Question

Log0,5(4+x)/(x-1)>=2

Answer 1

$log_{0,5} \frac{4+x}{x-1} \geq 2 2= log_{0,5} 0,5^{2} = log_{0,5} 0,25$
$log_{0,5} \frac{4+x}{x-1} \geq log_{0,5} 0,25$
основание логарифма  а=0,5.  0<0,5<1
 знак неравенства меняем:
$\left \{ {{ \frac{4+x}{x-1} \ \textgreater \ 0} \atop { \frac{4+x}{x-1} \leq 0,25 }} \right. \left \{ {{ \frac{4+x}{x-1} \ \textgreater \ 0} \atop { \frac{4+x}{x-1}-0,25 \leq 0 }} \right.$
$1. \frac{4+x}{x-1} \ \textgreater \ 0$,метод интервалов:
$\left \{ {{x+4=0} \atop {x-1 \neq 0}} \right. \left \{ {{x=-4} \atop {x \neq 1}} \right.$
      +              -                     +
-----------(-4)-----------(1)----------------->x
x∈(-4;∞)∪(1;∞)

$2. \frac{4+x}{x-1}-0,25 \leq 0.$
$\frac{4+x-0,25x+0,25}{x-1} \leq 0, \frac{0,75x+4,25}{x-1} \leq 0$
метод интервалов:
$\left \{ {{0,75x+4,25=0} \atop {x-1 \neq 0}} \right. \left \{ {{x= -\frac{17}{3} } \atop {x \neq 1}} \right.$
     +                       -                   +
----------[-17/3]-------------(1)--------------->x

x∈[-17/3;1)
                  / / / / / / / / / /  / / / / /  / 
------------[-17/3]------(-4)-----------(1)------------------>x
\ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \                       \ \ \  \ \ \ \  \ \ \  \ \

x∈[-17/3;-4)

Answer 2

To solve the inequality log base 0.5 of (4+x)/(x-1) >= 2, we need to apply logarithmic properties and algebraic manipulation. Here's the step-by-step solution:

Step 1: Rewrite the inequality using exponential form. The logarithmic inequality can be transformed to its equivalent exponential form: 0.5^2 ≤ (4+x)/(x-1)

Step 2: Simplify the right side of the inequality: 0.25 ≤ (4+x)/(x-1)

Step 3: Multiply both sides of the inequality by (x-1) to eliminate the fraction. Note that we need to consider the sign of (x-1) and ensure we do not multiply by a negative number (which would require reversing the inequality): 0.25(x-1) ≤ 4+x

Step 4: Distribute 0.25 on the left side: 0.25x - 0.25 ≤ 4 + x

Step 5: Move x terms to one side of the inequality and constant terms to the other side: 0.25x - x ≤ 4 + 0.25

Step 6: Combine like terms: -0.75x ≤ 4.25

Step 7: Divide both sides by -0.75. Since we are dividing by a negative number, we must reverse the inequality sign: x ≥ 4.25 / -0.75

Step 8: Perform the division: x ≥ -5.67

Therefore, the solution to the inequality log base 0.5 of (4+x)/(x-1) >= 2 is x ≥ -5.67.