Quiz 15 - s3.amazonaws.com€¦Solve using addition-subtraction. Give answer in interval notation. 7d – 6d < 6d – 6d 7d < 6d 8. d < 0 (–∞,0) Linear Inequalities Addition-Subtraction

Quiz 15

•  Get out paper and pencil or pen – 8.5x11 sheet of paper; fold vertically

•  Put your name outside at top •  Put notes away

•  You will have 5 minutes to complete the problem

Quiz 15

•  Add the symbol in the blank to make true: –4 __ –5

3/4 __ 3/5 |–4| __ –|–4|

√5 __ |5|

|–9| __|9|

> > >

< =

Here is the solution:

Math 72 Section 4.1

Identify Linear Inequalities

Addition-Subtraction for Linear Inequalities

Graphs and Tables for Linear Inequalities

Example Problem

•  The perimeter of the triangle shown must be less than 26 cm. Find the possible values for a.

a cm 8 cm

8 cm

Linear Inequalities

Verbally Algebraically Example Graph A linear

inequality in one variable is an

inequality that is ____ degree in that variable. first

For real constants A, B, and C, A ≠ 0.

Ax + B > C

Ax + B ≥ C

Ax + B < C

Ax + B ≤ C

x > 2

x ≥ 2

x < 2

x ≤ 2

( 2

[ 2

] 2

) 2

Linear Inequalities

Which of these are linear inequalities?

Yes, a linear inequality. No, it is an expresssion. No, the exponent is 2. No, it is an equation.

3x +1 < 10 3x + 1

3x2 + 1 > 10 3x + 1 = 10

1.

Linear Inequalities

•  A conditional inequality contains a variable and is true for _____, but not all, real values of the variable.

•  The solution of a linear inequality consists of all values that _____ the inequality. The solution of a conditional inequality will be an interval that contains an infinite set of values.

some

satisfy

Linear Inequalities

Determine whether x = 5 satisfies each inequality.

x < 5 x ≤ 5 x > 5 x ≥ 5 5 < 5 false

5 ≥ 5 true

5 > 5 false

5 ≤ 5 true

2.

Linear Inequalities

Determine whether either 4 or –4 satisfies the inequality 6x – 2 < 5x – 4

x = 4 6x – 2 < 5x – 4

x = –4 6x – 2 < 5x – 4

3.

6(4) – 2 < 5(4) – 4 22 < 16 False

x = 4 is not a solution

6(–4) – 2 < 5(–4) – 4 –26 < –24

True x = –4 is a solution

Linear Inequalities Addition-Subtraction Principle

Verbally Algebraically Numerically If the same

number is _____ to or subtracted from ____ sides of an inequality, the result is an

_________ inequality.

added

both

equivalent

If a, b, and c, are real numbers

then a < b

is equivalent to a + c < b + c

and to a – c < b – c

x – 2 < 5 is equivalent to

x – 2 + 2 < 5 + 2 and to x < 7


Solve using addition-subtraction. Give answer in interval notation.

x – 4 + 4 < 11 + 4

4.

x – 4 < 11

x < 15 (–∞,15)



3 – 3 ≤ 3 + x – 3

5.

3 ≤ 3 + x

0 ≤ x Or x ≥ 0

[0,∞)



3y – 2y ≤ 2y + 7 – 2y

6.

3y ≤ 2y + 7

y ≤ 7 (–∞,7]



7a – 6a ≥ 6a – 1 – 6a

7.

7a ≥ 6a – 1

a ≥ –1 [–1,∞)



7d – 6d < 6d – 6d 7d < 6d

8.

d < 0 (–∞,0)



3x – 2x – 1 ≤ 2x – 2x + 6 3x – 1 ≤ 2x + 6

9.

x – 1 + 1 ≤ 6 + 1 x ≤ 7

(–∞,7]



8x – 7x – 2 > 7x – 7x + 12 8x – 2 > 7x + 12

10.

x – 2 + 2 >12 + 2 x > 14 (14,∞)



5 – 3x + 4x < 6 – 4x + 4x 5 – 3x < 6 – 4x

11.

5 + x < 6 x + 5 – 5 < 6 – 5

x < 1 (–∞,1)



12x – 11x – 24 > 11x – 11x + 8 6(2x – 4) > 11x + 8

12.

x – 24 + 24 > 8 + 24 x > 32 (32,∞)



14x – 14 < 13x + 13 7(2x – 2) < 13(x + 1)

13.

14x – 13x – 14 < 13x – 13x + 13 x – 14 + 14 < 13 + 14

x < 27 (–∞,27)



6m +3 ≥ 5m – 10 3(2m + 1) ≥ 5(m – 1) – 5

14.

m + 3 ≥ – 10 m ≥ –13 [–13,∞)



3x + 6 ≥ 4x – 10 3(x + 2) ≥ 2(2x – 5)

15.

6 ≥ x – 10 16 ≥ x

Or x ≤ 16 (–∞,16]



–12 + 6x ≤ 4x – 15 + 3x –6(2 – x) ≤ 4x – 3(5 – x)

16.

–12 + 6x ≤ 7x – 15 –12 ≤ x – 15

3 ≤ x Or x ≥ 3

[3,∞)



y/4 – y/4 – 3/5 > 7/5 + 5y/4 – y/4

17.

– 3/5 > 7/5 + 4y/4 – 3/5 – 7/5 > 7/5 + y – 7/5

–10/5 > y –2 > y

Or y < –2 (–∞, –2)

Linear Inequalities Graphs & Tables

The graph represents monthly cost of two phone plans based on minutes. y1 for plan A and y2 for plan B.

a. Approximate the monthly cost of of A for 400 minutes. y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

$25

18.



b. Approximate the monthly cost of of B for 400 minutes. y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

$30

18.



c. Approximate the monthly cost of of A for 800 minutes. y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

$35

18.



d. Approximate the monthly cost of of B for 800 minutes. y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

$30

18.



e. How many minutes for both plans to have same cost? y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

600 minutes

18.



f. What is the monthly cost for 600 minutes? y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

$30

18.



g. Explain when you would choose plan A. y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

If you use less than 600 minutes.

18.



h. Explain when you would choose plan B. y

x0 100 200 300 400 500 600 700 800

60

40

20

0

y2

y1

If you use more than 600 minutes.

18.


Use the graph to solve: a.  y1 = y2

b.  y1 ≤ y2

c.  y1 > y2

-5

y

x 5

5

-5

y1

y2

x = 4

x ≤ 4 (–∞,4]

x > 4 (4,∞)

19.


Use the graph to solve: a.  y1 = y2

b.  y1 < y2

c.  y1 > y2

-5

y

x

5

5

-5

y1

y2

x = 1

x > 1 (1,∞)

x < 1 (–∞,1)

20.


Use the table to solve: a.  y1 = y2

b.  y1 < y2

c.  y1 ≥ y2

x y1 <, =, or >

y2

–4 1 –2 –3 2 0 –2 3 2 –1 4 4 0 5 6 1 6 8 2 7 10

> > > = < < <

x = –1

x > –1 (–1,∞)

x ≤ –1 (–∞,–1]

21.


Use the table to solve: a.  y1 = y2

b.  y1 ≤ y2

c.  y1 ≥ y2

x y1 <, =, or >

y2

4 –2 2 5 1 4 6 4 6 7 7 8 8 10 10 9 13 12

10 16 14

< < < < = > >

x = 8

x ≤ 8 (–∞,8]

x ≥ 8 [8,∞)

22.


Solve 4x – 5 > 3x – 2 by letting Y1 = 4x – 5 and Y2 = 3x – 2. Graph with window [–2, 6, 1] by [–5, 10, 1]. Use INTERSECT. Sketch. Use TABLE to complete the table: x y1 <, =,

or >

y2

0 1 2 3 4 5 6

< < < = > > >

–2 1 4 7 10 13 16

–5 –1 3 7 11 15 19

Graph: when is Y1 above Y2? For x right of 3

Y1

-2

y

x

8

11

-5 Y2

Solve 4x – 5 > 3x – 2 x – 5 > –2 x > 3 (3,∞)

Table: when Y1 > Y2? For x > 3

Do solutions all match? Yes

23.

Linear Inequalities

Solve algebraically.

x + 3 > –5 0.5x + 3 > –0.5x – 5

x > –8 (–8,∞)

24.

Linear Inequalities

Solve algebraically.

2x + 6 – 3x ≤ 5 – 2x 2(x + 3) – 3x ≤ 5 – 2x

–x + 6 ≤ 5 – 2x x + 6 ≤ 5

x ≤ –1 (–∞,–1]

25.

Linear Inequalities The tables represent two taxi service charges based on miles.

A charges $2.30 +$0.15/ ¼ mi. B charges $2.00 +$0.20/ ¼ mi. a. y1 = y2 A B

b. y1 < y2

c. y1 > y2

d.  Interpret

x Miles

y1 $ Cost

y2 $ Cost

0.50 2.60 2.40

1.00 2.90 2.80

1.50 3.20 3.20

2.00 3.50 3.60

2.50 3.80 4.00

3.00 4.10 4.40

x = 1.5

(1.5,∞)

(0,1.5) Why 0?

Both cost $3.20 for 1.5 miles. A less for more than 1.5 miles, B less for less than 1.5 miles.

x Miles

y1 $ Cost

y2 $ Cost

0.50 2.60 2.40

1.00 2.90 2.80

1.50 3.20 3.20

2.00 3.50 3.60

2.50 3.80 4.00

3.00 4.10 4.40

26.

x Miles

y1 $ Cost

y2 $ Cost

0.50 2.60 2.40

1.00 2.90 2.80

1.50 3.20 3.20

2.00 3.50 3.60

2.50 3.80 4.00

3.00 4.10 4.40

x Miles

y1 $ Cost

y2 $ Cost

0.50 2.60 2.40

1.00 2.90 2.80

1.50 3.20 3.20

2.00 3.50 3.60

2.50 3.80 4.00

3.00 4.10 4.40

Linear Inequalities

Phrase Inequality Notation

Interval Notation Graphical Notation

x is at least 5 x is at most 2 x exceeds –3

x is smaller than –1

x ≥ 5 [5,∞) [ 5

x ≤ 2 (–∞,2] ] 2

x > –3 (–3,∞) ( –3

x < –1 (–∞,–1) ) –1

Linear Inequalities

Write an algebraic inequality for the following statement using x to represent the number. Solve for x. Five less than three times a number is at least two times

the sum of the number and three. 3x – 5 ≥ 2(x + 3)

x – 5 ≥ 6 x ≥ 11 [11,∞)

Example Problem

•  The perimeter of the triangle shown must be less than 26 cm. Find the possible values for a.

a cm 8 cm

8 cm

P = 8 + 8 + a < 26 a + 16 < 26

a + 16 – 16 < 26 – 16 a < 10 (0,10)

Math 72 Section 4.1

•  Class Participation:

– Exercise 4.1, #68, 70

Quiz 15 - s3.amazonaws.com€¦Solve using addition-subtraction. Give answer in interval notation. 7d – 6d < 6d – 6d 7d < 6d 8. d < 0 (–∞,0) Linear Inequalities Addition-Subtraction

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