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Chapter 1 Expressions, Equations, and Functions

Chapter 1 Expressions, Equations, and Functions

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Chapter 1 Expressions, Equations, and Functions

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  1. Chapter 1Expressions, Equations, and Functions

  2. Sec 1.1Evaluate Expressions

  3. Variable – a letter used to represent one or more numbers • EX: x • Algebraic Expression – a collection of numbers, variables, and operations without an equal sign. • EX: 5x • EX: 16 / b • EX: 7 + y • EX: a – 17 • EX: 5x + 2

  4. Evaluate an Expression – to substitute a number for each variable and simplify • EX: Evaluate the expression. • 0.6x when x = 4 • 24 / y when y = 6 • (2/3)a when a = ½ • b – ½ when b = 5/6

  5. EX: Evaluate the Expression • You and your friend are going to see a movie. You pay for both admissions. Your total cost (in dollars) can be represented by the expression 2a, where a is the cost of admission. If each admission costs $7.75, what is your total cost?

  6. Power – an expression that represents repeated multiplication of the same factor. • EX: 43 • Base – the big, bottom number • Exponent – the small number that represent the number of times the base is being multiplied by itself. • EX:

  7. EX: Write the power in words and as a product. • 91 • n4 • (1/2)3 • (0.4)2

  8. EX: Evaluate the expression. • x3 when x = 8 • k2 when k = 2.5 • d4 when d = 1/3

  9. Exponents are used in the formulas for area and volume. • Area of a Square: A = s2 • Volume of a Cube = V = s3

  10. EX: • Each edge of the storage cube shown is 14 inches long. The storage cube is made so that it can be folded flat when not in use. Find the volume of the storage cube. • Find the area of the storage cube if it is folded flat.

  11. Sec 1.2Apply Order of Operations

  12. Order of Operations - the correct order for evaluating an expression • Order of Operations: • Parentheses – evaluate expressions inside the parentheses/grouping symbols • Exponents – evaluate all powers • Multiply and Divide – left to right • Add and Subtract – left to right • PEMDAS – “Please Excuse My Dear Aunt Sally”

  13. EX: Evaluate the expressions. • 27 ÷ 32· 2 – 3 • 20 – 42 • 2 · 32 + 4 • 32 ÷ 23 + 6 • 15 + 62 - 4

  14. Grouping Symbols • Parentheses ( ) • Brackets [ ] • Fraction Bar • Note: • Always start with the innermost grouping symbol and work your way out. • Always follow the order of operations inside each individual grouping symbol. • For a fraction bar, do what is above and below it before you divide.

  15. EX: Evaluate each expression. • 4(3 + 9) • 3(8 – 22) • 2[(9 + 3) ÷ 4]

  16. EX: Evaluate each expression when y = 8. • y2 – 3 • 12 – y – 1 • 10(y + 1) y - 3

  17. EX: • You join an online music service. The total cost (in dollars) of downloading 3 singles at $.99 each and 2 albums at $9.95 each is given by the expression 3(0.99) + 2(9.95). • Find the total cost. • You have $25 to spend. How much will you have left?

  18. Sec 1.3Write Expressions

  19. To translate verbal phrases into mathematical expressions, look for KEY WORDS. • Addition Key Words • Sum • Plus • Total • More than • Increased by • EX: The sum of 8 and a number x

  20. Subtraction Key Words • Difference • Less than *Subtract but reverse the order • Minus • Decreased by • EX: 7 less than a number y • NOTE: Order matters!

  21. Multiplication Key Words • Times • Product • Multiplied by • Of • EX: ½ of a number z

  22. Division Key Words • Quotient • Divided by • Divided into • EX: The quotient of 6 and a number a. • NOTE: Order matters!

  23. NOTE: Anytime you multiply or divide a sum or difference by something, you must put the sum or difference in parentheses. • EX: 3 times the sum of 7 and a number y

  24. EX: Write an expression for the situation. • A piece of rope L feet long is cut from a rope 10 feet long. Write an expression for the length of the remaining piece.

  25. Each person’s share if p people are to divide $90 evenly.

  26. EX: Translate the verbal phrase into an expression. • Three more than half of a number x • Product of four and a number y • 4 less than 6 times a number n

  27. The difference of 22 and the square of a number m • The quotient when the quantity 10 plus a number x is divided by 2

  28. Rate – a fraction that compares two quantities measured in different units. • EX: 100 miles / 2 hours • Unit Rate – when the denominator of the fraction is 1 unit. • EX: 50 miles / 1 hour • EX: 50 mph • Note: • Divide to turn a rate into a unit rate. • When comparing unit rates, make sure your units are the same.

  29. EX: Tell which rate is greater. • 1 1/8 miles in 1 minute and 30 seconds, or 1 3/16 miles in 1 minute and 15 seconds. • $3.50 for 25 ounces, or $4.75 for 40 ounces

  30. EX: Find the unit rate in feet per second. • Note: Multiply by a “form of one” to convert the units. • A car travels 150 miles in 3 hours. • 600 yards/1 minute

  31. EX: • You are ordering Detroit Lions tickets online. Each ticket costs $55 and there is a $6 charge no matter how many tickets are ordered. Write an expression for the cost (in dollars) of ordering the tickets. Then find the total cost if you order 8 tickets.

  32. EX: • Suppose you run 3 miles and then walk a cool down for 15 minutes at a rate of 0.1 miles per 90 seconds. What total distance will you cover?

  33. Sec 1.4Write Equations and Inequalities

  34. Equation – a mathematical sentence formed by placing the equal symbol (=) between two expressions. • EX: 2x + 6 = 18 • Inequality – a mathematical sentence formed by placing one of the symbols <, >, ≤, or ≥ between two expressions. • EX: 3x – 8 < 10 • Inequalities compare two numbers or expressions that are not necessarily equal.

  35. Symbols • = is equal to • < is less than • > is greater than • ≤ is less than or equal to • ≥ is greater than or equal to

  36. Combining Inequalities • Two individual inequalities can be combined to form one compound inequality. • EX: x > 8 and x < 15 is the same as 8 < x < 15 Read “x is greater than 8 and less than 15”

  37. EX: Write an equation or inequality. • The difference of twice a number k and 8 is 12. • The product of 6 and a number n is at most 24.

  38. The sum of y and 1 is no less than 5 and no more than 13. • The quotient of a number p and 12 is at least 30.

  39. To find a solution to an equation or inequality: • Substitute in the given value. • Simplify each side. • If the resulting statement is true, the number is a solution of the equation/inequality. If false, it is not a solution.

  40. EX: Check whether the number is a solution of the equation or inequality. • 9 – x = 4; 5 • r/3 – 4 > 4; 12 • 2n + 3 ≥ 21; 9

  41. EX: Solve the equation using mental math. • Think “What number will make this equation true?” • m + 6 = 11 • 5x = 40 • y/4 = 10

  42. EX: Write an equation or inequality and solve. • Your student senate budget is $300. You want to buy the members t-shirts for $6 each. Write an inequality that represents the number of shirts you can buy without going over budget.

  43. You have made $450 so far at your after school job. Your goal is to make $1000. How much more do you need to make?

  44. Sec 1.5Use a Problem Solving Plan