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## Chapter 1: Equations and inequalities

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**Chapter 1:Equations and inequalities**BIG IDEAS: Use properties to evaluate and simplify expressions Use problem solving strategies and verbal models Solve linear and absolute value equations and inequalities**What is the difference between a daily low temperature of -5**and a daily high temperature of 18?**Essential question**How are addition and subtraction related and how are multiplication and division related?**Opposite: additive inverse: the opposite of b is –b**• Reciprocal: multiplicative inverse: the reciprocal of a = 1/a. VOCABULARY**5**Graph the real numbers – and 3 on a number line. 4 5 Note that –= –1.25. Use a calculator to approximate 3 to the nearest tenth: 4 3 1.7. (The symbol means is approximately equal to.) 5 So, graph – between –2 and –1, and graph 3 between 1 and 2, as shown on the number line below. 4 EXAMPLE 1 Graph real numbers on a number line SOLUTION**1**7 + 4 = 4 + 7 13 = 1 13 EXAMPLE 3 Identify properties of real numbers Identify the property that the statement illustrates. SOLUTION Commutative property of addition SOLUTION Inverse property of multiplication**a + (2 – a)**= a + [2 + (– a)] EXAMPLE 4 Use properties and definitions of operations Use properties and definitions of operations to show that a + (2 – a) = 2. Justify each step. SOLUTION Definition of subtraction = a + [(– a) + 2] Commutative property of addition = [a + (– a)] + 2 Associative property of addition = 0 + 2 Inverse property of addition Identity property of addition = 2**(2 3) 9 = 2 (3 9)**15 + 0 = 15 for Examples 3 and 4 GUIDED PRACTICE Identify the property that the statement illustrates. SOLUTION Associative property of multiplication. SOLUTION Identityproperty of addition.**4(5 + 25) = 4(5) + 4(25)**1 500 = 500 for Examples 3 and 4 GUIDED PRACTICE Identify the property that the statement illustrates. SOLUTION Distributive property. SOLUTION Identityproperty of multiplication.**b (4 b) = 4 whenb = 0**1 1 1 b (4 b) = b (4 ) b b b = b ( 4) = (b ) 4 = 1 4 for Examples 3 and 4 GUIDED PRACTICE Use properties and definitions of operations to show that the statement is true. Justify each step. SOLUTION Def. of division Comm. prop. of multiplication Assoc. prop. of multiplication Inverse prop. of multiplication = 4 Identity prop. of multiplication**3x + (6 + 4x) = 7x + 6**3x + (6 + 4x) = 3x + (4x + 6) for Examples 3 and 4 GUIDED PRACTICE Use properties and definitions of operations to show that the statement is true. Justify each step. SOLUTION Comm. prop. of addition = (3x + 4x) + 6 Assoc. prop. of addition Combine like terms. = 7x + 6**Essential question**How are addition and subtraction related and how are multiplication and division related? They are inverses of one another. Subtraction is defined as adding the opposite of the number being subtracted. Division is defined as multiplying by the reciprocal of the divisor.**Jill has enough money for a total of 32 table decorations**and wall decorations. If n is the number of table decorations, write an expression for the number of wall decorations she can buy.**Essential question**When an expression involves more than one operation, in what order do you do the operations?**Power: an expression formed by repeated multiplication of**the same factor • Variable: a letter that is used to represent one or more numbers • Term: each part of an expression separated by + and – signs • Coefficient: the number that leads a variable • Identity: a statement that equates to two equivalent expressions VOCABULARY**= (–5) (–5) (–5) (–5)**(–5)4 = –(5 5 5 5) –54 EXAMPLE 1 Evaluate powers = 625 = –625**EXAMPLE 2**Evaluate an algebraic expression Evaluate –4x2– 6x+ 11 when x = –3. = –4(–3)2– 6(–3) + 11 –4x2– 6x+ 11 Substitute –3 for x. = –4(9) – 6(–3) + 11 Evaluate power. = –36 + 18 + 11 Multiply. = –7 Add.**Write an expression that shows your profit from**selling ccandles. EXAMPLE 3 Use a verbal model to solve a problem Craft Fair You are selling homemade candles at a craft fair for $3 each. You spend $120 to rent the booth and buy materials for the candles. • Find your profit if you sell 75 candles.**–**– 3 c 120 EXAMPLE 3 Use a verbal model to solve a problem SOLUTION STEP1 Write: a verbal model. Then write an algebraic expression. Use the fact that profit is the difference between income and expenses. An expression that shows your profit is 3c –120.**Your profit is $105.**ANSWER EXAMPLE 3 Use a verbal model to solve a problem STEP2 Evaluate: the expression in Step 1 when c = 75. 3c –120 = 3(75) – 120 Substitute 75 for c. = 225 – 120 Multiply. = 105 Subtract.**63**–26 for Examples 1, 2, and 3 GUIDED PRACTICE Evaluate the expression. SOLUTION 216 SOLUTION –64**(–2)6**5x(x –2) when x = 6 for Examples 1, 2, and 3 GUIDED PRACTICE SOLUTION 64 SOLUTION 120**3y2 – 4ywhen y = – 2**(z + 3)3when z = 1 for Examples 1, 2, and 3 GUIDED PRACTICE SOLUTION 20 SOLUTION 64**Your profit is $285.**ANSWER What If?In Example 3, find your profit if you sell 135 candles. for Examples 1, 2, and 3 GUIDED PRACTICE**8x + 3x**5p2 + p – 2p2 3(y + 2) – 4(y – 7) EXAMPLE 4 Simplify by combining like terms = (8 + 3)x Distributive property = 11x Add coefficients. = (5p2– 2p2) + p Group like terms. = 3p2 + p Combine like terms. = 3y + 6 – 4y + 28 Distributive property = (3y – 4y) + (6 + 28) Group like terms. = –y + 34 Combine like terms.**2x – 3y – 9x + y**EXAMPLE 4 Simplify by combining like terms = (2x – 9x) + (– 3y + y) Group like terms. = –7x – 2y Combine like terms.**for Example 5**GUIDED PRACTICE 8. Identify the terms, coefficients, like terms, and constant terms in the expression 2 + 5x – 6x2 + 7x – 3. Then simplify the expression. SOLUTION Terms: 2, 5x, –6x2 , 7x, –3 5 from 5x, –6 from –6x2 , 7 from 7x Coefficients: Like terms: 5xand 7x, 2 and –3 Constants: 2 and –3 Simplify: –6x2 +12x – 1**15m – 9m**2n – 1 + 6n + 5 for Example 5 GUIDED PRACTICE Simplify the expression. SOLUTION 6m SOLUTION 8n + 4**2q2 + q – 7q – 5q2**3p3 + 5p2–p3 for Example 5 GUIDED PRACTICE SOLUTION 2p3 + 5p2 SOLUTION –3q2– 6q**–4y –x + 10x + y**8(x – 3) – 2(x + 6) for Example 5 GUIDED PRACTICE SOLUTION 6x – 36 SOLUTION 9x –3y**Essential question**When an expression involves more than one operation, in what order do you do the operations? Order of operations: Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction**On a blank sheet of paper, complete #1-13 ODD on P16 in the**blue Quiz section. Please turn into the homework bin when finished.**Essential question**What are the steps for solving a linear equation?**Equation: a statement that two expressions are equal**• Linear equation: may be written in the form ax + b = 0; no exponents • Solution: a number that makes a true statement when substituted into an equation • Equivalent equations: two equations that have the same solutions VOCABULARY**Solve**x + 8 = 20. x + 8 = 20 x = (12) x = 12 4 5 Multiply each side by , the reciprocal of . 4 5 The solution is 15. ANSWER CHECK x = 15 in the original equation. 4 4 4 4 5 4 5 4 5 5 5 5 (15) + 8 = 12 + 8 = 20 x+ 8 = EXAMPLE 1 Solve an equation with a variable on one side Write original equation. Subtract 8 from each side. x = 15 Simplify.**Restaurant**During one shift, a waiter earns wages of $30 and gets an additional 15% in tips on customers’ food bills. The waiter earns $105. What is the total of the customers’ food bills? Write a verbal model. Then write an equation. Write 15% as a decimal. EXAMPLE 2 Write and use a linear equation SOLUTION**ANSWER**The total of the customers’ food bills is $500. EXAMPLE 2 Write and use a linear equation 105 = 30 + 0.15x Write equation. 75 = 0.15x Subtract 30 from each side. 500 = x Divide each side by 0.15.**3.**– x + 1 = 4 The solution is x = 3. The solution is x = 4. The solution is 5. ANSWER ANSWER ANSWER 3 5 for Examples 1 and 2 GUIDED PRACTICE Solve the equation. Check your solution. 1. 4x + 9 = 21 2. 7x – 41 = – 13**ANSWER**The agent must sell $950,000 in a year to each $ 60000 for Examples 1 and 2 GUIDED PRACTICE 4. REAL ESTATE A real estate agent’s base salary is $22,000 per year. The agent earns a 4% commission on total sales. How much must the agent sell to earn $60,000 in one year?**ANSWER**The correct answer is D EXAMPLE 3 Standardized Test Practice SOLUTION 7p + 13 = 9p – 5 Write original equation. 13 = 2p – 5 Subtract 7pfrom each side. 18 = 2p Add 5 to each side. 9 = p Divide each side by 2.**7(9) + 13 9(9) – 5**63 + 13 81 – 5 ? ? = = EXAMPLE 3 Standardized Test Practice CHECK 7p+ 13 = 9p– 5 Write original equation. Substitute 9 for p. Multiply. 76 = 76 Solution checks.**x =**2 2 ANSWER The solution 5 5 EXAMPLE 4 Solve an equation using the distributive property Solve 3(5x – 8) = –2(–x + 7) – 12x. 3(5x – 8) = –2(–x + 7) – 12x Write original equation. 15x – 24 = 2x – 14 – 12x Distributive property 15x – 24 = – 10x – 14 Combine like terms. 25x – 24 = –14 Add 10xto each side. 25x = 10 Add 24 to each side. Divide each side by 25 and simplify.**3(5 – 8) –2(– + 7) – 12**24 3(–6) –14 – 5 4 5 Substitute for x. 2 2 2 5 5 5 2 ? ? = = 5 EXAMPLE 4 Solve an equation using the distributive property CHECK Simplify. – 18 = – 18 Solution checks.**Car Wash**It takes you 8 minutes to wash a car and it takes a friend 6 minutes to wash a car. How long does it take the two of you to wash 7 cars if you work together? STEP 1 Write a verbal model. Then write an equation. EXAMPLE 5 Solve a work problem SOLUTION