1.1 Real Numbers and Number Operations The Real Number System Create a Venn Diagram that represents the real number system, be sure to include the following types of numbers: real, irrational, rational, integers, wholes, and natural/counting. Properties of Real Numbers Let a, b and c be real numbers. Property Commutative Associative Identity Addition a + b = b + a (a + b) + c = a + (b + c) a + 0 = a ; 0 + a = a Inverse a + (-­‐a) = 0 Distributive Property Identify the property illustrated. a.) 14 + 7 = 7 + 14 1
b.) 5⋅ = 1 5
2 ⎛ 2⎞
e.) + ⎜ − ⎟ = 0 3 ⎝ 3⎠
d.) 5 ( x + 2) = 5x + 10 Multiplication a · b = b · a (ab) · c = a · (bc) a ·1 = a , 1 · a = a 1
a ⋅ = 1 ; (a ≠ 0) a
a(b + c) = ab + ac c.) (5 + 3) + 2 = 5 + (3 + 2) f.) 1 · 5 = 5 Unit Analysis English Units 1 ft = 12 in 1 yd = 3 ft 1 mi = 5280 ft 1 lb = 16 oz 1 gal = 4 qt Conversion Reference Table Metric Units English / Metric Units 1 m = 100 cm 1 in = 2.54 cm 1 cm = 10 mm 1 mi = 1.61 km 1 kg = 1000 g 10 km = 6.2 mi 1 L = 1000 mL lb = 454 g 1 L = 1.057 qt Try the following conversions. 1.) 2.45 mi =_____________ ft 3.) 470 mi = _____________ km 5.) 526 yds/sec = _____________ mi/hr 7.) 10 gal = _____________ mL 2.) 36 g = _____________ kg 4.) 200 mi/hr = _____________ ft/sec 6.) 75.0 kg = _____________ lbs. 8.) 1.43 kg/L =_____________g/mL 1.1
Real Numbers and Number Operations Homework
1. 2. 3. 4. Give an example of a number that is both a rational number and a whole number. Give an example of a number that is both an integer and a natural number (counting number). Give an example of a number that is a rational number, but NOT an integer. Name the property illustrated in each example below: a) 5(x + 7) = 5x + 35 ______________________________ b) (59 + 7) + 62 = 59 + (7 + 62) ______________________________ c) 10 i
1
= 1 10
______________________________ 3.8 + 1.2 + 2.7 = 1.2 + 3.8 + 2.7 d) ______________________________ e) 387 + 0 = 387 ______________________________ f) 4(3) = 3(4) ______________________________ g) 0 = 5 + (-­‐5) ______________________________ 5. Perform the following conversions using conversion factors. Show your set up. 86 inches = ____ ft 5.17 lb/gal = _____ lb/qt 2.4 g/mL = ____ lb/gal 3.4 km/hr = ______ mi/hr Use what you have learned about conversion factors to solve the following application problems. 6. The elevator in the Washington Monument takes 75 seconds to travel 500 ft to the top floor. What is the speed of the elevator in miles per hour? 7. When you drive across the border into Quebec, Canada, all the speed limit signs are in km/hour. How might you get a quick, reasonable approximation of the equivalent speed in mi/hour? 8. As a nurse, you must determine the proper dose of a medication for your patient who weighs 160 pounds. The amount of medication is 20 mg/kg. (mg of medication per the patient’s weight in kilograms) How many milligrams of medication would be a proper dose for your patient? 9. A blood donor gives 1 pint of blood each time he donates. He learns from the American Red Cross that he will be honored at a special dinner for outstanding donors for his lifetime donation of 6 gallons of blood. How many times has he donated blood? (There are 2 pints per 1 quart of any liquid.) 1.2 Algebraic Expressions and Models & 1.3 Solving Linear Equations Order of Operations -­‐ PEMDAS 1.) Undo grouping symbols – parentheses, absolute value, braces, brackets 2.) Evaluate any exponents / powers 3.) Do multiplication and division as they appear left to right 4.) Do addition and subtraction as they appear left to right Evaluate 1. -­‐8 + 5(1 – (-­‐3))3 2. (−3)4 Evaluate −3x 2 − 5x + 7 when x = −2 1. 2. Simplifying Algebraic Expressions 1. 2. 5n 2 (n + 1) − 3n 3 − 2n Solving Equations 1. 3. 5(x − 2) = −4(2x + 7) + x 3. −34 2x 3 + 3x 2 + 27 when x = −4 2[(3n + 1)2 − n] 2. 1
1
1
x + = x − 3
4
6
4. 4
1⎞
⎛2
x = 4 ⎜ x + ⎟ ⎝3
5
5⎠
2⎛
6⎞ 1
3x +
= (5x − 1) 3⎝
5⎠ 5
5. 1⎞
⎛
6x + 2 − 4x = 3(2x + 1) − 2 ⎜ 2x + ⎟ ⎝
2⎠
6. −5(2x + 3) = 2(4 − 3x) − 4x Word Problem Practice 1.) Find four consecutive even integers that total 92. 2.) Find three consecutive integers such that if the sum of the first two is decreased by the third, the result will be 68. 3.) Find three consecutive even integers such that twice the smallest plus three times the largest will equal the middle integer increased by 82. 4.) Mrs. Bunker paid her $7.15 grocery bill with a ten-­‐dollar bill and asked for her change in dimes and quarters. If she received 21 coins in change, how many quarters did she receive? 5.) 65 students and teachers from Penncrest High School are going to a play. Tickets costing $4 for teachers and $3.60 for students were purchased at a total price of $244. How many $4 tickets were bought? 6.) A stamp collector has a group of ten cent, fifteen cent and twenty cent stamps that total $4.75 in value. If he has twice as many twenty-­‐cent stamps as ten cent stamps and three fewer fifteen cent stamps than 10 cent stamps, how many twenty cent stamps does he have? 1.2 and 1.3 Homework
Evaluate the following expressions. 1.) −(3) 2.) −(−2)4 3.) −4 2 4.) (−2)4 5.) (5 − 2)3 ÷ 9 − 6 6.) ((3 − 1) ⋅ 2 + (−3))5 7.) −12 − (−2)2 + (4 − 5)4 8.) x 2 − 4xy when x = −2 and y = −3 9.) 6 − x 2 + x when x = −2 Simplify the following expressions. 10.) 5(n 2 + n) − 3(n 2 − 2n) 11.) 8(y − x) − 2(x − y) Solve the following equations. 12.) 6(2x − 1) + 3 = 6(2 − x) − 1 13.) 5x + 2 = 2(2x+1) + x 1
5
1
19
14.) −5(2x + 3) = 2(4 − 3x) − 4x 15.) x − = − x + 2
3
2
4
4
5⎞
2⎛6
7 ⎞ 17
⎛7
16.) 2 ⎜ x − ⎟ = −2 17.) − ⎜ x − ⎟ =
⎝5
3⎠
3⎝ 5
10 ⎠ 20
18.) Find three consecutive integers such that if three times the smaller is decreased by the sum of the other two, the difference will be 46. 19.) Nancy has a bag of coins totaling $3.60 in value. If she has two more nickels than quarters, and twice as many quarters as dimes, how many of each coin does she have? 1.4 Rewriting Equations and Formulas Warm up: Solve the equation 3
2 7
1
x + = x + 2
3 2
6
Example 1: Solve for y: 11x − 9y
Example 2: Solve for y: 6x +
3
y = 21 4
Example 3: Solve for y. xy + 2x
= −4 = 20 Solve each formula for the indicated or underlined variable. a.) P = 2l + 2 w A = π r 2 b.) 1
c.) A = bh (solve for h) 2
d.) E = mc 2 (solve for m) 9
e.) F = C + 32 5
f.) S = C − rC (solve for C) g.) F =
Gm1m2
(for m2) r2
h.) I =
1
i.) A = h (b1 + b2 ) for b2 2
k.) S = 2WH + 2WL + 2LH (for H) j.) E
(for R) R+ r
1 1
1
= + (for R2) R R1 R2
1.4 Homework: Rewriting Equations and Formulas 1.) A = P + Prt (solve for t ) 2.) PV = nRT (solve for R) 3.)
P1V1 P2V2
=
(Solve for P2 ) T1
T2
4.) S = V0 t −16t 2 ( Solve for V0 ) 1
5.) V = πr 2 h (Solve for h) 3
6.)
1 1 1
= + (Solve for b) f a b
7.) P =
R−C
(Solve for R) n
8.) S = 2πr 2 + 2πrH (Solve for H) 1.4 Homework Day 2 1.6 Solving Linear Inequalities Graph the following inequalities: -­‐ 4 < x “And” inequalities x ≥ -­‐3 and x ≤ 4 “Or” inequalities x ≥ 4 or x ≤ 0 5 ≥ x x < 3 and x ≤ 6 x > 10 or x ≤ 6 Solve and Graph: 1.) -­‐ 11 y − 9 ≥ 13 ____________________________________________________ 2.) 2 x + 1 ≤ 6 x − 1 ____________________________________________________ 3.) −12 < −3x − 3 < 15 ____________________________________________________ 4.) − 2 x + 7 ≤ 3 or 3 x + 5 ≤ 2 ____________________________________________________ 1.6 Homework Match the inequality with its graph. 1.) x ≥ 4 2.) x < 4 4.) x ≥ 4 or x < -­‐4 5.) -­‐4 ≤ x ≤ 4 3.) – 4 < x ≤ 4 6.) x > 4 or x ≤ -­‐4 Determine whether the given number is a solution to the inequality. 1
7.) − x − 2 ≤ −4 Is 9 a solution to this inequality? 3
8.) −8 < x − 11 < −6 Is 5 a solution to this inequality? Solve the inequality. Then graph your solution. 1
9.) 5 + x ≤ 6 3
10.) 5 – 5x > 4(3 – x) 11.) −5 ≤ −n − 6 ≤ 0 2
12.) −8 < x − 4 < 10 3
13.) 3x + 2 < −10 or 2x − 4 > −4 1
14.) 3n + 1 > 10 and
n − 1 > 3 2
15.) 2(x + 3) < 14 or − 5 − x < 1 1.7 Absolute Value Equations and Inequalities General Form: ax + b = c The solution set for this equation can be found by solving these two equations: ax + b = c
ax + b = −c Example: 5x + 2 = 27 General Form: ax + b > c The solution set for this general form is the union of two sets, namely, ax + b > c
or
ax + b < −c When you solve inequalities of this form, rewrite the original inequality as the union of two separate inequalities. Solve and graph the union of the solution sets. Write the solution set using set notation. Example: −3x + 4 ≥ 5 General Form: ax + b < c The solution set for this general form is the intersection of two sets, namely, −c < ax + b
and
ax + b < c which can be written more simply as − c < ax + b < c When you solve inequalities of this form, rewrite the original inequality as the union of two separate inequalities. Solve and graph the intersection of the solution sets. Write the solution set using set notation. Example: 4 x + 3 ≤ 7 Practice Problems: 1
1.) x − 3 = 2 2.) 2 x + 7 < 11 `2
3.) −3x + 10 > 7 4.) −x + 5 ≤ 6 5.) 2 x − 1 − 4 ≥ 2 6.) 4 2 y − 7 + 5 < 9 7.) 3 x − 5 < −1 9.) 2 x − 1 ≤ 0 10.) x − 3 > 0 8.) 3 x − 5 > −1