![1 + \frac{1}{2} \log_{ \sqrt{3} } { ( \frac{x+5}{x+3} ) } \geq \log_9 {(x+1)^2} \](https://tex.z-dn.net/?f=+1+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Clog_%7B+%5Csqrt%7B3%7D+%7D+%7B+%28+%5Cfrac%7Bx%2B5%7D%7Bx%2B3%7D+%29+%7D+%5Cgeq+%5Clog_9+%7B%28x%2B1%29%5E2%7D+%5C+ "1 + \frac{1}{2} \log_{ \sqrt{3} } { ( \frac{x+5}{x+3} ) } \geq \log_9 {(x+1)^2} \")
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ОДЗ:
0 \ , \\\\ (x+1)^2 > 0 \ ; \end{array}\right " alt=" \left\{\begin{array}{l} \frac{x+5}{x+3} > 0 \ , \\\\ (x+1)^2 > 0 \ ; \end{array}\right " align="absmiddle" class="latex-formula">
0 \ , \\\\ x \neq -1 \ ; \end{array}\right " alt=" \left\{\begin{array}{l} x \neq -3 \ , \\\\ \frac{x+5}{x+3} (x+3)^2 > 0 \ , \\\\ x \neq -1 \ ; \end{array}\right " align="absmiddle" class="latex-formula">
0 \ ; \end{array}\right " alt=" \left\{\begin{array}{l} x \notin \{ -3 , -1 \} \ , \\\\ ( x + 3 ) ( x + 5 ) > 0 \ ; \end{array}\right " align="absmiddle" class="latex-formula">
![\left\{\begin{array}{l} x \notin \{ -3 , -1 \} \ , \\ x \notin [ -5 ; -3 ] \ ; \end{array}\right](https://tex.z-dn.net/?f=+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D+x+%5Cnotin+%5C%7B+-3+%2C+-1+%5C%7D+%5C+%2C+%5C%5C+x+%5Cnotin+%5B+-5+%3B+-3+%5D+%5C+%3B+%5Cend%7Barray%7D%5Cright+ "\left\{\begin{array}{l} x \notin \{ -3 , -1 \} \ , \\ x \notin [ -5 ; -3 ] \ ; \end{array}\right")
![x \notin \{ [ -5 ; -3 ] \cup \{ -1 \} \} \ ;](https://tex.z-dn.net/?f=+x+%5Cnotin+%5C%7B+%5B+-5+%3B+-3+%5D+%5Ccup+%5C%7B+-1+%5C%7D+%5C%7D+%5C+%3B+ "x \notin \{ [ -5 ; -3 ] \cup \{ -1 \} \} \ ;")
Решение:
![1 + \log_{ \sqrt{3} } { \sqrt{ \frac{x+5}{x+3} } } \geq \log_{ \sqrt{9} } { \sqrt{ ( x + 1 )^2 } } \](https://tex.z-dn.net/?f=+1+%2B+%5Clog_%7B+%5Csqrt%7B3%7D+%7D+%7B+%5Csqrt%7B+%5Cfrac%7Bx%2B5%7D%7Bx%2B3%7D+%7D+%7D+%5Cgeq+%5Clog_%7B+%5Csqrt%7B9%7D+%7D+%7B+%5Csqrt%7B+%28+x+%2B+1+%29%5E2+%7D+%7D+%5C+ "1 + \log_{ \sqrt{3} } { \sqrt{ \frac{x+5}{x+3} } } \geq \log_{ \sqrt{9} } { \sqrt{ ( x + 1 )^2 } } \")
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![\log_3 {3} + \log_3 { \frac{x+5}{x+3} } \geq \log_3 { |x+1| } \](https://tex.z-dn.net/?f=+%5Clog_3+%7B3%7D+%2B+%5Clog_3+%7B+%5Cfrac%7Bx%2B5%7D%7Bx%2B3%7D+%7D+%5Cgeq+%5Clog_3+%7B+%7Cx%2B1%7C+%7D+%5C+ "\log_3 {3} + \log_3 { \frac{x+5}{x+3} } \geq \log_3 { |x+1| } \")
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![\log_3 { ( 3 \cdot \frac{x+5}{x+3} ) } \geq \log_3 { |x+1| } \](https://tex.z-dn.net/?f=+%5Clog_3+%7B+%28+3+%5Ccdot+%5Cfrac%7Bx%2B5%7D%7Bx%2B3%7D+%29+%7D+%5Cgeq+%5Clog_3+%7B+%7Cx%2B1%7C+%7D+%5C+ "\log_3 { ( 3 \cdot \frac{x+5}{x+3} ) } \geq \log_3 { |x+1| } \")
;
![3 \cdot \frac{x+5}{x+3} \geq |x+1| \](https://tex.z-dn.net/?f=+3+%5Ccdot+%5Cfrac%7Bx%2B5%7D%7Bx%2B3%7D+%5Cgeq+%7Cx%2B1%7C+%5C+ "3 \cdot \frac{x+5}{x+3} \geq |x+1| \")
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-1 \ , \\ 3 (x+5) \geq (x+1)(x+3) \ ; \end{array}\right \end{array}\right " alt=" \left[\begin{array}{l} \left\{\begin{array}{l} x < -5 \ , \\ 3 ( x + 5 ) \leq -(x+1)(x+3) \ ; \end{array}\right \\\\ \left\{\begin{array}{l} -3 < x < -1 \ , \\ 3 ( x + 5 ) \geq -(x+1)(x+3) \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x > -1 \ , \\ 3 (x+5) \geq (x+1)(x+3) \ ; \end{array}\right \end{array}\right " align="absmiddle" class="latex-formula">
-1 \ , \\ 3x + 15 \geq x^2 + 4x + 3 \ ; \end{array}\right \end{array}\right " alt=" \left[\begin{array}{l} \left\{\begin{array}{l} x < -5 \ , \\ 3x + 15 + x^2 + 4x + 3 \leq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} -3 < x < -1 \ , \\ 3x + 15 + x^2 + 4x + 3 \geq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x > -1 \ , \\ 3x + 15 \geq x^2 + 4x + 3 \ ; \end{array}\right \end{array}\right " align="absmiddle" class="latex-formula">
-1 \ , \\ x^2 + x - 12 \leq 0 \ ; \end{array}\right \end{array}\right " alt=" \left[\begin{array}{l} \left\{\begin{array}{l} x < -5 \ , \\ x^2 + 7x + 18 \leq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} -3 < x < -1 \ , \\ x^2 + 7x + 18 \geq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x > -1 \ , \\ x^2 + x - 12 \leq 0 \ ; \end{array}\right \end{array}\right " align="absmiddle" class="latex-formula">
-1 \ , \\ x^2 + 2 \cdot x \cdot \frac{1}{2} + ( \frac{1}{2} )^2 - 12 - \frac{1}{4} \leq 0 \ ; \end{array}\right \end{array}\right " alt=" \left[\begin{array}{l} \left\{\begin{array}{l} x < -5 \ , \\ x^2 + 2 \cdot x \cdot \frac{7}{2} + ( \frac{7}{2} )^2 + 18 - \frac{49}{4} \leq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} -3 < x < -1 \ , \\ x^2 + 2 \cdot x \cdot \frac{7}{2} + ( \frac{7}{2} )^2 + 18 - \frac{49}{4} \geq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x > -1 \ , \\ x^2 + 2 \cdot x \cdot \frac{1}{2} + ( \frac{1}{2} )^2 - 12 - \frac{1}{4} \leq 0 \ ; \end{array}\right \end{array}\right " align="absmiddle" class="latex-formula">
-1 \ , \\ ( x + \frac{1}{2} )^2 \leq \frac{49}{4} \ ; \end{array}\right \end{array}\right " alt=" \left[\begin{array}{l} \left\{\begin{array}{l} x < -5 \ , \\ ( x + \frac{7}{2} )^2 + 18 - 12 \frac{1}{4} \leq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} -3 < x < -1 \ , \\ ( x + \frac{7}{2} )^2 + 18 - 12 \frac{1}{4} \geq 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x > -1 \ , \\ ( x + \frac{1}{2} )^2 \leq \frac{49}{4} \ ; \end{array}\right \end{array}\right " align="absmiddle" class="latex-formula">
<img src="
https://tex.z-dn.net/?f=+%5Cleft%5B%5Cbegin%7Barray%7D%7Bl%7D+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D+x+%3C+-5+%5C+%2C+%5C%5C+%28+x+%2B+%5Cfrac%7B7%7D%7B2%7D+%29%5E2+%5Cleq+-+5+%5Cfrac%7B3%7D%7B4%7D+%5C+%3B+%5Cend%7Barray%7D%5Cright+%5C%5C%5C%5C+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D+-3+%3C+x+%3C+-1+%5C+%2C+%5C%5C+%28+x+%2B+%5Cfrac%7B7%7D%7B2%7D+%29%5E2+%5Cgeq+-+5+%5Cfrac%7B3%7D%7B4%7D+%5C+%3B+%5Cend%7Barray%7D%5Cright+%5C%5C%5C%5C+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D+x+%3E+-1+%5C+%2C+%5C%5C+%7C+x+%2B+%5Cfrac%7B1%7D%7B2%7D+%7C+%5Cleq+%5Cfrac%7B7%7D%7B2%7D+%5C+%3B+%5Cend%7Barray%7D%5Cright+%5Cend%7Barray%7D%5Cright+" id="TexFormula16" title=" \left[\begin{array}{l} \left\{\begin{array}{l} x <