Chapter 1 Basic Algebra PDF

Chapter 1 BASIC ALGEBRA Chapter 1, Introduction Course of MATH 111 Basic Sciences Department, Science Track 1 1.2. Real Numbers and Basic Arithmetic Operations 1.2.1 Sets of Numbers 1.2.2 Algebraic Operati...

Chapter 1 BASIC ALGEBRA Chapter 1, Introduction Course of MATH 111 Basic Sciences Department, Science Track 1 1.2. Real Numbers and Basic Arithmetic Operations 1.2.1 Sets of Numbers 1.2.2 Algebraic Operations and Expressions 1.2.3 Computing with Fractions 1.2.4 Natural Number Exponents 1.2.5 Grouping Symbols and Order of Operations Course of MATH 111 Basic Sciences Department, Science Track 2  1.2.1 SETS OF NUMBERS: Definitions ℕ = {1, 2, 3, 4, … } Natural Numbers W={0, 1, 2, 3, … } Whole Numbers ℤ = {…,−2, −1, 0, 1, 2,… } Integers 𝑎 ℚ = { 𝑟∣𝑟 = , 𝑎,𝑏 ∈ ℤ, 𝑏 ≠ 0 } Rational Numbers 𝑏 1100 2 10000 ℚ = { …,− ,…, , … , 3, … , , …} Rational Numbers 141 21 47 The real number line Every point on the real line corresponds to a real number. 5 ℝ = {… , − 7, … , −2, … , − , … , 0, … 𝜋, … } Real numbers 3 An irrational number is a real number that is not rational. 𝐼 = 𝑥 𝑥 ∊ ℝ, 𝑥 ∉ ℚ Irrational Numbers Some examples of irrational numbers are 2 and 𝜋 Course of MATH 111 Basic Sciences Department, Science Track 3  1.2.1 SETS OF NUMBERS: Relationships among the subsets of Real Numbers Real Numbers Rational numbers Irrational numbers 4 5 11 ,− , 9 8 7 2 ℕ⊂𝑊⊂ℤ⊂ℚ⊂ℝ Integers 𝐼⊂ℝ -11, -6, -3, -2, -1 15 ℝ=ℚ∪𝐼 Whole numbers - 8 ℚ∩𝐼 =∅ 0 𝜋 Natural numbers 1, 2, 3, 4, 5, 𝜋 4 Course of MATH 111 Basic Sciences Department, Science Track 4  1.2.1 SETS OF NUMBERS: Evaluating Statements about Numbers Example 1: Determine whether the statements are true or false. a) Every integer is a rational number. 22 b) The number 𝜋 is a rational number because 𝜋 =. 7 Solution: 𝑎 a) True: If 𝑎 is an integer then 𝑎 = Since the numerator and. 1 the denominator are integers, the number is rational. 22 b) False: The number 𝜋 is irrational. The number is an 7 approximation of 𝜋 and it is not equal to 𝜋. Course of MATH 111 Basic Sciences Department, Science Track 5  1.2.1 SETS OF NUMBERS: Decimal Representation of Real Numbers Every real number can be written in decimal notation. It is called decimal representation of the number. For example, 3 = 0.75 (read 0.75 as “seventy-five hundredths”). 4 7 = 2.3333 … (read 2.333 … as “two point three 3 three…”). 2 = 1.41421356237309 … 𝜋 = 3.14159265358979323846 … Course of MATH 111 Basic Sciences Department, Science Track 6  1.2.1 SETS OF NUMBERS: Decimal Representation of Real Numbers Numbers such as 0.75 and 2.25 are called terminating decimals Numbers such as 0.333 … and 3.75454 … are called repeating decimals. Decimal Representation of a Real Number ▪ The decimal representation of a rational number is either repeating or terminating. ▪ The decimal representation of an irrational number is neither repeating nor terminating Course of MATH 111 Basic Sciences Department, Science Track 7  Activity: Classifying Real Numbers Put an ✕ in the box if the number is an element of the set at the top of the column Real Rational Irrational Whole Integers numbers numbers numbers numbers −3 22 7 −𝜋 3.14 2.718281 … 1331 11 110.916 5625 Course of MATH 111 Basic Sciences Department, Science Track 8  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Basic Algebraic Operations on Real Numbers Operation Symbol Word Addition + Plus Multiplication × Times Subtraction − Minus Division ÷ Divided by Variables 𝑥, 𝑦, 𝑧, 𝑎, 𝑏, 𝑐 or other letters/symbols Algebraic Expressions 𝜋+𝑡 𝑥 + 𝑦, , 2𝑥 − 𝑦𝑧 − 5. 𝑠−1 Sum of Real Numbers 𝑥+𝑦 “ the sum of x and y” OR “x plus y” Commutative Law of Addition 𝑎+𝑏=𝑏+𝑎 Associative Law of Addition (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) Course of MATH 111 Basic Sciences Department, Science Track 9  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Additive Identity 𝑥+0= 0+𝑥 =𝑥 0 is the additive identity Additive Inverse 𝑥 + −𝑥 = −𝑥 + 𝑥 = 0 – 𝑥 is the additive inverse of 𝑥 Difference of Real Numbers 𝑥−𝑦 “𝑥 minus 𝑦” Example 3: Indicate whether the following statements are true or false. Subtraction is commutative. Subtraction is associative Solution: False: 1 − 2 ≠ 2 − 1. Subtraction is not commutative. False: 1 − 1 − 1 ≠ 1 − 1 − 1. Subtraction is not associative. Course of MATH 111 Basic Sciences Department, Science Track 10  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Product of Real Numbers 𝑎×𝑏 “𝑎 times 𝑏” or “𝑎 multiplied by 𝑏” “𝑎 𝐚𝐧𝐝 𝑏 are factors of the 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 𝑎 × 𝑏” Commutative Law of Multiplication 𝑎𝑏 = 𝑏𝑎 Associative Law of Multiplication (𝑎𝑏)𝑐 = 𝑎(𝑏𝑐) Example 4: Apply the Commutative and Associative Laws of Multiplication to multiply: 2 ⋅ 129 ⋅ 5. Do not use calculator Solution: Use commutative law of “×”. 2 ⋅ 129 ⋅ 5 = 2 ⋅ 5 ⋅ 129 Use associative law of “×”. (2 ⋅ 5) ⋅ 129 = 10 ⋅ 129 Find the product. 10 ⋅ 129 = 1290 Course of MATH 111 Basic Sciences Department, Science Track 11  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Multiplicative Identity 𝑥∙1=1∙𝑥 =𝑥 1 is the multiplicative identity 1 1 1 Multiplicative Inverse 𝑥⋅ = ⋅𝑥 =1 is the multiplicative inverse 𝑥 𝑥 𝑥 (reciprocal) of 𝑥 (𝑥 ≠ 0). Zero Factor Property If 𝑥𝑦 = 0 then 𝑥 = 0 or 𝑦 = 0 Example 5: Let 𝑥 be a real number. Suppose that 𝑥 − 1 𝑥 + 1 = 0. What are the possible values of 𝑥? (Use the zero factor property) Solution: By the zero factor property, 𝑥 − 1 = 0 or 𝑥 + 1 = 0, and by the additive inverse property 𝑥 = 1 or 𝑥 = −1. Course of MATH 111 Basic Sciences Department, Science Track 12  1.2.2 Algebraic Operations and Expressions: Definitions and Properties The Distributive Law of Multiplication Over Addition If 𝑎, 𝑏 and 𝑐 are real numbers then 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 (left distributive law of “×” over “+”). 𝑎 + 𝑏 𝑐 = 𝑎𝑐 + 𝑏𝑐 (right distributive law of “×” over “+”). Example 6: Remove the parentheses from each of the following expressions. a) 5(x+4) b) (2+x)x c)(2x+3)(3x+2) Solution: a) Use the right distributive law of “×” over “+”. 5 x + 4 = 5x + 20 b) Use the left distributive law of “×” over “+”. 2 + x x = 2x + x 2 Course of MATH 111 Basic Sciences Department, Science Track 13  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Example 6: Remove the parentheses from each of the following expressions. a) 5(x+4) b) (2+x)x c) (2x+3)(3x+2) Solution: C) (2𝑥 + 3)(3𝑥 + 2) Use the left distributive = 2𝑥 + 3 3𝑥 + (2𝑥 + 3)(2) law of “×” over “+”. Use the right distributive = 2𝑥 3𝑥 + 3 3𝑥 law of “×” over “+”. + 2𝑥 2 + 3(2) Simplify. = 6𝑥 2 + 9𝑥 + 4𝑥 + 6 = 6𝑥 2 + 13𝑥 + 6 Course of MATH 111 Basic Sciences Department, Science Track 14  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Properties of Negatives If 𝑎 and 𝑏 are real numbers, then a)− −𝑎 = 𝑎 b) −1 𝑎 = −𝑎 c) −𝑎 −𝑏 = 𝑎𝑏 d) −𝑎 𝑏 = 𝑎 −𝑏 = −(𝑎𝑏) NOTATION: We usually write −(𝑎𝑏) as – 𝑎𝑏, but the product must be performed before taking the negative. Course of MATH 111 Basic Sciences Department, Science Track 15  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Example 7: Evaluate each of the following expressions: a) (−2)3 b) (−2)(−3) c) −2 × 3 d) −(−5) e) (−2)2 f) −22 g) (−2)(−3)(−4) h) −5 −2 −3 (−4) Solution: a) −2 3 = − 2 × 3 = −6 b) −2 −3 = 2 × 3 = 6 c) −2 × 3 = − 2 × 3 = −6 d) − −5 = 5 2 e) −2 = −2 −2 = 4 f) −22 = − 2 × 2 = −4 g) −2 −3 −4 = −24 h) −5 −2 −3 −4 = 120 Course of MATH 111 Basic Sciences Department, Science Track 16  1.2.2 Algebraic Operations and Expressions: Definitions and Properties The Distributive Law of Multiplication Over Subtraction If 𝑎 and 𝑏 are real numbers, then 𝑎 𝑏 − 𝑐 = 𝑎𝑏 − 𝑎𝑐 (left distributive law of “×” over “−”). 𝑎 − 𝑏 𝑐 = 𝑎𝑐 − 𝑏𝑐 (right distributive law of “×” over “−”). Example 8: Remove the parentheses from the following expressions and simplify. a) 2(2𝑤 − 5) b) (2𝑥 − 3)(𝑥 − 7) Solution: a) Use the right distributive law of “×” 2 2𝑤 − 5 = 2 2𝑤 − 2 5 over “−” then simplify. = 4𝑤 − 10 Course of MATH 111 Basic Sciences Department, Science Track 17  1.2.2 Algebraic Operations and Expressions: Definitions and Properties Example 8: Remove the parentheses from the following expressions and simplify. a) 2(2𝑤 − 5) b) (2𝑥 − 3)(𝑥 − 7) Solution: b) 2𝑥 − 3 𝑥 − 7 = 2𝑥 − 3 𝑥 + 2𝑥 − 3 −7 = 2𝑥 𝑥 − 3𝑥 + 2𝑥 (−7) − 3(−7) = 2𝑥 2 − 3𝑥 − 14𝑥 + 21 = 2𝑥 2 − 17𝑥 + 21 NOTE: Parentheses preceded by “−” sign can be removed very quickly by changing the sign of each term inside the parentheses and dropping the parentheses. Course of MATH 111 Basic Sciences Department, Science Track 18  1.2.2 Algebraic Operations and Expressions: Simplifying Expression Involving Negatives Example 9: Simplify the following expressions. a) 4𝑥 − 3 − 4𝑥 b) −4𝑥 − (3 + 4𝑥) Solution: a) b) 4𝑥 − 3 − 4𝑥 = 4𝑥 − 3 + 4𝑥 −4𝑥 − (3 + 4𝑥) = −4𝑥 − 3 − 4𝑥 = 8𝑥 − 3 = −8𝑥 − 3 Try & Check: Simplify the following expressions. a) 5𝑦 − 5 −𝑥 − 𝑦. b) 5𝑦 − (−𝑥 + 𝑦). Course of MATH 111 Basic Sciences Department, Science Track 19  1.2.3 Computing with Fractions 𝑎 Division of Real Numbers “𝑎 divided by 𝑏” or “𝑎 over 𝑏” 𝑏 𝑎 Important: If a number is written as , it is also called a fraction. 𝑏 0 𝑎 Division involving zero i) = 0 for 𝑎 ≠ 0 ii) is undefined 𝑎 0 𝑎 𝑐 Checking Equivalent Fractions = if and only if 𝑎𝑑 − 𝑏𝑐 = 0 𝑏 𝑑 𝑎𝑐 𝑎 𝑎÷𝑐 𝑎 Cancellation Law for Fractions = , and = , (𝑐 ≠ 0). 𝑏𝑐 𝑏 𝑏÷𝑐 𝑏 3 2 2 Extra example 9: Determine whether = is true. 3𝜋+1 𝜋+1 Solution: False because 3 2 𝜋 + 1 − 3𝜋 + 1 2 = 3𝜋 2 + 3 2 − 3𝜋 2 − 2 = 2 2 ≠ 0 Course of MATH 111 Basic Sciences Department, Science Track 20 1.2.3 Computing with Fractions: Properties of Fractions Let 𝑎, 𝑏, 𝑐 and 𝑑 be real numbers. Sum of two fractions. 𝑎 𝑐 𝑎+𝑐 + = , 𝑏 ≠ 0; 𝑏 𝑏 𝑏 𝑎 𝑐 𝑎𝑑 + 𝑏𝑐 + = , 𝑏, 𝑑 ≠ 0. 𝑏 𝑑 𝑏𝑑 Opposite of a fraction. 𝑎 −𝑎 𝑎 − = = , 𝑏 ≠ 0. 𝑏 𝑏 −𝑏 Difference of two fractions. 𝑎 𝑐 𝑎−𝑐 − = , 𝑏 ≠ 0; 𝑏 𝑏 𝑏 𝑎 𝑐 𝑎𝑑 − 𝑏𝑐 − = , 𝑏, 𝑑 ≠ 0. 𝑏 𝑑 𝑏𝑑 Reciprocal of a fraction. 𝑎 −1 𝑏 = , 𝑎, 𝑏 ≠ 0. 𝑏 𝑎 Quotient of two fractions. 𝑎 𝑐 𝑎 𝑑 𝑎𝑑 ÷ = = , 𝑏, 𝑐, 𝑑 ≠ 0. 𝑏 𝑑 𝑏 𝑐 𝑏𝑐 Course of MATH 111 Basic Sciences Department, Science Track 21  1.2.3 Computing with Fractions Example 10: Perform the indicated operations. Write your answer as a single fraction and simplify as you can. Do not use calculator. 1 3 7 3 1 3 a) + c) − f) − 2 5 2 5 6 8 3 5 11 4 2 5 g) − − i) ÷ l) − ÷ − 5 3 5 3 3 6 Solution: 1 3 1 5 +2 3 11 7 3 7 5 −2 3 35−6 29 a) + = = c) − = = = 2 5 2 5 10 2 5 2 5 10 10 1 3 1 3 3 1 3 5 3 5 f) − =− ⋅ = − = − g) − − = ⋅ =1 6 8 6 8 48 16 5 3 5 3 11 4 11 3 11⋅3 33 2 5 2 5 i) ÷ = = = l) − ÷ − = ÷ 5 3 5 4 5⋅4 20 3 6 3 6 2 6 4 = = 3 5 5 Course of MATH 111 Basic Sciences Department, Science Track 22  Try & Check Perform the indicated operations. Write your answer as a single fraction and simplify as you can. Do not use calculator 1 1 1 1 1 2 a) + b) − c) − 3 2 2 3 9 27 2 14 2 14 1 d) ⋅ e) − f) 5− 7 3 7 3 5 1 1 2 14 g) 𝑎 ÷ , 𝑎 real. h) ÷ 𝑎, 𝑎 ≠ 0 i) − − 3 3 7 3 2 14 1 1 5 j) ÷ k) − ÷ − l) 3 7 3 3 2 7 Course of MATH 111 Basic Sciences Department, Science Track 23  1.2.4 Natural Number Exponents Exponential Expression If 𝑎 is a real number and 𝑛 is a natural number then 𝑎𝑛 = 𝑎 ⋅ 𝑎 ⋯ 𝑎 , 𝑎 appears as a factor 𝑛 times. 𝑛 times The expression 𝑎𝑛 is read as “𝑎 to the 𝑛th-power”. The number 𝑎 is the base and 𝑛 is the exponent. The expression itself is called an exponential expression Vocabulary: Let 𝑎 be a real number. We read 𝑎2 as “𝑎 to the second power” or “𝑎 squared”. We read 𝑎3 as “𝑎 to the third power” or “𝑎 cubed”. We read 𝑎𝑛 as “𝑎 to the 𝑛”. Course of MATH 111 Basic Sciences Department, Science Track 24  1.2.4 Natural Number Exponents Example 11: Read and evaluate each of the following expressions. Indicate the base and the exponent. a) 13 b) −2 4 c) −24 Solution: a) We read 13 as “1 to the third power” or “1 cubed”. The base is 1 and the exponent is 3. 13 = 1 ⋅ 1 ⋅ 1 = 1. b) We read −2 4 as “negative 𝟐 to the 4th power” or “negative 2 to the 4”. The base is −2 and the exponent is 4. −2 4 = −2 −2 −2 −2 = 16. c) We read −24 as “the negative of 𝟐 to the 𝟒”. The power is 24 , the base is 2 and the exponent is 4. We first evaluate the power then we take the negative: −24 = − 24 = −16 Course of MATH 111 Basic Sciences Department, Science Track 25 1.2.4 Grouping Symbols and Order of Operations Grouping Symbols: By grouping symbols we mean: parentheses ( ), brackets [ ], braces { } and fraction bar. The Order of Operations Step 1. Working from left to right, evaluate expressions in parentheses or in other grouping symbols (whichever comes first). Start with the innermost grouping symbol and work outwards. Step 2. Working from left to right, evaluate exponential expressions (whichever comes first). Step 3. Working from left to right, do multiplication or division (whichever comes first). Step 4. Working from left to right, do addition or subtraction (whichever comes first). Course of MATH 111 Basic Sciences Department, Science Track 26  1.2.4 Natural Number Exponents: Applying the Order of Operations Example 12: Evaluate: 15 + 40 ÷ 5 ⋅ 8 − 6 − 11 2. Solution: 15 + 40 ÷ 5 ⋅ 8 − 6 − 11 2 Evaluate inside parentheses. = 15 + 40 ÷ 5 ⋅ 8 − −5 2 Evaluate the power. = 15 + 40 ÷ 5 ⋅ 8 − 25 Do the division. = 15 + 8 ⋅ 8 − 25 Do the multiplication. = 15 + 64 − 25 Do the addition and then the = 79 − 25 subtraction. = 54 Course of MATH 111 Basic Sciences Department, Science Track 27  1.2.4 Natural Number Exponents: Applying the Order of Operations Example 13: Remove the all delimiters from the following expression and simplify: 𝐴 = −2 3 𝑥 − 2𝑦 + 7 + 3 2 − 5𝑥 + 10 − 7 −2 𝑥 − 3 + 5 Solution: From left to right, inside brackets, remove the parentheses. 𝐴 = −2 3𝑥 − 6𝑦 + 7 + 3 2 − 5𝑥 + 10 − 7 −2𝑥 + 6 + 5 From left to right, simplify within brackets. 𝐴 = −2 3𝑥 − 6𝑦 + 7 + 3 12 − 5𝑥 − 7 −2𝑥 + 11 From left to right, apply the distributive law 𝐴 = −6𝑥 + 12𝑦 − 14 + 36 − 15𝑥 + 14𝑥 − 77 Simplify. 𝐴 = −7𝑥 + 12𝑦 − 55 Course of MATH 111 Basic Sciences Department, Science Track 28  Try & Check Simplify the expression −2 −5 𝑦 − 𝑥 − 2 3 2𝑥 − 5 − 7 2 − 4 − 3 − 7 Course of MATH 111 Basic Sciences Department, Science Track 29 1.2.4 Grouping Symbols and Order of Operations: Simplifying Fractions Simplifying Fractions To simplify a fraction, simplify the numerator and the denominator of the fraction separately. Then simplify the fraction, if possible. Examples: −36 𝑦𝑥 2 𝑧 = −4 = 𝑥 2 , 𝑦 ≠ 0 𝑎𝑛𝑑 𝑧 ≠ 0 9 𝑦𝑧 Course of MATH 111 Basic Sciences Department, Science Track 30 1.2.4 Grouping Symbols and Order of Operations: Evaluating Algebraic Expression by Substitution Example 14: If 𝑥 = −2, 𝑦 = −4 and 𝑧 = 5, evaluate the expression 𝑥 2 + 2𝑦𝑧. 3 𝑥+𝑧 Solution: Substitute −2 for 𝑥, 𝑥 2 +2𝑦𝑧 −2 2 +2 −4 5 = −4 for 𝑦 and 5 for 𝑧. 3 𝑥+𝑧 3 −2+5 In the numerator, evaluate the power. 4+2 −4 5 = 3 −2+5 In the numerator, do the multiplication, 4+(−40) −36 = = then do the addition 3 −2+5 3 −2+5 In the denominator, evaluate within −36 −36 = = = −4 parentheses, do the multiplication and 3 3 9 simplify the fraction Course of MATH 111 Basic Sciences Department, Science Track 31  Try & Check − 𝑥+3 2 −9 If 𝑥 = −2 and 𝑧 = −4, evaluate 6−𝑧 Course of MATH 111 Basic Sciences Department, Science Track 32  Review of New Notations ℤ = {…,−2, −1, 0, 1, 2,… } Integers 𝑎 ℚ = { 𝑟∣𝑟 = , 𝑎,𝑏 ∈ ℤ, 𝑏 ≠ 0 } Rational Numbers 𝑏 5 ℝ = {… , − 7, … , −2, … , − , … , 0, … 𝜋, … } Real numbers 3 𝐼 = 𝑥 𝑥 ∊ ℝ, 𝑥 ∉ ℚ Irrational Numbers "+" Addition "−" Subtraction "×" Multiplication "÷" Division 𝑎 " "𝑏≠0 Fraction 𝑏 𝑛 "𝑎 " Exponential expression Course of MATH 111 Basic Sciences Department, Science Track 33  Worksheet Course of MATH 111 Basic Sciences Department, Science Track 34