Question

Решите неравенство log 0,5(1-0,5x)>-3

Answer 1

log0,5(1-0,5x)>-3

ОДЗ:

1-0,5x>0

-0,5x>-1

x<2

x∈(-∞;2)

log0,5(1-0,5x)>-3log0,5(0,5)

log(1-0,5x)>log0,5(0,5)^(-3)

log(1-0,5x)>log0,5(8)

1-0,5x<8

-0,5x<7

x>-14

Ответ: х∈(-14;2)

Answer 2

To solve the inequality log₀.₅(1 - 0.5x) > -3, we need to apply the properties of logarithms. The base of the logarithm is 0.5 in this case. Remember that for any positive base "b" and positive numbers "a" and "c":

1. log_b(a) > c is equivalent to a > b^c.

Let's apply this property to solve the inequality:

log₀.₅(1 - 0.5x) > -3

Using the property, we get:

1 - 0.5x > 0.5^(-3)

Now, calculate 0.5^(-3):

0.5^(-3) = 1/(0.5³) = 1/(0.125) = 8.

So, the inequality becomes:

1 - 0.5x > 8

Now, isolate the variable "x" by moving terms around:

-0.5x > 8 - 1

-0.5x > 7

Finally, divide both sides by -0.5 (remembering that when dividing or multiplying by a negative number, you need to reverse the inequality sign):

x < 7 / -0.5

x < -14

Therefore, the solution to the inequality is:

x < -14