linear inequalities - metropolitan community college · linear inequalities joseph lee metropolitan...
TRANSCRIPT
![Page 1: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/1.jpg)
Linear Inequalities
Joseph Lee
Metropolitan Community College
Joseph Lee Linear Inequalities
![Page 2: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/2.jpg)
Example 1.a
Solve.x − 4 > 3
Solution.x − 4 > 3
x > 7
Joseph Lee Linear Inequalities
![Page 3: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/3.jpg)
Example 1.a
Solve.x − 4 > 3
Solution.x − 4 > 3
x > 7
Joseph Lee Linear Inequalities
![Page 4: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/4.jpg)
Example 1.a
Solve.x − 4 > 3
Solution.x − 4 > 3
x > 7
Joseph Lee Linear Inequalities
![Page 5: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/5.jpg)
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
![Page 6: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/6.jpg)
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
![Page 7: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/7.jpg)
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
![Page 8: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/8.jpg)
Example 1.b
Solve.2x − 3 ≤ 5
Solution.2x − 3 ≤ 5
2x ≤ 8
x ≤ 4
Joseph Lee Linear Inequalities
![Page 9: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/9.jpg)
Example 2.a
Solve.−2x < 8
Solution.−2x < 8
x ≥ −4
Joseph Lee Linear Inequalities
![Page 10: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/10.jpg)
Example 2.a
Solve.−2x < 8
Solution.−2x < 8
x ≥ −4
Joseph Lee Linear Inequalities
![Page 11: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/11.jpg)
Example 2.a
Solve.−2x < 8
Solution.−2x < 8
x ≥ −4
Joseph Lee Linear Inequalities
![Page 12: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/12.jpg)
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
![Page 13: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/13.jpg)
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
![Page 14: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/14.jpg)
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
![Page 15: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/15.jpg)
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
![Page 16: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/16.jpg)
Example 2.b
Solve.−4(x − 3) ≥ −8
Solution.−4(x − 3) ≥ −8
−4x + 12 ≥ −8
−4x ≥ −20
x ≤ 5
Joseph Lee Linear Inequalities
![Page 17: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/17.jpg)
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
![Page 18: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/18.jpg)
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
![Page 19: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/19.jpg)
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
![Page 20: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/20.jpg)
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
![Page 21: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/21.jpg)
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
![Page 22: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/22.jpg)
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
![Page 23: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/23.jpg)
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
![Page 24: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/24.jpg)
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
![Page 25: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/25.jpg)
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
![Page 26: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/26.jpg)
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
![Page 27: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/27.jpg)
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
![Page 28: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/28.jpg)
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
![Page 29: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/29.jpg)
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
![Page 30: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/30.jpg)
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
![Page 31: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/31.jpg)
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
![Page 32: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/32.jpg)
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
![Page 33: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/33.jpg)
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
![Page 34: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/34.jpg)
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
![Page 35: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/35.jpg)
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
![Page 36: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/36.jpg)
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
![Page 37: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/37.jpg)
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
![Page 38: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/38.jpg)
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
![Page 39: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/39.jpg)
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
![Page 40: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/40.jpg)
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
![Page 41: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/41.jpg)
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
![Page 42: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/42.jpg)
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
![Page 43: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/43.jpg)
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
![Page 44: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/44.jpg)
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
![Page 45: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/45.jpg)
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
![Page 46: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/46.jpg)
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
![Page 47: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/47.jpg)
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
![Page 48: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/48.jpg)
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
![Page 49: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/49.jpg)
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
![Page 50: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/50.jpg)
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
![Page 51: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/51.jpg)
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
![Page 52: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/52.jpg)
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
![Page 53: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/53.jpg)
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
![Page 54: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/54.jpg)
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
![Page 55: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/55.jpg)
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
![Page 56: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/56.jpg)
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
![Page 57: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/57.jpg)
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
![Page 58: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/58.jpg)
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
![Page 59: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/59.jpg)
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
![Page 60: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/60.jpg)
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
![Page 61: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/61.jpg)
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
![Page 62: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/62.jpg)
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
![Page 63: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/63.jpg)
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
![Page 64: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/64.jpg)
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
![Page 65: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/65.jpg)
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
![Page 66: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/66.jpg)
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
![Page 67: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/67.jpg)
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
![Page 68: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/68.jpg)
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
![Page 69: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/69.jpg)
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
![Page 70: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/70.jpg)
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
![Page 71: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/71.jpg)
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
![Page 72: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/72.jpg)
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
![Page 73: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/73.jpg)
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
![Page 74: Linear Inequalities - Metropolitan Community College · Linear Inequalities Joseph Lee Metropolitan Community College Joseph Lee Linear Inequalities. Example 1.a Solve. x 4 > 3](https://reader030.vdocuments.site/reader030/viewer/2022040807/5e4ac512e0dd316c2b7a38a9/html5/thumbnails/74.jpg)
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