Al7

Page 1

№ 1421 | 2 x + 1 – x| + |x – 2 x + 2| = 7; | 2 x –x+1|+|–( 2 x –x+1)+3| = 7; | 2 x – x + 1 = a; |a| + |3 – a| = 7. а) a > 3; a + a – 3 = 7; a = 5. б) 0 ≤ a ≤ 3; a + 3 – a = 7; a ∈ ∅. в) a<o;–a+3–a=7; a=–2; а) 2 x –x+1=5; 2 x =x+4; 4x=x2+8x + 16; x2 + 4 x + 16 = 0 – действ. корней нет. б) 2 x – x + 1 = –2; 2 x = x – 3; x ≥ 0; 4x = x2 – 6x + 9; x2 – 10x + 9 = 0; x1, 2 = 5 ±

25 − 9 = 5 ± 4 ; x1 = 9, x2 = 1. 1

x = 1 не является решением, т.к. |2 + 1 – 1| + |1 – 2+ 2| ≠ 7. Ответ: x=9.

№ 1422 1) 9·41/x+5·61/x=4·91/x, 32·41/x+5·21/x·31/x=4·32/x, 32·22/x+5·21/x·31/x=22·32/x 2 x

2

1

 2 x  2 x разделим все на (3) ≠ 0 , 9  +   − 4 = 0 ; 3 3 −5 + 13 4 = ; t2 9t2 + 5t – 4 = 0; D = 25 + 144, t1 = 18 9 1

1

 2 x   = t; t > 0 3 −5 − 13 = = −1 18

2

1 1  2 x 4  2  = 2; x = .   = =  ; 3 9 3 x 2     2 2) log2(x – 3) – log2(6x – 10) + 1 = 0,  2  2 x 2 − 3 1 2 x 2 − 6 − 6 x + 10 = ; = 0 ; 2x − 6x + 4 = 0 ; x − 3x + 2 = 0 ; 2(6 x − 10 ) 6 x − 10 2 x ≠ 10 / 6 x ≠ 10 / 6 2 x1 = 1, x2 = 2, т.к. x – 3 > 0 и 6x – 10 > 0, то x = 1 не является решением. Ответ: x = 2. 1 3) 2log2x – 2 log 2 = 3 log 2 x ; 2log2x + 1 = 3 log 2 x , x > 0. 2 1 log 2 x = a ≥ 0; 2a2 – 3a + 1 = 0; a1 = 1, a2 = ; log2x = 1, x = 2; 2 1 2 . Ответ: x = 2 . log2x = , x = 4 4) logx(2x2 – 3x – 4) = 2; 2x2 – 3x – 4 > 0, x > 0, x ≠ 1; logx(2x2 – 3x – 4) = logxx2; 2x2 – 3x – 4 = x2; x2 – 3x – 4 = 0; x1 = –1, x2 = 4; x1 < 0, следовательно не является решением. Ответ: x = 4.

№ 1423 1) 1 + logx(5 – x) = log74 · logx7; 1 + logx(5 – x) = log x 7log7 4 ; 272


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