1. 3^(x+2) + 3^(x-1) <28<br>3² * 3^x + 3^(-1) * 3^x <28<br>3^x (3² + 1/3) <28<br>3^x (9 + 1/3) <28<br>3^x (28/3) < 28
3^x < 28 * (3/28)
3^x < 3
x< 1
3.
9^x - 3^x -6 >0
Пусть 3^x=y 9^x=y²
y²-y-6>0
y²-y-6=0
D=1+24=25
y1=1-5 = -2
2
y2= 1+5 = 3
2
+ - +
--------- -2---------- 3 ---------------
\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\
y< -2
y>3
При у<-2<br>3^x< -2
нет решений.
При y>3
3^x>3
x>1
Ответ: x>1
4. 5^(2x+1) +4*5^(x) -1 >0
5*5^(2x) + 4*5^(x) -1>0
Пусть 5^(x)=y 5^(2x) =y²
5y²+4y-1>0
5y²+4y-1=0
D=16+20=36
y1= -4-6 = -1
10
y2 = -4+6 =1/5
10
+ - +
------- -1 ------------ 1/5 -------------
\\\\\\\\\ \\\\\\\\\\\\\\\
y< -1
y> 1/5
При y<-1<br>5^(x)<-1<br>нет решений
При y> 1/5
5^(x) > 1/5
5^(x) > 5^(-1)
x> -1
Ответ: х> -1
5. log(1/2) (2x+3) > log(1/2)(x+1)
{2x+3>0
{x+1>0
{2x+3 < x+1
2x+3>0
2x>-3
x>-1.5
x+1>0
x> -1
2x+3< x+1
2x-x< 1-3
x< -2
{x>-1.5
{x> -1
{x< -2
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
------ -2 --------- -1.5 ---------- -1 ---------------
\\\\\\\\\ \\\\\\\\\\\\\\\\\\\
Нет решений.
Ответ: нет решений.
6.
log(6) (x²-3x+2)≥1
log(6) (x²-3x+2)≥log(6) 6
{x²-3x+2>0
{x²-3x+2≥6
Решаем только последнее неравенство:
x²-3x+2≥6
x²-3x+2-6≥0
x²-3x-4≥0
x²-3x-4=0
D=9+16=25
x1= 3-5 = -1
2
x2= 3+5 =4
2
+ - +
-------- -1 ----------- 4 --------------
\\\\\\\\\\\ \\\\\\\\\\\\\\\\
x∈(-∞; -1] U [4; +∞)
Ответ: (-∞; -1] U [4; +∞)
7. log(1/6) (10-x) + log(1/6) (x-3) ≥ -1
log(1/6) [(10-x)(x-3)] ≥ log(1/6) 6
{10-x>0
{x-3>0
{(10-x)(x-3)≤6
10-x>0
-x>-10
x<10<br>
x-3>0
x>3
(10-x)(x-3)≤6
10-x²-30+3x-6≤0
-x²+13x-36≤0
x²-13x+36≥0
x²-13x+36=0
D=13² -4*36 =169-144=25
x1= 13-5 = 4
2
x2 = 13+5 =9
2
+ - +
----------- 4 ---------- 9 -------------
\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\
x∈(-∞; 4]U [9; +∞)
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
--------- 3 -----4 ------------- 9----- 10 --------------
\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\
х∈(3; 4] U [9; 10)
Ответ: (3; 4] U [9; 10)
8. lgx - log(0.1) (x-1) > log(0.1) 0.5
log(10^(-1)) (x^(-1)) - log(0.1) (x-1) > log(0.1) 0.5
- log(0.1) x - log(0.1) (x-1) > log(0.1) 0.5
-(log(0.1) x + log(0.1) (x-1)) > log(0.1) 0.5
log(0.1) (x(x-1)) <- log(0.1) 0.5<br> log(0.1) (x(x-1)) < log(0.1) 0.5^(-1)
log(0.1) (x(x-1)) < log(0.1) 2
{x>0
{x-1>0
{x(x-1)>2
x-1>0
x>1
x(x-1)>2
x²-x-2>0
x²-x-2=0
D=1+8=9
x1= 1-3 = -1
2
x²= 1+3 =2
2
+ - +
------ -1 ---------- 2 --------------
\\\\\\\\\ \\\\\\\\\\\\\\\
x<-1<br>x>2
{x>0
{x>1
{x<-1<br>{x>2
x>2
Ответ: х>2