Basic Equations and Inequalities CHAPTER 3 Basic Equations and Inequalities

3.1 Solving Equations by Adding or Subtracting 1) Equation = a mathematical sentence that contains an equal sign. 2) Solution = is a number that, when substituted for a variable, makes a mathematical sentence true. 3) Inverse Operations = operations that undo one another (opposites). the variable always wants to be alone the opposite of addition is subtraction and the opposite of multiplication is division whatever you do to one side, you must do to the other

Example 1 No, it is an expression Is this an equation? If so, what is the solution? 2x + 3 No, it is an expression

Example 2 Is this an equation? If so, what is the solution? x + 3 = 4 −3 −3 yes; x = 1

Example 3 Solve 47 = m + 3, and check the solution. m + 3 = 47 −3 −3 m = 44

Example 4 Write each word phrase as a mathematical equation. Use n for the variable. 1) four more than a number is ten n + 4 = 10 2) a number increased by ten is negative six n + 10 = −6

Example 4 n − 5 = 3 n − 7 = −9 3) five less than a number is three 4) a number decreased by seven is negative nine n − 7 = −9

Topic: Solving Equations by Adding & Subtracting HOMEWORK Topic: Solving Equations by Adding & Subtracting examples on pages 84-88 Assignment: Lesson 3.1 in book on pages 89-90 7-37 odd (16 total)

Topic: Solving Equations by Adding & Subtracting HOMEWORK Topic: Solving Equations by Adding & Subtracting examples on pages 84-88 Assignment: Lesson 3.1 in book on pages 89-90 8-38 even (16 total)

3.2 Solving Equations by Multiplying or Dividing 1) Multiplication & Division are inverse operations (opposites). Whatever you do to one side of the equation must be done to the other side (this keeps both sides equal). 2) Multiplication Property of Equality = it states that if you multiply both sides of an equation by the same number, the sides remain equal.

Example 1 Solve = 14 x −4 (−𝟒)𝒙 −𝟒 =𝟏𝟒 (−𝟒) x = −56

Example 2 Solve 9y = 72 𝟗𝒚 𝟗 = 𝟕𝟐 𝟗 y = 8

Example 3 x = Joyce’s age 2x = Jon’s age x or = Eileen’s age 1 4 x 4 Write an expression for the ages of Jon and Eileen. Jon’s age is twice Joyce’s, and Eileen’s age is one-fourth of Joyce’s. x = Joyce’s age 2x = Jon’s age x or = Eileen’s age 1 4 x 4

Example 4 Find out how many nickels each person has if Sergei has 8 more than Evan, and Kimberly has 6 fewer than Evan. Together they have a total of 62 nickels. Use x as the variable. Evans = x Sergei = x + 8 Kimberly =x − 6 x + (x + 8) + (x − 6) = 62

Example 4 continued… Evans = x Sergei = x + 8 Kimberly =x − 6 x + (x + 8) + (x − 6) = 62 Evans = 20 Sergei = 20 + 8 = 28 Kimberly = 20 − 6 = 14 x + x + 8 + x − 6 = 62 3x + 2 = 62 −2 −2 3x = 60 /3 /3 x = 20

Topic: Solving Equations by Multiplying or Dividing HOMEWORK Topic: Solving Equations by Multiplying or Dividing examples on pages 93-96 Assignment: Lesson 3.2 in book on pages 96-97 9-20 all, 27-33 odd (16 total)

Topic: Solving Equations by Multiplying or Dividing HOMEWORK Topic: Solving Equations by Multiplying or Dividing examples on pages 93-96 Assignment: Lesson 3.2 in book on pages 96-97 9-20 all, 28-34 even (16 total)

3.3 Solving Two-Step Equations When solving equations with two or more terms: Determine the order in which the operations have been performed on the variable. Perform the “order of operations” in reverse on both sides of the equation in the following order: (1) add/subtract, (2) multiply/divide, (3) exponents, (4) grouping 1 STEP EQUATIONS 2 STEP EQUATIONS 1 TERM 2 TERMS 2n = 6 3x + 6 = 15

Example 1 2n + 7 = 25 PEMDAS: First n is multiplied by 2; then 7 is added to the product. OPPOSITE: Subtract 7 from both sides; then divide both sides by 2.

Example 1 2n + 7 = 25 −𝟕 −𝟕 𝟐𝒏=𝟏𝟖 𝟐𝒏 𝟐 = 𝟏𝟖 𝟐 𝒏=𝟗

OPPOSITE: Multiply both sides by 2; then add 3 to both sides. Example 2 n − 3 2 = 9 PEMDAS: First 3 is subtracted from n; then the difference is divided by 2. OPPOSITE: Multiply both sides by 2; then add 3 to both sides.

Example 2 n − 3 2 = 9 (𝟐)(𝒏 −𝟑) 𝟐 =𝟗(𝟐) 𝒏−𝟑=𝟏𝟖 +𝟑 +𝟑 𝒏=𝟐𝟏

Example 3 x 4 + 6 = 20 − 𝟔 −𝟔 𝒙 𝟒 =𝟏𝟒 (𝟒)𝒙 𝟒 =𝟏𝟒(𝟒) 𝒙=𝟓𝟔

Topic: Solving Two-Step Equations HOMEWORK Topic: Solving Two-Step Equations examples on pages 98-102 Assignment: Lesson 3.3 in book on pages 102-103 5-33 odd (15 total)

Topic: Solving Two-Step Equations HOMEWORK Topic: Solving Two-Step Equations examples on pages 98-102 Assignment: Lesson 3.3 in book on pages 102-103 6-34 even (15 total)

3.4 Simplifying Before Solving Like Terms – are terms that contain the same variables raised to the same power. Only the coefficients of like terms can be different. Operator + – Coefficient = a number by which a variable is multiplied 2n + 5 Constant = is a number whose value does not change (no variable) Terms = can be a number, variable, or a product of a number and variable (separated by an operator) Variable = symbol that represents a value (can be any letter in the alphabet)

Example 1 3x2 + 2x − 4 How many terms? 3 Any like terms? No What is the constant? 3 No –4

Example 2 2xy2 + 3xy − xy2 How many terms? 3 Any like terms? Yes What is the constant? 3 Yes None

Example 3 2a2b3c4 + 10 How many terms? 2 Any like terms? No What is the constant? 2 No 10

Example 4 Solve 2x − 9x = 343. –𝟕𝒙=𝟑𝟒𝟑 –𝟕𝒙 –𝟕 = 𝟑𝟒𝟑 –𝟕 𝒙=−𝟒𝟗

Example 5 Let s = the number of small sodas. The concession stand sold twice as many 16 oz. sodas as they did 12 oz. sodas and gave two large sodas (16 oz.) to the referees. If they used 30.5 gal. (3,904 oz.) of soda during the games, how many sodas of each size did they sell? Let s = the number of small sodas. 2s = the number of large sodas sold.

Example 5 Each small soda was 12 oz, each large soda was 16 oz, and the sodas sold plus the two large sodas for the referees totaled 3,904 oz. 12s + 16(2s) + 2(16) = 3,904 Let s = the number of small sodas. 2s = the number of large sodas sold.

Example 5 12s + 16(2s) + 2(16) = 3,904 Simplify the equation. Subtract 32 from both sides. 44s = 3,872 Divide both sides by 44. s = 88

Topic: Simplifying Before Solving HOMEWORK Topic: Simplifying Before Solving examples on pages 104-106 Assignment: Lesson 3.4 in book on page 107 11-37 odd (14 total)

Topic: Simplifying Before Solving HOMEWORK Topic: Simplifying Before Solving examples on pages 104-106 Assignment: Lesson 3.4 in book on page 107 12-38 even (14 total)

3.5 Using Equations R.E.S.T. Read – the problem carefully (2x-3x). Evaluate – how to represent the words with an algebraic equation (look for total & variable). Solve – the equation. Try again – by plugging in the answer (for the variable) to verify if it is correct.

3.5 Using Equations

Example 1 A contractor charges $50 for a service call plus $60 per hour for his work. How long will he have to work to earn $350? Let x = the number of work hours $50 + $60x = $350

Example 1 50 + 60x = 350 − 50 − 50 60x = 300 60 x = 5 hours

Example 2 Jackie is at an airport and walks at a rate of 6 ft./sec. on a moving walkway traveling 10 ft./sec. How long will it take her to go 160 ft.? d = s • t d = 160 s = 6 + 10 s = 16 ft./sec.

Example 2 d = s • t 160 = 16t 160 16 16t 16 = t = 10 seconds

Example 3 120 feet of fence is needed to enclose a rectangular garden that is 20 feet wide. What is the length of the garden? x 20 Let x = the length 2x + 2(20) = 120

Example 3 2x + 2(20) = 120 2x + 40 = 120 − 40 − 40 2x = 80 2 x = 40 feet

Topic: Using Equations HOMEWORK Topic: Using Equations examples on pages 110-112 Assignment: Lesson 3.5 in book on page 113 1-12 all (12 total)

3.7 Solving Linear Inequalities Natural Numbers – have no negative numbers and no fractions and are defined as N = {0, 1, 2, 3, ...} Integers – are positive and negative whole numbers, which are numbers that are not fractions or decimals. Inequality – is the relation between two quantities that may or may not be equal (≤ ≥ ≠ < > ).

3.7 Solving Linear Inequalities Number Lines Less Than Greater Than (goes left) (goes right)

3.7 Solving Linear Inequalities RULES always read inequalities starting from the variable if the alligator is eating the variable, it is the greater than sign if the alligator is not eating the variable, it is the less than sign if > or < the circle is an empty circle (not filled in) if ≥ or ≤ the circle is a solid dot (filled in) when both sides are multiplied or divided by a negative number, you must flip the inequality sign

Example 1 Graph x > 6. 2 3 4 5 6 7 8

Example 2 Graph x ≤ −3. −5 −4 −3 −2 −1 1

Example 3 Graph x ≠ 2. −2 −1 1 2 3 4

Example 4 Graph x = 3. −3 −1 1 3 5 −2 2 4

Example 5 Solve x − 5 ≥ −2 and graph. + 5 + 5 x ≥ 3 1 2 3 4 5 6

Example 6 Solve 15 ≤ −5x − 25 and graph. +25 +25 40 ≤ −5x −5 −5 −8 ≥ x +25 +25 REMINDER: when both sides are multiplied or divided by a negative number, you must flip the inequality sign 40 ≤ −5x −5 −5 −8 ≥ x −10 −9 −8 −7 −6 −5 −4

Example 7 x 4 Solve − + 7 ≥ 15 and graph. − 7 − 7 x 4 − ≥ 8 x 4 − 7 − 7 − ≥ 8 x 4 −4( ) (8)−4 − ≥ x 4 x ≤ −32 −35 −34 −33 −32 −31 −30 −29

Topic: Solving Linear Inequalities HOMEWORK Topic: Solving Linear Inequalities examples on pages 120-124 Assignment: Lesson 3.7 in book on pages 124-125 7-20 graph all (14 total)

3.8 Using Inequalities Key Terms more than = Greater Than > at least = Greater Than or Equal To ≥ less than = Less Than < at most = Less Than or Equal To ≤

Example 1 n + 6 > 4 –6 –6 n > –2 –3 –2 –1 Write an inequality, solve, and graph: A number increased by six is more than four. n + 6 > 4 –6 –6 n > –2 –3 –2 –1

Example 2 Write an inequality, solve, and graph: Four added to a number is at least negative five. n + 4 ≥ –5 –4 –4 n ≥ –9 –10 –9 –8

Example 3 Write an inequality, solve, and graph: Seven times a number is less than thirty-five. 7n < 35 7 7 n < 5 4 5 6

Example 4 Write an inequality, solve, and graph: Three multiplied by a number is at most negative twenty-seven. 3n ≤ –27 3 3 n ≤ –9 –10 –9 –8

Example 5 250n ≥ 1000 250 250 n ≥ $4 per watermelon 3 4 5 Write an inequality, solve, and graph: Mr. Acker wants to sell 250 watermelons and make at least $1,000. How much should he sell each of them for? 250n ≥ 1000 250 250 n ≥ $4 per watermelon 3 4 5

Topic: Using Inequalities 1-10 equation, solve, & graph all (10 total) HOMEWORK Topic: Using Inequalities examples on pages 126-129 Assignment: Lesson 3.8 in book on page 129 1-10 equation, solve, & graph all (10 total)

3.8 Using Inequalities R.E.S.T. Read – the problem carefully. Evaluate – how to represent the words with an algebraic equation. Solve – the equation. Try again – by plugging in the answer (for the variable) to verify if it is correct. RULES always read inequalities starting from the variable if the alligator is eating the variable, it is the greater than sign if the alligator is not eating the variable, it is the less than sign if > or < sign, the circle on the line is an empty circle (not filled in) if ≥ or ≤ sign, the circle on the line is a solid dot (filled in) when both sides are multiplied or divided by a negative number, you must flip the inequality sign