Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3

Linear Inequalities

Joseph Lee

Metropolitan Community College

Joseph Lee Linear Inequalities

Example 1.a

Solve.x − 4 > 3

Solution.x − 4 > 3

x > 7


Example 1.a

Solve.x − 4 > 3


x > 7


Example 1.a

Solve.x − 4 > 3


x > 7


Example 1.b

Solve.2x − 3 ≤ 5

Solution.2x − 3 ≤ 5

2x ≤ 8

x ≤ 4


Example 1.b

Solve.2x − 3 ≤ 5


2x ≤ 8

x ≤ 4


Example 1.b

Solve.2x − 3 ≤ 5


2x ≤ 8

x ≤ 4


Example 1.b

Solve.2x − 3 ≤ 5


2x ≤ 8

x ≤ 4


Example 2.a

Solve.−2x < 8

Solution.−2x < 8

x ≥ −4


Example 2.a

Solve.−2x < 8

Solution.−2x < 8

x ≥ −4


Example 2.a

Solve.−2x < 8

Solution.−2x < 8

x ≥ −4


Example 2.b

Solve.−4(x − 3) ≥ −8

Solution.−4(x − 3) ≥ −8

−4x + 12 ≥ −8

−4x ≥ −20

x ≤ 5


Example 2.b

Solve.−4(x − 3) ≥ −8

Solution.−4(x − 3) ≥ −8

−4x + 12 ≥ −8

−4x ≥ −20

x ≤ 5


Example 2.b

Solve.−4(x − 3) ≥ −8

Solution.−4(x − 3) ≥ −8

−4x + 12 ≥ −8

−4x ≥ −20

x ≤ 5


Example 2.b

Solve.−4(x − 3) ≥ −8

Solution.−4(x − 3) ≥ −8

−4x + 12 ≥ −8

−4x ≥ −20

x ≤ 5


Example 2.b

Solve.−4(x − 3) ≥ −8

Solution.−4(x − 3) ≥ −8

−4x + 12 ≥ −8

−4x ≥ −20

x ≤ 5


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.a

Solve.−2x + 3 > 3(x − 4)

Solution.−2x + 3 > 3(x − 4)

−2x + 3 > 3x − 12

3 > 5x − 12

15 > 5x

3 > x

x < 3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3


Example 3.b

Solve.2(x − 1)− 3x ≤ 8x − 5

Solution.2(x − 1)− 3x ≤ 8x − 5

2x − 2− 3x ≤ 8x − 5

−x − 2 ≤ 8x − 5

−2 ≤ 9x − 5

3 ≤ 9x

1

3≤ x

x ≥ 1

3Joseph Lee Linear Inequalities

Example 4.a

Solve.1

3(x − 5) < 10

Solution.1

3(x − 5) < 10

3

[1

3(x − 5)

]< 3(10)

x − 5 < 30

x < 35


Example 4.a

Solve.1

3(x − 5) < 10

Solution.1

3(x − 5) < 10

3

[1

3(x − 5)

]< 3(10)

x − 5 < 30

x < 35


Example 4.a

Solve.1

3(x − 5) < 10

Solution.1

3(x − 5) < 10

3

[1

3(x − 5)

]< 3(10)

x − 5 < 30

x < 35


Example 4.a

Solve.1

3(x − 5) < 10

Solution.1

3(x − 5) < 10

3

[1

3(x − 5)

]< 3(10)

x − 5 < 30

x < 35


Example 4.a

Solve.1

3(x − 5) < 10

Solution.1

3(x − 5) < 10

3

[1

3(x − 5)

]< 3(10)

x − 5 < 30

x < 35


Example 4.b

Solve.1

4x − 3

5≥ 1

2

Solution.1

4x − 3

5≥ 1

2

20

(1

4x − 3

5

)≥ 20

(1

2

)5x − 12 ≥ 10

5x ≥ 22

x ≥ 22

5


Example 4.b

Solve.1

4x − 3

5≥ 1

2

Solution.1

4x − 3

5≥ 1

2

20

(1

4x − 3

5

)≥ 20

(1

2

)5x − 12 ≥ 10

5x ≥ 22

x ≥ 22

5


Example 4.b

Solve.1

4x − 3

5≥ 1

2

Solution.1

4x − 3

5≥ 1

2

20

(1

4x − 3

5

)≥ 20

(1

2

)

5x − 12 ≥ 10

5x ≥ 22

x ≥ 22

5


Example 4.b

Solve.1

4x − 3

5≥ 1

2

Solution.1

4x − 3

5≥ 1

2

20

(1

4x − 3

5

)≥ 20

(1

2

)5x − 12 ≥ 10

5x ≥ 22

x ≥ 22

5


Example 4.b

Solve.1

4x − 3

5≥ 1

2

Solution.1

4x − 3

5≥ 1

2

20

(1

4x − 3

5

)≥ 20

(1

2

)5x − 12 ≥ 10

5x ≥ 22

x ≥ 22

5


Example 4.b

Solve.1

4x − 3

5≥ 1

2

Solution.1

4x − 3

5≥ 1

2

20

(1

4x − 3

5

)≥ 20

(1

2

)5x − 12 ≥ 10

5x ≥ 22

x ≥ 22

5


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]

−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15

− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.c

Solve.

−2

3(2x − 1) <

1

2(x + 5)

Solution.

−2

3(2x − 1) <

1

2(x + 5)

6

[−2

3(2x − 1)

]< 6

[1

2(x + 5)

]−4(2x − 1) < 3(x + 5)

−8x + 4 < 3x + 15

−11x + 4 < 15− 11x < 11

x > −1


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)

2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 4.d

Solve.x − 2

3≤ x + 5

6

Solution.x − 2

3≤ x + 5

6

6

(x − 2

3

)≤ 6

(x + 5

6

)2(x − 2) ≤ x + 5

2x − 4 ≤ x + 5

x − 4 ≤ 5

x ≤ 9


Example 5.a

Solve.2x − 4 + 3x ≤ 5(x + 1)

Solution.2x − 4 + 3x ≤ 5(x + 1)

5x − 4 ≤ 5x + 5

−4 ≤ 5

We arrive at the inequality −4 ≤ 5, which is true. That is to say,for any value of x , the inequality holds. Thus, the solution to thisinequality is all real numbers.


Example 5.a

Solve.2x − 4 + 3x ≤ 5(x + 1)

Solution.2x − 4 + 3x ≤ 5(x + 1)

5x − 4 ≤ 5x + 5

−4 ≤ 5



Example 5.a

Solve.2x − 4 + 3x ≤ 5(x + 1)

Solution.2x − 4 + 3x ≤ 5(x + 1)

5x − 4 ≤ 5x + 5

−4 ≤ 5



Example 5.a

Solve.2x − 4 + 3x ≤ 5(x + 1)

Solution.2x − 4 + 3x ≤ 5(x + 1)

5x − 4 ≤ 5x + 5

−4 ≤ 5



Example 5.a

Solve.2x − 4 + 3x ≤ 5(x + 1)

Solution.2x − 4 + 3x ≤ 5(x + 1)

5x − 4 ≤ 5x + 5

−4 ≤ 5



Example 5.b

Solve.7x − 2− 3x > 2(x + 3) + 2x

Solution.7x − 2− 3x > 2(x + 3) + 2x

4x − 2 > 2x + 6 + 2x

4x − 2 > 4x + 6

−2 > 6

We arrive at the inequality −2 > 6, which is false. That is to say,for any value of x , the inequality fails. Thus, this inequality has nosolution.


Example 5.b

Solve.7x − 2− 3x > 2(x + 3) + 2x

Solution.7x − 2− 3x > 2(x + 3) + 2x

4x − 2 > 2x + 6 + 2x

4x − 2 > 4x + 6

−2 > 6



Example 5.b

Solve.7x − 2− 3x > 2(x + 3) + 2x

Solution.7x − 2− 3x > 2(x + 3) + 2x

4x − 2 > 2x + 6 + 2x

4x − 2 > 4x + 6

−2 > 6



Example 5.b

Solve.7x − 2− 3x > 2(x + 3) + 2x

Solution.7x − 2− 3x > 2(x + 3) + 2x

4x − 2 > 2x + 6 + 2x

4x − 2 > 4x + 6

−2 > 6



Example 5.b

Solve.7x − 2− 3x > 2(x + 3) + 2x

Solution.7x − 2− 3x > 2(x + 3) + 2x

4x − 2 > 2x + 6 + 2x

4x − 2 > 4x + 6

−2 > 6



Example 5.b

Solve.7x − 2− 3x > 2(x + 3) + 2x

Solution.7x − 2− 3x > 2(x + 3) + 2x

4x − 2 > 2x + 6 + 2x

4x − 2 > 4x + 6

−2 > 6



Example 5.c

Solve.2(3x − 4) < x − 8

Solution.2(3x − 4) < x − 8

6x − 8 < x − 8

5x − 8 < −8

5x < 0

x < 0


Example 5.c

Solve.2(3x − 4) < x − 8

Solution.2(3x − 4) < x − 8

6x − 8 < x − 8

5x − 8 < −8

5x < 0

x < 0


Example 5.c

Solve.2(3x − 4) < x − 8

Solution.2(3x − 4) < x − 8

6x − 8 < x − 8

5x − 8 < −8

5x < 0

x < 0


Example 5.c

Solve.2(3x − 4) < x − 8

Solution.2(3x − 4) < x − 8

6x − 8 < x − 8

5x − 8 < −8

5x < 0

x < 0


Example 5.c

Solve.2(3x − 4) < x − 8

Solution.2(3x − 4) < x − 8

6x − 8 < x − 8

5x − 8 < −8

5x < 0

x < 0


Example 5.c

Solve.2(3x − 4) < x − 8

Solution.2(3x − 4) < x − 8

6x − 8 < x − 8

5x − 8 < −8

5x < 0

x < 0


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Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3