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1·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
1. ª√–‚¬§¿“…“·≈–ª√–‚¬§ —≠≈—°…≥å
°‘®°√√¡∑’Ë 1
1. „Àâπ—°‡√’¬π‡¢’¬π —≠≈—°…≥å·∑π¢âÕ§«“¡µàÕ‰ªπ’È
(1) ÀⓇ∑à“¢Õß®”π«π®”π«πÀπ÷ËßπâÕ¬°«à“ 15
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(2) “¡‡∑à“¢Õß®”π«π®”π«πÀπ÷Ëß¡“°°«à“¬’Ë ‘∫‡ÕÁ¥
...........................................................................................................................................
(3) ®”π«π®”π«πÀπ÷Ëß∫«°¥â«¬ 9 ¡“°°«à“ 15
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(4) x ≈∫¥â«¬ 8 πâÕ¬°«à“ 1
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(5) º≈§Ÿ≥¢Õß 9 °—∫ a ‰¡à‡°‘π 20
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(6) ‡»…Àπ÷Ëß à«π “¡¢Õß b ‰¡àπâÕ¬°«à“ 5
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Õ ¡°“√1
2 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(7) ’ˇ∑à“¢Õߺ≈∫«°¢Õß®”π«πÀπ÷Ëß°—∫ 3 ¡“°°«à“ 23
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(8) º≈∫«°¢Õß®”π«π§’Ë “¡®”π«π‡√’¬ß°—π‰¡à‡°‘π 39
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(9) ‡»… “¡ à«πÀâ“¢Õß®”π«π®”π«πÀπ÷Ëß¡“°°«à“ 7
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(10) À°‡∑à“¢Õß®”π«π®”π«πÀπ÷Ëß≈∫¥â«¬ 4 πâÕ¬°«à“ 8
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(11) º≈§Ÿ≥¢Õß®”π«ππ—∫ Õß®”π«π‡√’¬ß°—π‰¡à‡∑à“°—∫ 30
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(12) º≈§Ÿ≥¢Õß 5 °—∫ y ¡“°°«à“ 100
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2. „Àâπ—°‡√’¬π‡ª≈’ˬπª√–‚¬§µàÕ‰ªπ’ȇªìπª√–‚¬§¿“…“
(1) x 5 2
x ≈∫¥â«¬ 5 ............................ 2
(2) 12
x 5
‡»…Àπ÷Ëß à«π ÕߢÕß x ............................ 5
(3) x 7 10
º≈∫«°¢Õß x °—∫ 7 ............................ 10
(4) 3x 5x 40
“¡‡∑à“¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ ................................................................................
(5) 2(m 4) 12
Õ߇∑à“¢Õߺ≈µà“ߢÕß m ≈∫¥â«¬ 4 ............................................................................
(6) n2 4 9
...........................................................................................................................................
3·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(7) 3(x 5) 2x 18
...........................................................................................................................................
(8)
13
15
x x 20
...........................................................................................................................................
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3. „Àâπ—°‡√’¬πæ‘®“√≥“¢âÕ§«“¡„¥‡ªìπ ¡°“√·≈–¢âÕ§«“¡„¥‡ªìπÕ ¡°“√ À√◊Õ‰¡à‡ªìπ
∑—Èß ÕßÕ¬à“ß
¡°“√ Õ ¡°“√ ‰¡à‡ªìπ ¡°“√·≈–Õ ¡°“√
(1) 2 3 3 5 .............................................................................................
(2) 4x 8 .............................................................................................
(3) 3x 2 5 .............................................................................................
(4) 4x x .............................................................................................
(5) 5x 11 x 1 .............................................................................................
(6) 3x 8 .............................................................................................
(7) 2x 5 x 3 .............................................................................................
(8) 3x 11 2x 7 .............................................................................................
(9) 4x 3m 5 .............................................................................................
(10) 4k 8 0 .............................................................................................
4. „Àâπ—°‡√’¬πÀ“®”π«π¡“·∑πµ—«·ª√‡æ◊ËÕ„Àâª√–‚¬§µàÕ‰ªπ’ȇªìπ®√‘ß
(1) 2x 18 ∑ÿ°®”π«π¬°‡«âπ 9
(2) 3x 15 ..............................................................................................
(3) 3m 21 ..............................................................................................
(4) 3m 2x 4 ..............................................................................................
(5) 3m 5 m 11 ..............................................................................................
(6) 4x 5 2x 1 ..............................................................................................
4 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(7) 8x 56 ..............................................................................................
(8) 2x 1 x 2 ..............................................................................................
(9)15
x 3 ..............................................................................................
(10) 34
x 9 ..............................................................................................
(11) x 5 13 ..............................................................................................
(12) 2x 3 x 1 ..............................................................................................
(13) 3x 4 2x 7 ..............................................................................................
(14) 7 x 3 ..............................................................................................
(15) 10 x 5 ..............................................................................................
(16) 5x 5 ..............................................................................................
(17)25
x 6 ..............................................................................................
(18) 10 2 8x ..............................................................................................
(19) 12 5x 12 4x ..............................................................................................
(20)
x7
12
914
..............................................................................................
5. „Àâπ—°‡√’¬π‡ª≈’ˬπª√–‚¬§µàÕ‰ªπ’ȇªìπª√–‚¬§ —≠≈—°…≥å ·≈â«À“®”π«π¡“·∑π
µ—«·ª√‡æ◊ËÕ„Àâª√–‚¬§‡ªìπ®√‘ß
(1) º≈∫«°¢Õß x ·≈– 5 ¡“°°«à“ 12
x 5 12
x 7 [∑ÿ°®”π«π∑’Ë¡“°°«à“ 7]
(2) y ≈∫¥â«¬ 8 ‰¡à‡∑à“°—∫ 1
...........................................................................................................................................
...........................................................................................................................................
(3) º≈§Ÿ≥¢Õß 5 °—∫ a ‰¡à‡°‘π 15
...........................................................................................................................................
...........................................................................................................................................
5·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
2. §”µÕ∫¢ÕßÕ ¡°“√
°‘®°√√¡∑’Ë 2
1. „Àâπ—°‡√’¬πµ√«® Õ∫¥Ÿ«à“ §”µÕ∫∑’Ë°”Àπ¥„Àâ„π«ß‡≈Á∫ [ ] ‡ªì𧔵Õ∫¢Õß ¡°“√
À√◊ÕÕ ¡°“√∑’Ë°”Àπ¥„ÀâÀ√◊Õ‰¡à
‡ªìπ ‰¡à‡ªìπ
(1) x 1 3 [x 4] ................................................
(2) 3y 3 33 [y 12] ................................................
(3) 2m 1 3m 2 [m 3] ................................................
(4) 8t 9 6t 7 [t 8] ................................................
(5) 7p 5 2 [p 1] ................................................
(6) 4q 2 10 [q 3] ................................................
(7) 5r r 8 [r 2] ................................................
(8) 3s 2s 15 [s 5] ................................................
(9) 4w 1 2w 3 [w 3] ................................................
(10) 3z 9 0 [z 3] ................................................
(4) ’ˇ∑à“¢Õß®”π«π®”π«πÀπ÷Ëß¡“°°«à“ 5 ‰¡à‡°‘π 3
...........................................................................................................................................
...........................................................................................................................................
(5) ‡»… Õß à«π “¡¢Õß®”π«π®”π«πÀπ÷Ë߉¡àπâÕ¬°«à“ 8
...........................................................................................................................................
...........................................................................................................................................
6 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
‡ªìπ ‰¡à‡ªìπ
(11) 2c 5 1 [c 2] ................................................
(12) 8d 12 d 2 [d 2] ................................................
(13) 12e 11 e 22 [e 2] ................................................
(14) 2f 13 f 3 [f 10] ................................................
(15) 3g 15 2g 3 [g 18] ................................................
(16) h 1 3 4h [h 4] ................................................
(17) 4x 5 x 10 [x 5] ................................................
(18) 3x 2 2x 7 [x 5] ................................................
(19) 5x 3 2x 3 [x 2] ................................................
(20) 7x 7 3 3x [x 1] ................................................
2. „Àâπ—°‡√’¬πÀ“§”µÕ∫¢Õß ¡°“√À√◊Õ§”µÕ∫¢ÕßÕ ¡°“√µàÕ‰ªπ’È
(1)x2
3 ..............................................................................................
(2)23
x 2 ..............................................................................................
(3)x3
1 3 ..............................................................................................
(4)
x8
32
..............................................................................................
(5)
34
32
x ..............................................................................................
(6) 10 5 13
x ..............................................................................................
(7) 1 1
234
x
..............................................................................................
(8)35
x 1 7 ..............................................................................................
(9)x3
5 4 ..............................................................................................
(10)x3
x 6 ..............................................................................................
(11) 2 x4
..............................................................................................
(12)x7
4 ..............................................................................................
(13) x2 81 ..............................................................................................
7·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(14) x3 1,000 ..............................................................................................
(15) a 12 20 ..............................................................................................
(16) 11a 12 133 ..............................................................................................
(17) 3a 5 2a 5 ..............................................................................................
(18) 11a 7 6a 32 ..............................................................................................
(19) 0.2a 6 0.8a ..............................................................................................
(20) 3a 7 12a 29 ..............................................................................................
(21) b 2 3 ..............................................................................................
(22) b 5 4 ..............................................................................................
(23) b 11 9 ..............................................................................................
(24) b 12 11 ..............................................................................................
(25) b 1 12 ..............................................................................................
(26) c 3 5 ..............................................................................................
(27) c 7 8 ..............................................................................................
(28) 4 c 8 ..............................................................................................
(29) d 5 7 ..............................................................................................
(30) d 7 3 ..............................................................................................
(31) 8 d 12 ..............................................................................................
(32) 2t 3 7 ..............................................................................................
(33) 23
w 3 6 ..............................................................................................
(34)115
p 2 13 ..............................................................................................
(35) 7.3 r 6.8 ..............................................................................................
(36) 513
y 12 28 ..............................................................................................
8 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
3. °√“ø· ¥ß§”µÕ∫
°‘®°√√¡∑’Ë 3
1. „Àâπ—°‡√’¬π‡¢’¬π°√“ø· ¥ß§”µÕ∫¢Õß ¡°“√·≈–Õ ¡°“√µàÕ‰ªπ’È
(1) x 3 4
(2) x 5 2
(3) x 4 1
(4) x 3 3
(5) x 1 3
(6) x 1
9·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(7) 2 x 3
(8) 4 x 4
(9) x 4, 3, 1, 2
(10) x 3 À√◊Õ x 3
(11) x 2 À√◊Õ x 3
(12) 3 x 3
2. °√“øµàÕ‰ªπ’È· ¥ß®”π«πÕ–‰√∫â“ß
(1)
...........................................................................................................................................
(2)
...........................................................................................................................................
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
10 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(3)
...........................................................................................................................................
(4)
...........................................................................................................................................
(5)
...........................................................................................................................................
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
4. Õ ¡°“√
°‘®°√√¡∑’Ë 4
1. „Àâπ—°‡√’¬π‡µ‘¡‡§√◊ËÕßÀ¡“¬ , À√◊Õ ≈ß„π™àÕß«à“߇æ◊ËÕ∑”„Àâª√–‚¬§‡ªìπ®√‘ß
(1) ∂â“ 3 5 ·≈– 5 x ·≈â« 3 ..................... x
(2) ∂â“ 5 8 ·≈– 8 y ·≈â« 5 ..................... y
(3) ∂â“ a 10 ·≈– 10 b ·≈â« a ..................... b
(4) ∂â“ a 1 b ·≈â« a ..................... b
(5) ∂â“ x 2 y ·≈â« x ..................... y
(6) ∂â“ a b ·≈â« 2a ..................... 2b
(7) ∂â“ a b ·≈â« a5
..................... b5
11·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(8) ∂â“ a b ·≈â« 5a ..................... 5b
(9) ∂â“ a b ·≈â« a−3
..................... b
−3
(10) ∂â“ x y ·≈– y 0 ·≈â« xy ..................... 0
(11) ∂â“ m 0 ·≈â« m ..................... 0
(12) ∂â“ n 0 ·≈â« n2
..................... 0
(13) ∂â“ x 0 ·≈â« x−5
..................... 0
(14) ∂â“ x 1 5 ·≈– 5 y ·≈â« x 1 ..................... y
(15) ∂â“ x y ·≈– y 10 ·≈â« x ..................... 10
2. „Àâπ—°‡√’¬π·°âÕ ¡°“√µàÕ‰ªπ’È·≈–‡¢’¬π°√“ø· ¥ß§”µÕ∫
(1) 5x 5 4x 4
«‘∏’∑” 5x 5 4x 4
5x 5 5 4x 4 5
5x 4x 1
5x 4x 4x 1 4x
...........................................................................................................................................
(2) 6x 5 4x 3
«‘∏’∑” 6x 5 4x 3
6x 4x 3 5
...........................................................................................................................................
...........................................................................................................................................
12 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(3) x 3 12
x 2
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(4) 4x 35
2x 75
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(5)x2
7 4
x3
12
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
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13·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(6) 5(x 1) 3(5x 3)
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(7)x2
3 x 2
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(8) x 3 5 12
x
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
14 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(9) 5 4(1 2x) 3(5 3x)
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(10)x2
4 x6
5 x3
«‘∏’∑” .................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
3. „Àâπ—°‡√’¬π·°âÕ ¡°“√µàÕ‰ªπ’Èæ√âÕ¡∑—È߇¢’¬π§”µÕ∫
(1) x 1 2
x 1 1 2 1
x 3
∑ÿ°®”π«π∑’ËπâÕ¬°«à“ .......................................................................................................
(2) 3x 5 4
3x 5 5 4 5
3x 1
...........................................................................................................................................
∑ÿ°®”π«π∑’Ë¡“°°«à“ ........................................................................................................
15·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
(3) 5x 4 4x 3
5x 4x 3 4
...........................................................................................................................................
...........................................................................................................................................
(4)13
(2x 1) 13
x
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(5)
2 13
12
x x+ + 1 x + 5
3
π” 6 §Ÿ≥∑—Èß Õߢâ“ß
6 2 1
3x +
6 ( )x + 1
2 6 1 6
( )x + 53
2(2x 1) 3(x 1) 6 2(x 5)
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(6)14
(x 3) 13
12
(x 1)
π” 12 §Ÿ≥∑—Èß Õߢâ“ß
12 14
(x 3) 12 13
12 12
(x 1)
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
16 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
(7)
13
3 25
4x −
25
(x 1)
13
3 2 205
x − +
25
(x 1)
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
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5. °“√À“§”µÕ∫¢ÕßÕ ¡°“√
µ—Èß·µà ÕßÕ ¡°“√¢÷Èπ‰ª
°‘®°√√¡∑’Ë 5
„Àâπ—°‡√’¬πÀ“§”µÕ∫¢ÕßÕ ¡°“√æ√âÕ¡‡¢’¬π°√“ø· ¥ß§”µÕ∫
1. x 5 4 ·≈– 2x 8 4
x 4 5 2x 4 8
x 1 2x 12
...........................................................................
17·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
§”µÕ∫¢Õß∑—Èß ÕßÕ ¡°“√§◊Õ 6 x 1
§”µÕ∫¢Õß∑—Èß ÕßÕ ¡°“√∑’ˇªìπ®”π«π‡µÁ¡§◊Õ .................................................................
2. 21 x 2x 3 6 x
21 x 2x 3 ·≈– 2x 3 6 x
21 3 ...................... 2x x ...............................................
...................................................................................................................................................
...................................................................................................................................................
§”µÕ∫¢Õß∑—Èß ÕßÕ ¡°“√§◊Õ ..............................................................................................
3.13
x 4 16
x 12 x 2
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
18 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
4.12
x 3 13
x 6 x 7
12
x 3 13
x 6 ·≈– 13
x 6 x 7
π” 6 §Ÿ≥∑—Èß Õߢâ“ß π” 6 §Ÿ≥∑—Èß Õߢâ“ß
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
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°‘®°√√¡∑’Ë 6
1. Àπ—ß ◊Õ‡≈à¡Àπ÷Ëß¡’ 250 Àπâ“ ∂â“∑—»‰π¬Õà“πÀπ—ß ◊Õ‡≈à¡π’È∑ÿ°«—π «—π≈–‰¡à∂÷ß 25 Àπâ“
∑—»‰π¬®–µâÕßÕà“πÀπ—ß ◊ÕÕ¬à“ßπâÕ¬°’Ë«—π®÷ß®∫‡≈à¡
«‘∏’∑” „Àâ∑—»‰π¬Õà“πÀπ—ß ◊Õπ“π x «—π®÷ß®∫‡≈à¡
Àπ—ß ◊Õ¡’ ............................................................Àπâ“
µâÕßÕà“π«—π≈– ...................................................Àπâ“
Õà“π«—π≈–‰¡à‡°‘π 25 Àπâ“
Õ ¡°“√§◊Õ 250x
25
6. °“√·°â‚®∑¬åªí≠À“‡°’ˬ«°—∫
Õ ¡°“√‡™‘߇ âπµ—«·ª√‡¥’¬«
19·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
2. æ’Ë¡’Õ“¬ÿ¡“°°«à“πâÕß 6 ªï ·µàæ’ËπâÕߧŸàπ’È¡’Õ“¬ÿ√«¡°—π‰¡à∂÷ß 36 ªï πâÕß®–¡’Õ“¬ÿ¡“°∑’Ë ÿ¥
°’˪ï
«‘∏’∑” „ÀâπâÕß¡’Õ“¬ÿ x ªï
æ’Ë¡’Õ“¬ÿ¡“°°«à“πâÕß ........................................ªï
æ’Ë¡’Õ“¬ÿ ..............................................................ªï
æ’ËπâÕߧŸàπ’È¡’Õ“¬ÿ√«¡°—π‰¡à∂÷ß ...........................ªï
Õ ¡°“√§◊Õ .................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
3. æ‘‡πµ¡’‡À√’¬≠Àâ“∫“∑·≈–‡À√’¬≠ ‘∫∫“∑√«¡°—π 60 ‡À√’¬≠ §‘¥‡ªìπ‡ß‘ππâÕ¬°«à“
550 ∫“∑ ®ßÀ“«à“æ‘‡πµ¡’‡À√’¬≠Àâ“∫“∑Õ¬à“ßπâÕ¬∑’Ë ÿ¥°’ˇÀ√’¬≠·≈–¡’‡À√’¬≠ ‘∫∫“∑
Õ¬à“ß¡“°∑’Ë ÿ¥°’ˇÀ√’¬≠
«‘∏’∑” „Àâæ‘‡πµ¡’‡À√’¬≠Àâ“∫“∑ x ‡À√’¬≠
§‘¥‡ªìπ‡ß‘π ........................................................∫“∑
∑’ˇÀ≈◊Õ§‘¥‡ªìπ‡À√’¬≠ ‘∫∫“∑ ........................ ‡À√’¬≠
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20 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
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4. √∂¬πµå§—π·√°·≈àπ¥â«¬Õ—µ√“‡√Á« 60 °‘‚≈‡¡µ√µàÕ™—Ë«‚¡ß √∂¬πµå§—π∑’Ë Õß·≈àπ¥â«¬
Õ—µ√“‡√Á« 50 °‘‚≈‡¡µ√µàÕ™—Ë«‚¡ß ∂â“√∂¬πµå Õߧ—π·≈àπÕÕ°‡«≈“‡¥’¬«°—πæ√âÕ¡°—π
‰¥â√–¬–∑“ß√«¡°—π¡“°°«à“ 330 °‘‚≈‡¡µ√ ®ßÀ“«à“√∂¬πµå∑—Èß Õߧ—π„™â‡«≈“‡∑à“‰√
«‘∏’∑” „Àâ√∂¬πµå·µà≈–§—π„™â‡«≈“ t ™—Ë«‚¡ß
√∂¬πµå§—π·√°‡≈àπ‰¥â∑“ß 60t °‘‚≈‡¡µ√
√∂¬πµå§—π∑’Ë Õß·≈àπ‰¥â∑“ß ...........................°‘‚≈‡¡µ√
√∂¬πµå 2 §—π ·≈àπ‰¥â∑“ß√«¡°—π¡“°°«à“ 330 °‘‚≈‡¡µ√
Õ ¡°“√§◊Õ .................................................................................................................
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¥—ßπ—Èπ ..........................................................................................................................
5. √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“√ŸªÀπ÷Ëß¡’¥â“𬓫¬“«‡ªìπ Õ߇∑à“¢Õߥâ“π°«â“ß ∂Ⓡ âπ√Õ∫√Ÿª¢Õß
√Ÿª ’ˇÀ≈’ˬ¡√Ÿªπ’Ȭ“«‰¡à‡°‘π 36 ‡´πµ‘‡¡µ√ ®ßÀ“§«“¡¬“«¢Õߥâ“π°«â“ß·≈–¥â“𬓫
«‘∏’∑” .....................................................................................................................................
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21·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
µÕπ∑’Ë 1 „Àâπ—°‡√’¬π‡≈◊Õ°§”µÕ∫∑’Ë∂Ÿ°µâÕß
·∫∫∑¥ Õ∫º≈ —¡ƒ∑∏‘Ï∑“ß°“√‡√’¬πª√–®”Àπ૬
1. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√
2x 5 1
1. x 3 2. x 3
3. x 52
4. x 3
2. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√
4x 13 1
1. x 3 2. x 3
3. x 4 4. x 12
3. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√5x 5 3x 3
1. x 2 2. x 3
3. x 4 4. x 6
4. ¢âÕ„¥µàÕ‰ªπ’È∂Ÿ°µâÕß
1. ∂â“ 3x 3 12 ·≈â« x 6
2. ∂â“ 4x 1 13 ·≈â« x 4
3. ∂â“ 2 3x 7 ·≈â« x 3
4. ∂â“ 2x 13 27 ·≈â« x 7
5. ¢âÕ„¥µàÕ‰ªπ’È∂Ÿ°µâÕß1. ∂â“ 5x 2 3x 10 ·≈â«
x 5
2. ∂â“ 3x 5 13 3x ·≈â«x 3
3. ∂â“ 4x 4 2x 2 ·≈â« x 2
4. ∂â“ 2x 13 x 27 ·≈â«x 7
6. ¢âÕ„¥µàÕ‰ªπ’È∂Ÿ°µâÕß
1. ∂â“ x − 13
5 ·≈â« x 10
2. ∂â“ x + 43
6 x ·≈â« x 7
3. ∂â“ x − 23
34 ·≈â« x 4
4. ∂â“ x + 23
2 x ·≈â« x 1
7. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√4 3x 2 11
1. 2 x 9
2. 2 x 3
3. 1 x 3
4. 4 x 9
8. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√2x 3 9 ·≈– 3x 9 12 ‡¡◊ËÕx ‡ªìπ®”π«π‡µÁ¡1. 3, 4 ·≈– 5 2. 4, 5 ·≈– 63. 5, 6 ·≈– 7 4. 6, 7 ·≈– 8
9. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√3x 13 4x 1 ·≈– 2x 5
3x 7 ‡¡◊ËÕ x ‡ªìπ®”π«π‡µÁ¡1. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 12
2. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 12,
13
3. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 13
4. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 13,
14
22 ·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
µÕπ∑’Ë 2 „Àâπ—°‡√’¬π· ¥ß«‘∏’∑”
1. æ’Ë¡’Õ“¬ÿ¡“°°«à“πâÕß 5 ªï ·µàæ’ËπâÕߧŸàπ’È¡’Õ“¬ÿ√«¡°—π‰¡à∂÷ß 35 ªï πâÕß®–¡’Õ“¬ÿ¡“°∑’Ë ÿ¥
°’˪ï
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10. ¢âÕ„¥µàÕ‰ªπ’ȇªì𧔵Õ∫¢ÕßÕ ¡°“√
2x 1 11 ·≈– 3x 4 17
1. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 5
2. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 5, 6
3. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 6, 7
4. ®”π«π‡µÁ¡∑ÿ°®”π«π¬°‡«âπ 7
11. ®”π«π∑ÿ°®”π«π∑’Ë¡“°°«à“ 11
‡ªì𧔵Õ∫¢ÕßÕ ¡°“√„π¢âÕ„¥
1. 5x 10 3x 12
2. 4x 7 3x 11
3. 5(x 2) 3(x 2)
4. 4(x 5) 3(x 6)
12. ®”π«π∑ÿ°®”π«π∑’ˉ¡à‡∑à“°—∫ 1 ‡ªì𧔵Õ∫¢ÕßÕ ¡°“√
„π¢âÕ„¥
1.5 3
4x −
2x 1 2.3 1
2x −
4x 3
3.x + 1
2 3x 2 4.
2 34
x + 3x 4
23·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 2
2. √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“√ŸªÀπ÷Ëß ¥â“𬓫¬“«°«à“ Õ߇∑à“¢Õߥâ“π°«â“ß 5 ‡´πµ‘‡¡µ√ ∂â“
¥â“𬓫¬“«‰¡à‡°‘π 15 ‡´πµ‘‡¡µ√ ¥â“π°«â“ß®–¬“«‡∑à“‰√
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3. ®”π«π§Ÿà∫«° 2 ®”π«π‡√’¬ß°—π√«¡°—π‰¥â‰¡à‡°‘π 120 ®ßÀ“®”π«π§Ÿà∫«°®”π«π¡“°
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