(Sec 3) Linear Inequalities Applications and Word Problems Solutions



(Sec 3) Linear Inequalities ¨C Applications and Word Problems Solutions

1(a)

2 x ? 3( x ? 7) ? 1

2 x ? 3 x ? 21 ? 1

2 x ? 3 x ? 1 ? 21

5 x ? 22

22

x?

5

2

x?4

5

the smallest integer of x is 5 (Ans)

(b)

( x ? 3) ? 3 x ? 7 ? 2 x ? 5

x ? 3 ? 3x ? 7, 3 x ? 7 ? 2 x ? 5

3 ? 7 ? 3 x ? x, 3 x ? 2 x ? 5 ? 7

10 ? 2 x, x ? 12

5 ? x, x ? 12

5 ? x ? 12

the smallest integer of x is 6 (Ans)

(c)

3x ? 5

x ?1

? 4x ? 5 ?

2

3

3x ? 5

x ?1

? 4 x ? 5, 4x ? 5 ?

2

3

3x ? 5

x ?1

? (4 x ? 5) ? 0, (4x ? 5) ?

?0

2

3

3x ? 5 2

3

x ?1

? (4 x ? 5) ? 0,

(4x ? 5) ?

?0

2

2

3

3

(3 x ? 5) ? 2(4 x ? 5) ? 0,

3 x ? 5 ? 8 x ? 10 ? 0,

3(4x ? 5) ? ( x ? 1) ? 0

12x ? 15 ? x ? 1 ? 0

?5 x ? 15 ? 0, 11x ? 16 ? 0

15 ? 5 x, 11x ? 16

15

16

x? , x?

5

11

5

x ? 3, x ? 1

11

5

1 ? x?3

11

the smallest integer of x is 2 (Ans)

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2(a)

2x ?1 ? 7

2x ? 8

x?4

The greatest integer of x = 3 (Ans)

(b)

x ? 5 ? 9x ? 3

5 ? 3 ? 9x ? x

8 ? 8x

x ?1

The greatest integer of x = 0 (Ans)

1

1

( x ? 5) ? ( x ? 3)

3

2

2( x ? 5) ? 3( x ? 3)

(c)

2 x ? 10 ? 3x ? 9

10 ? 9 ? 3x ? 2 x

19 ? x

x ? 19

The greatest integer of x = 18 (Ans)

3.

?60 < 5x ,

5x ¡Ü 60

?12 < x

x > ?12

x ¡Ü 12

?12 < x ¡Ü 12

Ans:

(a) Greatest possible value of x is 12

(b) Least possible value of x is ?11

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4.

2( x ? 7) ? 3( x ? 5) ? 7( x ? 3)

2 x ? 14 ? 3 x ? 15 ? 7 x ? 21

2 x ? 14 ? 10 x ? 36

14 ? 36 ? 10 x ? 2 x

50 ? 8x

50

?x

8

25

?x

4

25

x?

4

(a) the greatest rational value of x ?

25

4

(b) the greatest integer value of x = 6

(a) the greatest prime number of x = 5

5.

x ?3 x ?4 x ?5

?

?

2

3

4

?12, 6( x ? 3) ? 4( x ? 4) ? 3( x ? 5)

6 x ? 18 ? 4 x ? 16 ? 3 x ? 15

6 x ? 18 ? x ? 1

6 x ? x ? ?1 ? 18

5 x ? 17

17

5

2

x?3

5

x?

(a) the greatest integer value of x = 3

(b) the largest prime number of x = 3

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6.

x ? 7 3 ? 2x x ? 6

?

?

2

5

3

3 ? 2x x ? 6

x ? 7 3 ? 2x

?

?

2

5

5

3

5( x ? 7) ? 2(3 ? 2 x)

3(3 ? 2 x) ? 5( x ? 6)

5 x ? 35 ? 6 ? 4 x

9 ? 6 x ? 5 x ? 30

9 x ? 41

?21 ? 11x

x?

41

9

x?4

?

5

9

?1

??1

Ans:

21

?x

11

10

?x

11

10

5

?x?4

11

9

(a) the least integer value of x = ?1

(b) the greatest prime number of x = 3

7.

Ans:

(a)

the greatest possible value of x + y = (?1) + (7) = 6

(b)

the least possible value of xy = (?7) ? (7) = ?49

(c)

the greatest value of (x? y)? = ( -7 ¨C 7)? = 196

(d)

the greatest value of

(e)

the greatest value of y? ? x? = (7)? ? (?1)? = 48

y2

(0) 2

=

=0

x

?7

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8(a)

5 ? 2 ? 10

Let x be the point obtained in the 3rd game

10 ? x

?6

3

10 ? x ? 18

x ? 18 ? 10

x?8

Annie must obtain at least 8 points to win a prize (Ans)

8(b)

(i)

( x ? 5) ? 9 ? 74

(ii)

( x ? 5) ?

74

9

74

?5

9

2

x ?8 ?5

9

2

x ? 13

9

greatest whole number = 13 (Ans)

x?

9(a)

greatest possible value of x = 7

greatest possible value of y = 10

greatest possible perimeter = 7 + 7 + 10 + 10 = 34 cm (Ans)

(b)

smallest possible value of x = 3

smallest possible value of y = 5

smallest possible area of the rectangle = 3 ? 5 = 15 cm? (Ans)

10.

Let x be the price of a VIP ticket

120 x + 5880 (x ¨C 165) ¡Ý 1 000 000

120 x + 5880 x ¨C 970 200 ¡Ý 1 000 000

6000 x ¡Ý 1 970 200

x ¡Ý 328.37

Minimum price is $328.37 (Ans)

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