chapter 1: expressions, equations, & inequalities sections 1.3 – 1.6 1

Chapter 1: Expressions, Equations, & Inequalities

Sections 1.3 – 1.6

1

1.3 Algebraic Expressions

Algebraic Expression: contains numbers, variables, and mathematical signs (no equal sign)

Equation: contains numbers, variables, mathematical signs, and an EQUAL SIGN

2


Write an algebraic expression

1. one less than the product of six and w

6w – 1

3


2. You are on a bicycle trip. You travel 52 miles on the first day. Since then, your average rate has been 12 miles per hour. What algebraic expression models the distance traveled? Let h be the number of hours traveled.

52 + 12h

4


Evaluate the following expressions

3. 2r + 5(s+6) – 1 if r = 3, s = – 9

2(3) + 5(– 9+6) – 1

2(3) + 5(–3) – 1

6 + – 15 – 1

– 9 – 1

– 10

5


4. c³ - d/8 if c = ¼ , d = 1

(¼)³ – 1/8

1/64 – 1/8

1/64 – 8/64

– 7/64

6


5. Tickets to a museum are $8 for adults, $5 for children, and $6 for seniors

a.) What algebraic expression models the total number of dollars collected in ticket sales?

8a + 5c + 6s

7


b.) If 20 adults, 16 children, and 10 senior tickets are sold one morning, how much money is collected in all?

8(20) + 5(16) + 6(10)

160 + 80 + 60

300

8


Simplify

6. 2a² + 3b² + 6b² + 5a²

7a² + 9b²

9


7. –(x + 4y) + 5(3x – y)

– x – 4y + 15x – 5y

14x – 9y

Assign pgs: 22 – 23, #10 – 19, 20 – 26 even, 30 – 44 even, 52

(23 problems)

10

1.4 Solving Equations

Reflexive: a = a Symmetric: if a = b then b = a Transitive: if a = b and b = c, then a = c Addition: if a = b then a + c = b + c Subtraction: if a = b then a - c = b – c Multiplication: if a = b then a(c) = b(c) Division: if a = b then a ÷ c = b ÷ c

11


Solve the following equations

1. x – 8 = -10

+8 +8

x = – 2

12


2. – 2(y – 1) = -16 + y – 2y + 2 = – 16 + y +2y +2y 2 = – 16 + 3y +16 +16

18 = 3y 3 3 y = 6Assign Pg. 23 – 24, #53, 55, 63 – 66

Pg. 30, #10 – 24 even(14 problems)13

1.4 Solving Equations Cont’d

Solve

1. 1 + 5x -6 = 6x – 5 – x

5x – 5 = 5x – 5

– 5x – 5x

– 5 = – 5 which means…infinite number of solutions or all real numbers

14


2. –x + 2(5x – 1) = 2(3x+4) + x

– x + 10x – 2 = 6x + 8 + x

9x – 2 = 7x + 8

– 7x – 7x

2x – 2 = 8

+ 2 +2

2x = 10

x = 515


3. What is t in terms of A in A = 1000(1+0.05t) A = 1000 + 50t

– 1000 – 1000 A – 1000 = 50t 50 50 t = A – 20 50

16


4. Solve A = ½ (b + c) for b 2(A) = 2 (½)(b + c) 2A = b + c – c – c

2A - c = b b = 2A – c

Assign pgs: 30–31, #28 – 36, 38, 41, 42, 46, 48, 49, 6116 problems

17

1.5 Part 1 Solving Inequalities

Transitive: if a > b and b > c, then a > c Addition: if a > b then a + c > b + c Subtraction: if a > b then a - c > b – c Multiplication: if a > b and c > 0 then a(c) > b(c)

if a > b and c < 0 then a(c) < b(c) Division: if a > b and c > 0 then a ÷ c > b ÷ c

if a > b and c < 0 then a ÷ c < b ÷ c

18


*If you multiply or divide by a negative number, FLIP THE ARROW!

Graphing:

19

>, < mean open dots

≥, ≤ mean closed dots

Graph x > 3.

Graph 3 < x.

Graph 4 < x.


20

3

3

4


1. Solve the inequality and graph the solution.

4(x – 7) > −20

4x – 28 > –20

+28 +28

4x > 8

4 4

x > 2

21 2


2. 4(−n – 2) – 6 >18

– 4n – 8 – 6 > 18

– 4n – 14 > 18

+ 14 +14

– 4n > 32

– 4 – 4

n < – 8

22 −8


Solve.

3. 3(x + 3) ≥ 4(2 + x)

3x + 9 ≥ 8 + 4x

– 3x – 3x

9 ≥ 8 + x

1 ≥ x which can also be

written as x ≤ 1Assign pgs.38-40: #14-23 all, 68,69,71-78 all

Reminder: QUIZ (1.3 – 1.4) TOMORROW!!!!

23

1

1-5 Part 2 Solving Inequalities

4. What inequality represents the sentence?

a.5 fewer than the product of seven and a number is no more than 50.

7n – 5 < 50

24


What inequality represents the sentence?

b.The quotient of a number and 6 is at least 10.

25

106

n


5. −½(y + 3) ≥ 1/3y – 4

26


27 y ≤ 3

5 (cont’d)

5 (cont’d)


28

y < 3

Assign pgs 38 – 39: # 10-13 all, 24,27,45,46 8 problems


5. −½(y + 3) ≥ 1/3y – 4

29

43

1

2

3

2

1

yy

43

1

1

6

2

3

2

1

1

6yy

–3y – 9 ≥ 2y – 24 –2y –2y


5 (cont’d)

–5y – 9 ≥ –24

+9 +9

–5y ≥ –15

y ≤ –3

30

Assign pgs 38 – 39: # 10-13 all, 24,27,45,46 8 problems


Solve

6. 9 – x – 5 < -x + 4

– x + 4 < – x + 4

+ x + x

4 < 4 which means…

No Solution

31


Solve

7. 9 – x – 5 ≤ − x + 4

– x + 4 ≤ – x + 4

+ x + x

4 ≤ 4 which means…

All Real Numbers

32


Compound Inequality:

Two inequalities joined together by the word “and” or the word “or”

33


“and”

The solution must be true for both inequalities at the same time. (usually shades in the middle)

34


2(½a) < 2(3)

a < 6

– 3a + 5 < 8

−5 −5

– 3a < 3

– 3 – 3

a > – 1

35

8. ½a < 3 and – 3a + 5 < 8

a < 6 and


36

8. (cont’d) ½a < 3 and – 3a + 5 < 8

– 1 < a

a < 6 and a > − 1

a < 6

Smallest number

− 1 6

This is the solution!!


“or”

The solution will make any or all parts of the inequalities true. (usually shades to the outside)

37


9. ½a > 3 or – 3a + 5 > 8

38− 1 6

a > 6 or a < − 1

All of this is the solution!!!

All of this is the solution!!!


Now try these problems on your own!

Solve and graph.

10. 5x ≥ −15 and 2x < 4

11.−2x > 10 or x + 6 ≥ 7

Assign: p.38-40 #29-43 odd, 47

39


12. 1 < 2x + 3 < 9

− 3 − 3 − 3

− 2 < 2x < 6

2 2 2

− 1 < x < 3

40

− 1 3


Assign:

p.38-40 #28-42 even, 55,59,67

41

1.6 Absolute Value Equations

Absolute value: the distance from 0 on a number line

│5 │= 5 │−5 │= 5

Notice that either a number OR its opposite have the same absolute value.

42


To Solve Absolute Value Equations:1. Get the absolute value on a side by itself.2. Set the expression inside the absolute bars

equal to its value (the number on the other side).3. Set the opposite of the expression inside the

absolute bars equal to its value (the number on the other side).

4. Solve and check.

43


x = 5 − x = 5

− 1 − 1

x = −5

44

1. Solve.|x| = 5

x = 5,− 5

SOLUTION

x = ± 5


Solve.

2. │2x + 5 │= 9

45

2x + 5 = 9 2x = 4 x = 2

−(2x + 5) = 9 −2x − 5 = 9 − 2x = 14 x = − 7


3. ½│2x − 4 │ − 2 = 6

+2 +2

½│2x − 4 │= 8

2 ∙ (½│2x − 4 │) = 2 ∙ (8)

│2x − 4 │= 16

Continued on next slide…

46


3. continued │2x − 4 │= 16

47

2x – 4 = 16 2x = 20 x = 10

− (2x – 4) = 16 −2x + 4 = 16 −2x = 12 x = −6

x = 10, − 6


4. |3x| = −9

48

3x = − 9 x = − 3

− 3x = −9 x = 3

NO SOLUTION!!!

WHY ???????????


Assignment

pgs.46 #10 – 18 all, 22

49

1-6 Part 2 (Abs. Value)

Less than (and)

Greater (or)50

an an

er or


5. |x| < 5

51

− 5 5

− 5 < x x < 5 x < 5 AND x > − 5

x < 5 AND ‒x < 5


6. |x| > 5

52

x > 5 OR − x > 5x < − 5

− 5 5

x > 5 OR x < − 5


2x + 6 > 10

2x > 4

x > 2

−(2x + 6) > 10

−2x − 6 > 10

−2x > 16

x < −8

53

7. 2│2x + 6 │+ 6 ≥ 26 2│2x + 6 │ ≥ 20 │2x + 6 │ ≥ 10

OR

x < −8 OR x > 2

2−8


4x + 3 < 5

4x < 2

x < ½

− (4x + 3) < 5

− 4x − 3 < 5

− 4x < 8

x > − 2

54

8. Solve and graph. │4x + 3 │< 5

AND

− 2 < x x < ½

−2 1/2


Assignment:

pgs.46 #23, 25 – 36 all

55

chapter 1: expressions, equations, & inequalities sections 1.3 – 1.6 1

Documents

y slide

graph x

c b c multiplication

contd slide

c d8if c

c subtraction

c addition

c assign pgs