: Do the following operations Practice: Do the following operations a) 2 3 b 3 4 c 12 3 a) (-2 3 b) 6 1 Solution: a) 2 3 b) 3 4 c) 12 3 = = = = = 16 2 = 5 9 =8 =3 a) 9 b) 4 8 4  Chapter 5 1 Decimal Numbers The decimal Notation of 9723.456 Thousands hundreds tens ones and tenths hundredths thousandths Example: Write the word name of 23.456 Solution: Twenty three and four hundred fifty six thousandths Example: Write the word name of 325.87 Solution: Three hundred twenty five and eighty seven hundredths Practice: Write the word name of 456.341 Solution: ………………………………………………………………………………….…………….. 2 1  Convert between decimal and fraction Example: Example: Find the fraction Notation of Find decimal Notation of a) 71.789 b) 0.05 c) 3.7 a) ଶହ଺ b) ଻ c) ଷ ଵ଴ ଶ଴ ଵଵ Solution: Solution: ଻ଵ଻଼ଽ a) 71.789 = ଶହ଺ ଵ଴଴଴ a) = 25.6 ଵ଴ ହ b) 0.05 = ଻ ଵ଴଴ b) = 0.35 ଶ଴ ଷ଻ c) 3.7 = ଷ ଵ଴ c) ൌ 0.272727…. = 0.27 ଵଵ 3 Order: < or > To compare two decimal numbers, we will compare them digit by digit according to the place value. For example, to compare 2.301 and 2.4: The ones digit for both is 2, The tenth digit for the first number is 3, and the tenth digit for the second number is 4: Then 2.301 < 2.4 Example: Use < or > to write a true statement a) 2.362 2.319 b) 0.002 0.0008 c) -0.35 -0.7 d) 0.02031 0.03987 Solution: a) 2.362 > 2.319 b) 0.002 > 0.0008 c) -0.35 > -0.7 d) 0.02031 0.03987 4 2  Rounding 1. We first locate the digit, and then 2. We check the digit next to it (on its right)  If it is less than 5, we keep the digit as it is  If it is 5 or more, we add one the main digit 3. Keep all the digits on the left and replace all the digits on the right by 0. Example: Round 537.18 to the nearest ten. Example: Round 5.483 to the nearest tenth. Solution: Solution: The ten is 3. The digit on its right is 7 The tenth is 4. The digit on its right is 8 Add 1 to 3 and replace the digit on its right by 0. Add 1 to 4 and replace the digit on its right by 0. So the answer is 540.00 = 540 So the answer is 5.500 = 5.5 5 Rounding Example: Round 234.718 to the nearest tenth. Solution: 234.700 = 234.7 Example: Round 518.239 to the nearest hundredth. Solution: 518.240 = 518.24 Practice: Round 518.239 to the nearest ten. Practice: Round 34.159 to the nearest tenth. Solution: Solution: 6 3  ESTIMATING Example: Estimate the following by rounding first to the nearest one: a) 2.5 + 13.09 b) 13.08 - 2.63 c) 25.14 ൊ 5.03 d) 19.94 ൈ 2.7 Solution: a) 2.5 + 13.09 Round to the nearest one c) 25.14 ൊ 5.03 Round to the nearest one 3 + 13 = 16 25 ൊ 5 = 5 b) 13.08 - 2.63 Round to the nearest one d) 19.94 ൈ 2.7 Round to the nearest one 13 – 3 = 10 20 ൈ 3 = 60 Practice: Estimate the following by rounding first to the nearest ten: a) 82.5 + 17.1 b) 53.08 - 8.13 c) 85.4 ൊ 27.3 Solution: 7 4  Chapter 7 1 AVERAGES, MODES, AND MEDIAN  Average: Add the numbers and then divide by the number of data.  Mode: the most repeated number  Median: Arrange the numbers and then the middle number will be the median. Example: Find the average, the modes, and the median 16, 12, 12, 14, 13 ଵ଺ ା ଵଶ ା ଵଷ ା ଵସ ା ଵଷ Solution: Average = = 13.4 Mode = 12 ହ Median: arrange the numbers: 12, 12, 13, 14, 16 Median = 13 Example: Find the average, the modes, and the median of 9, 4, 2, 3, 6, 3, 5, 5, 6, 6 ଽ ାସ ାଶ ାଷ ା଺ ାଷ ାହ ାହ ା଺ ା଺ Solution: Average = = 4.9 Mode = 6 ଵ଴ ସ ାହ Median: Arrange the numbers 2, 3, 3, 4, 4, 5, 6, 6, 7, 9 Median = = 4.5 ଶ Practice: Find the average, the modes, and the median of 10, 14, 12, 10, 12, 10, 12, 10 2 1  INTERPRETING DATA FROM TABLES AND GRAPHS Example: Based on the table, a) Find the amount of Salt in crispix b) Find the amount of Sugar in Shredded wheat c) Find the amount of fat in Weetabix d) which cereals do we have 6g fiber or more? Solution: a) 1.25 g b) 0.25 g c) Less than 5g d) Shredded Wheat and weetabix 3 Extract and interpret data from graphs Example: Answer the following questions: Earnings of Apple Valley Juice a) In what year, the earnings go below 300 million. b) Find the earning for 2001. c) Find the earning for 2005. d) How many more earning in 2000 than in 1999? Solution: a) 2001 b) 200 million dollars c) 350 million dollars d) 450 – 300 = 150 million dollars 4 2  Extract and interpret data from histograms Grades Example: Based on The histogram, 40 a) How many students scored between 60 and 69? 35 Number of Students 30 Number of Tests b) How many students scored between 50 and 69? 25 c) How many students scored between 70 and 89? 20 15 Solution: 10 5 a) 5 0 b) 10 + 5 =15 50‐59 60‐69 70‐79 80‐89 90‐99 Test Grades c) 20 + 30 = 50 5 INTERPRETING AND DRAWING BAR GRAPHS AND LINE GRAPHS Example: The Bar graph to the right represents the Enrollment enrollment per program: a) Find the enrollment in AAS. AA BUSINESS b) Which program has the highest enrollment? c) Which program has the least enrollment? AAS Solution: AS a) 1000 AA b) AA c) AS 0 500 1000 1500 2000 2500 Enrollment 6 3  Extract and interpret data from line graphs Example: Consider the line graph. Lake Mead Average Water Level a) What is the water level in 1960? 1240 b) What is the water level in 1980? 1220 Height in feet 1200 c) In what year, the water level is the highest? 1180 1160 d) In what year, the water level is the least? 1140 1120 Solution: 1100 1940 1950 1960 1970 1980 1990 2000 2005 a) 1170 Year b) 1200 c) 2000 d) 2005 7 INTERPRETING CIRCLE GRAPHS Example: Consider the circle graph. Favorite Breakfasts a) What is the percentage who prefer doughnuts? other 6% b) What is the percentage who don’t like eggs? eggs 20% c) What is the percentage who prefer eggs or toast? cereal 42% d) What is the breakfast with the higher percentage? doughnuts 8% e) From 2000 persons, how many would prefer Pancakes? toast 6% Solution: waffles pancakes 11% 7% a) 8% b) 100% - 20% = 80% c) 20% + 6% = 26% d) Cereal e) 2000 x 0.11 = 220 8 4  Chapter 8 1 Perimeter of a Polygon Perimeter = the sum of the lengths of all the external sides. Example: Find the perimeter of a rectangle with Length 10 cm, and Width 6 cm. Solution: P = 10 + 10 + 6 + 6 = 32 cm 10 m Example: Find the perimeter this rectangle. Solution: 4m P = 10 + 10 + 4 + 4 = 28 m 12 cm Example: Find the perimeter of the following polygon. Solution: 7 cm 8 cm P = 12 8 5 5 7 37 cm 2 1  8.3 Area (Rectangle and Square) Area of a rectangle: A = 𝑙 𝑤 (𝑙 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑎𝑛𝑑 𝑤 𝑖𝑠 𝑡ℎ𝑒 𝑤𝑖𝑑𝑡ℎ) Area of a square: A = 𝑠 𝑠 (𝑠 is the length of the side) Example: Find the Area of a rectangle with length 6cm, and width 4cm. Solution: A=𝑙 𝑤 6 4 24𝑐𝑚 Example: Find the Area of a square with side 5 ft. Solution: A=5 5 25𝑓𝑡 3 Area of a Parallelogram Area of a parallelogram: 𝐴 𝑏 ℎ (𝑏 is the base and ℎ is the height) h b Example: Find the Area of a parallelogram with base 8cm and height 6cm. Solution: A = 8 6 48𝑐𝑚 4 2  Area of a Triangle Area of a Triangle: 𝐴 (b is the base and h is the height) h b Example: Find the Area of a triangle with base 10in and height 6in. Solution: 𝐴 = 30 𝑖𝑛 5 Area of a Trapezoid a Area of a Trapezoid: 𝐴 ℎ (a and b are the bases and h is the height) h b Example: Find the Area of a trapezoid with bases 4 cm and 6 cm, and height 7 cm. Solution: 𝐴 7 = 35 𝑐𝑚 8 cm Practice: Find the Area of a trapezoid. Solution: 3 cm 6 cm 10 cm 6 7 cm 3  Chapter 9 1 Evaluate algebraic expressions Example: Evaluate Practice: Evaluate a) 3𝑥 5𝑦 for 𝑥 2 and 𝑦 7 a) 12 2𝑥𝑦 15 for 𝑥 3 and 𝑦 4 b) 2𝑥2 3𝑦 5 for 𝑥 2 and 𝑦 3 b) 𝑥2 3𝑦3 12 for 𝑥 2 and 𝑦 2 c) for 𝑥 2 and 𝑦 5 Solution: Solution: a) 3𝑥 5𝑦 3 2 5 7 6 35 = 29 b) 2𝑥2 3𝑦 5 = 2 (-2)2 + 3(3) – 5 =2(4) + 9 – 5 =12 c) = Undefined 2 1  Translate phrases to algebraic expressions Addition Subtraction Multiplication Division Add Subtract Multiply Divide Plus Minus Times Divided by Added to Subtracted from Product Ratio Increased by Decreased by Multiplied by Quotient More Less Greater than Smaller than Sum Example: Translate into algebraic expressions. “ a number “ a) A number increased by 10 b) A number decreased by 12 variable letter, for c) The sum of 10 and twice a number example x Solution: a) 𝑥 10 b) 𝑥 12 c) 10 2𝑥 3 Translate phrases to algebraic expressions Example: Translate into algebraic expressions. Practice: a) A number added to 5 1) 25 subtracted from a number b) A number subtracted from 10 2) The product of 5 and twice a number c) 5 less than a number 3) Twice a number decreased by 7 d) The sum of 3 and twice a number 4) Five times a number minus eight e) The quotient of 100 and a number 5) The ratio of a number and 9 f) 5 more than 7 times a number Solution: g) 5 times a number minus 8 times another number Solution: a) 𝑥 5 b) 10 𝑥 c) 𝑥 5 d) 2𝑥+3 e) f) 7𝑥 5 g) 5𝑥 8𝑦 4 2  Absolute Value Absolute value of a number is the distance of this number from zero on the number line. Absolute value of a positive number is the same as the number. Absolute value of a negative number is the opposite of the number. Example: Find Practice: Find a) 3 b) 4 c) 13 6 d) 24 3 5 6 a) 15 13 6 Solution: a) 3 3 b) 4 4 c) 13 6 7 7 b) 10 3 7 1 d) 24 3 5 6 24 9 24 9 15 5 Operations of real numbers Example: Find the opposite of a) 12 b) -8 c) 5 d) -7 Solution: a) -12 b) 8 c) -5 d) 7 Example: Evaluate a) 3 + (-5) b) -3 + (-5) c) -3 - (-5) d) -2 + (-3) - (-9) + 5 - (-4) Solution: a) -2 b) -8 c) 2 d) 13 The multiplication or division of two positive (negative) numbers is positive The multiplication or division of two numbers with opposite signs is negative. Example: Evaluate a) 2 3 b) 3 4 c) 4 2 d)10 5 Solution: a) 6 b) 12 c) 2 d) -2 6 3  Distributive Law and Factoring The distributive law: 𝑎 𝑏 𝑐 𝑎 𝑏 𝑎 𝑐 Example: Multiply a) 2 𝑎 5𝑏 𝑏 3 4 𝑥 𝑐 5 3𝑥 𝑦 𝑑 3𝑎 2𝑏 Solution: a) 2 𝑎 5𝑏 2𝑎 10𝑏 b) 3 4 𝑥 12 3𝑥 c) 5 3𝑥 𝑦 15𝑥 5𝑦 d) 3𝑎 2𝑏 3𝑎 2𝑏 Example: Factor a) 2𝑎 6𝑏 𝑏 3𝑎 9 𝑐 10𝑥 15𝑦 d) 21𝑥 12𝑦 3 Solution: a) 2𝑎 6𝑏 2 𝑎 3𝑏 b) 3𝑎 9 3 𝑎 3 c) 10𝑥 15𝑦 5 2𝑥 3𝑦 d) 7 Collect like terms Like terms: terms with the same variable and same power for the variable. Like terms can be combined in order to have a simplified form of the expression. Example: Combine like terms Practice: Combine like terms a) 2x + 3y – 9 - 4x + 7y + 1 1) 5x - 13y – 19 - 9x + 7y + 1 b) -3x2 + 5xy + 7x2 - 3xy + 2 2) -7x2 - 12xy – 6 + 15x2 - 3xy + 2 c) 5.9x - 1.3y – 2.4x + 7y 3) 𝑥 𝑥 a) Solution: Solution: a) 2x + 3y – 9 - 4x + 7y + 1 = -2x + 10y - 8 b) -3x2 + 5xy + 7x2 - 3xy + 2 = 4x2 + 2xy + 2 c) 5.9x - 1.3y – 2.4x + 7y = 3.5x + 5.7y 8 4  Simplifying expressions Example: Simplify Practice: Simplify a) (2x + 3y - 5) + 2(3x - 8y) 1) 3 (2x + 5y - 2) - 5(2x - 2y - 1) b) (x + y - 5) - 3(2x - 5y + 2) 2) 2 [ 10x -3(2x - 5)] c) 𝑥 −[2𝑥 − 3(2𝑥 − 4)] Solution: Solution: a) (2x + 3y - 5) + 2(3x - 8y) b) (x + y - 5) - 3(2x - 5y + 2) = 2x+3y-5+6x-16y = x+y-5-6x+15y-6 = 8x - 13y - 5 = -5x+16y-11 c) 𝑥 −[2𝑥 − 3(2𝑥 − 4)] 𝑥 2𝑥 6𝑥 12] 𝑥 2𝑥 6𝑥 12 5x - 12 9 Order of Operations 1. Brackets 15 – 5 × 2 = ……. 2. Powers 3. Multiplication and Division (we start from left to right If both are there). 4. Addition and Subtraction. Example: Simplify a) 3 2 2 2 5 b) 2 2 2 5 4 3 Solution: a) 3 2 2 2 5 b) 2 2 2 5 4 3 4 2 5 2 8 12 4 2 5 2 5 16 12 -3 4 10 5  Order of Operations Example: Simplify Practice: Simplify a) 62 (10 -7)2 2 - 8 b) 32 3 8 2 18 13 3(7-5)3 (15 - 9) 5 – 12 Solution: a) 62 32 2−8 b) 32 3 8 2 18 13 36 9 2 8 32 11 2 5 Practice: Simplify 4 2 8 = 32 22 5 3 7 3 2 2 5 6 8 8 = 10 5 0 2 11 6  8/23/2023 Chapter 10 1 Solving Equations Example: Determine whether 7 is a solution of x + 9 = 15 Solution: We substitute x by 7: 7 + 9 ≠ 15 Then, 7 is not a solution of the equation Example: Determine whether 3 is a solution of 2x – 9 = -3 Solution: We substitute x by 3: 6 - 9 = -3 Then, 3 is a solution of the equation Practice: Determine whether -5 is a solution of 3x + 11 = 4 Solution: 2 1  8/23/2023 Solving Equations Example: Solve a) 2𝑥 + 6= 40 b) 3x – 7 = 5 c) 0.5y – 3.7 = 16.3 d)2 + 5x = 2x + 29 Solution: a 2𝑥 40 - 6 b 3𝑥 5 7 c 0.5y 16.3 3.7 d 5x – 2x 29 - 2 2𝑥 34 3𝑥 12 0.5y 20 3x 27 x 9 0.5y 20 𝑥 17 𝑥 4 0.5 0.5 y 40 3 Solving Equations Solving Equations Example: Solve a 2 - 5x 29 - 2x b 3 x-2 5 5 x c 2x 5 15 – 2x Solutions: a - 5x 2x 29 - 2 b 3x - 6 25 5x c 2x 75 – 10x -3x 27 3x – 5x 25 6 2x 10x 75 x -9 -2x 31 12x 75 x - -15.5 x 6.25 4 2  8/23/2023 Solving Equations Example: Solve 𝑥+ Example: Solve 𝑥+4= - 𝑥 Solution: Solution: 𝑥 - 𝑥+ 𝑥 -4 2 4 𝑥= 𝑥 3 9 𝑥= = × = 𝑥= = × 5 TWO SPECIAL CASES Example: Solve 8x – 18 = 6 + 8(x – 3) Example: Solve 3 – 4(x + 6) = –4(x – 1) – 3. Solution Solution 8x – 18 = 6 + 8(x – 3) 3 – 4(x + 6) = –4(x – 1) – 3 8x – 18 = 6 + 8x – 24 3 – 4x – 24 = –4x + 4 – 3 8x – 18 = 8x – 18 –4x – 21 = –4x + 1 8x – 8x = – 18 + 18 –4x + 4x = 1 + 21 0=0 0 = 22 All real numbers No solutions 6 3  8/23/2023 Practice Solving Equations Solve a 3 – 5 3x – 1 36 - 8x b 3 x-2 6 5 3x c 𝑥 𝑥 2 Solution: 7 Evaluating Formula and Solving for a specific letter Example: If d = rt2, find d for r=10 and t=2 Example: If P = 2L+2W, find P for L=6 and W=4 Solution: Solution: d = 10 22 = 40 P = 2(6) + 2 (4) = 20 Example: Solve a+b+c 5x − 3y + 5 a) 2x + y = 10 for x b) m = for a c) A = for y d) V = for x 3 12 Solution: a b c c 7A 7 a 2x 10 - y b 3m 3 3 x 3m a b c 7A 5xy a 3m - b - c y 8 4  8/23/2023 Applications of Percent What 𝑥 What Percent is = of × n% What is 8% of 200? 12 is 40% of What? What percent of 80 is 50? Solution: Solution: Solution: 𝑥= 200 12 = 𝑥 80 = 50 𝑥 = 16 12 = 0.40 𝑥 80 𝑥 5000 𝑥 = 30 𝑥 = 62.5% What is 10% of 54? 24 is 25% of What? What percent of 32 is 8? Solution: Solution: Solution: 𝑥= 54 24 = 𝑥 32 = 8 𝑥 = 5.4 24 = 0.25 𝑥 32 𝑥 = 800 𝑥 = 96 𝑥 = 25% 9 Practice What 𝑥 What Percent is = of × n% What is 12% of 500? 28 is 70% of What? What percent of 400 is 140? Solution: Solution: Solution: 10 5  8/23/2023 Problems About Consecutive Integers Type of Integer Translation Consecutive integers x, x+1, x+2, x+3,... Consecutive odd integers x, x+2, x+4, x+6,... Consecutive even integers x, x+2, x+4, x+6,... Example: The sum of two consecutive Example: The sum of two consecutive integers is 345, find these integers. even integers is 38, find these integers. Solution: Solution: x x+1 x x+2 x + (x + 1) = 345 x + (x + 2) = 38 2x + 1 = 345 2x + 2 = 38 2x = 344 2x = 36 x = 172 x = 18 The integers are 172 and 173 The integers are 18 and 20 11 Problems About Consecutive Integers Type of Integer Translation Consecutive integers x, x+1, x+2, x+3,... Consecutive odd integers x, x+2, x+4, x+6,... Consecutive even integers x, x+2, x+4, x+6,... Example: The sum of three consecutive Practice: The sum of three consecutive odd integers is 87, find these integers. even integers is 276, find these integers. Solution: Solution: x x+2 x+4 x + (x + 2) + (x + 4) = 87 3x + 6 = 87 3x = 81 x = 27 The integers are 27, 29, and 31 12 6  8/23/2023 problems about the angles in a triangle The sum of angles in a triangle is 180° Example: In a triangle, The second angle is 2 times Practice: In a triangle, The second angle is the first. The third angle is 10 degrees more than the 5 times the first. The third angle is 4 times first. Find the measure of each angle. the first. Find the measure of each angle. Solution: Solution: x 2x x + 10 x + 2x + x + 10 = 180° 4x = 170° x = 42.5° First angle: x = 42.5 Second angle: 2x 2 42.5 85° Third angle: x + 10 = 42.5 +10 = 52.5° 13 problems related to length Example: A pipe is cut into 2 pieces. One Practice: A pipe is cut into 2 pieces. One piece is 4 times the other. The total length piece is 20 more than the other. The total is 120. Find the length of each piece. length is 120. Find the length of each piece. Solution: Solution: x 4x x + 4x = 120 5x = 120 x = 24 The first piece is x = 24 The second piece is 4x = 4 24 96 14 7  8/23/2023 Solving Inequalities < less than less than or equal > greater than  greater than or equal Example: Graph a) x < 3 b) y  4 Solution: a) The numbers to the left of 3. b) The numbers to the right of 4. ) [ Example: Example: Determine if 3 is a solution of x +5 < 7 Determine if 8 is a solution of 2x +5 < 7 Solution: Solution: Substitute x by -3 Substitute x by 8 3 + 5 < 7 is true Then 3 is a solution. 2(8) +5 < 7 is false Then 8 is not a solution. 15 Solving Inequalities Example: Solve each of the following inequalities a) 2x + 7 > 3 b) 3x - 3 12 c) -4y - 8 < 20 Practice d) 3x -12  20 + 7x Solution: a) 2x + 7 > 3 b) 3x 3 12 c) -4y - 8 < 20 2x > 3 - 7 3x 12 3 -4y < 20 + 8 2x > - 4 3x 15 -4y < 28 x>-2 x 5 y > -7 16 8  8/23/2023 Solving Inequalities Example: Solve a) 3x + 9 < 5x - 7 b) 9x  2  3(x  5) + 25 Practice c) 2(x  3)  3x  3(x  5) + 10 Solution: a) 3x + 9 < 5x - 7 b) 9x  2  3(x  5) + 25 3x  5x <  7 – 9 9x  2  3x  15 + 25 - 2x <  16 9x  3x   15 + 25 +2 x>8 6x  12 x2 17 9  Chapter 11 1 Cartesian Coordinate (Plot points) Second quadrant, First quadrant Horizontal axis: x-axis Vertical axis: y-axis x0 x>0 and y>0 To plot a point (x , y), move x horizontally and y vertically Example: Plot each of the points a) (-3,2) b) (4,-3) c) (3, 0) d) (-2, -5) (-3,2) Solution: a) Starting from the origin, we move 3 steps left (-3), and then we move 2 steps up. b) Starting from the origin, we move 4 steps right, and then (4, -3) we move 3 steps down (-3). Third quadrant, Fourth quadrant, x