Linear Inequalities
Joseph Lee
Metropolitan Community College
Joseph Lee Linear Inequalities
Example 1.a
Solve.x − 4 > 3
Solution.x − 4 > 3
x > 7
Joseph Lee Linear Inequalities
Example 1.a
Solve.x − 4 > 3
Solution.x − 4 > 3
x > 7
Joseph Lee Linear Inequalities
Example 1.a
Solve.x − 4 > 3
Solution.x − 4 > 3
x > 7
Joseph Lee Linear Inequalities
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
Example 2.a
Solve.−2x < 8
Solution.−2x < 8
x ≥ −4
Joseph Lee Linear Inequalities
Example 2.a
Solve.−2x < 8
Solution.−2x < 8
x ≥ −4
Joseph Lee Linear Inequalities
Example 2.a
Solve.−2x < 8
Solution.−2x < 8
x ≥ −4
Joseph Lee Linear Inequalities
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph 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
Joseph 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
Joseph 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
Joseph 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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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.
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities
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
Joseph Lee Linear Inequalities