Question

Log2(18-9x)-log2(x+2)>log2(x²-6x+8)​

Answer 1

Держи
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Answer 2

To solve the inequality involving logarithms, we need to follow certain steps. First, we'll combine the logarithms on the left side of the inequality using logarithmic properties. Then we'll solve for the values of x that satisfy the inequality. Keep in mind that the logarithm is only defined for positive values, so we must ensure that the expressions inside the logarithms are positive.

The given inequality is:

log2(18 - 9x) - log2(x + 2) > log2(x² - 6x + 8)

Step 1: Combine the logarithms using the quotient rule of logarithms.

log2((18 - 9x)/(x + 2)) > log2(x² - 6x + 8)

Step 2: Convert the inequality to exponential form.

2^(log2((18 - 9x)/(x + 2))) > 2^(log2(x² - 6x + 8))

Step 3: Simplify the exponential expressions.

(18 - 9x)/(x + 2) > x² - 6x + 8

Step 4: Clear the fraction by multiplying both sides by (x + 2) (remember, x cannot be -2 as it would make the denominator zero, which is not allowed).

(x + 2) * ((18 - 9x)/(x + 2)) > (x + 2) * (x² - 6x + 8)

Step 5: Cancel out (x + 2) on the left side and distribute on the right side.

18 - 9x > x³ - 4x² - 6x + 16

Step 6: Move all the terms to the left side of the inequality to set it to zero.

x³ - 4x² - 6x + 16 + 9x - 18 > 0

Step 7: Combine like terms.

x³ - 4x² + 3x - 2 > 0

Now, we have a cubic inequality. To solve it, we can use various methods, such as plotting the graph or checking the intervals. In this case, we'll use the trial and error method by checking the sign of the expression in intervals.

Step 8: Find critical points by setting the expression to zero and solving for x.

x³ - 4x² + 3x - 2 = 0

Using synthetic division or another method, we find that x = 1 and x = 2 are the critical points.

Step 9: Test intervals:

1. For x < 1: Choose x = 0 (arbitrary value less than 1): 0³ - 4(0)² + 3(0) - 2 = -2 (negative) The expression is negative in this interval.

2. For 1 < x < 2: Choose x = 1.5 (arbitrary value between 1 and 2): 1.5³ - 4(1.5)² + 3(1.5) - 2 ≈ 0.125 (positive) The expression is positive in this interval.

3. For x > 2: Choose x = 3 (arbitrary value greater than 2): 3³ - 4(3)² + 3(3) - 2 = 11 (positive) The expression is positive in this interval.

Step 10: Determine the solution:

The inequality x³ - 4x² + 3x - 2 > 0 is satisfied when x is less than 1 and when x is greater than 2.

So, the solution to the inequality is:

x < 1 or x > 2