Question

2㏒₀,₅(x-2) - ㏒₀,₅(x²-x-2) ≥ 1

Answer 1

Ответ:

Объяснение:

ответ на фото\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Answer 2

To solve the inequality 2㏒₀,₅(x-2) - ㏒₀,₅(x²-x-2) ≥ 1, we need to find the values of x that satisfy the inequality.

Let's start by simplifying the logarithms using the change of base formula:

2㏒₀,₅(x-2) = 2 * (㏒(x-2) / ㏒(0.5)) ㏒₀,₅(x²-x-2) = ㏒(x²-x-2) / ㏒(0.5)

Now, the inequality becomes:

2 * (㏒(x-2) / ㏒(0.5)) - (㏒(x²-x-2) / ㏒(0.5)) ≥ 1

To get rid of the denominators, we can multiply both sides of the inequality by ㏒(0.5):

2 * ㏒(x-2) - ㏒(x²-x-2) ≥ ㏒(0.5)

Now, we can use the properties of logarithms to combine them:

㏒((x-2)²) - ㏒(x²-x-2) ≥ ㏒(0.5)

Now, using the property of logarithms, ㏒(a) - ㏒(b) = ㏒(a/b), we can simplify further:

㏒(((x-2)²) / (x²-x-2)) ≥ ㏒(0.5)

Now, we can eliminate the logarithms and solve for the expression inside:

((x-2)²) / (x²-x-2) ≥ 0.5

To solve this inequality, we need to find the critical points by setting the numerator and denominator equal to zero:

(x - 2)² = 0 -> x = 2 x² - x - 2 = 0 -> x ≈ -1.618, x ≈ 2.618

Now, we can create a sign chart for the expression ((x-2)²) / (x²-x-2) to find the intervals that satisfy the inequality:

scssCopy code
x < -1.618       (-1.618, 2)        2      (2, 2.618)         x > 2.618

+---------------+------------------+-------+-------------------+ | ((x-2)²) / (x²-x-2) | - | + | - | +---------------+------------------+-------+-------------------+

From the sign chart, we can see that the expression is greater than or equal to 0.5 in the intervals (-1.618, 2) and (2, 2.618).

Therefore, the solution to the inequality is:

-1.618 < x < 2 OR 2 < x < 2.618