(x - 5)/(2x - 3) ≤ ⅔

(x - 5)/(2x - 3) - ⅔ ≤ 0

3(x - 5)/(2x - 3) - 2 ≤ 0

( 3(x - 5) - 2(2x - 3) ) / (2x - 3) ≤ 0

(-x - 9)/(2x - 3) ≤ 0

(x + 9)/(2x - 3) ≥ 0


x ≠ 3/2 as that would mean 2x - 3 = 0 and the denominator would be 0, which makes the function and the inequality undefined.

The denominator is positive when x > 3/2 (because it makes 2x - 3 > 0) and negative when x < 3/2 (because it makes 2x - 3 < 0)

The numerator is positive when x > -9 (because it makes x + 9 > 0) and negative when x < -9 (because it makes x + 9 < 0)


Now if x = -9, then x + 9 = 0, which makes the numerator 0. So the expression becomes 0.
So x = 9 satisfies the inequality.


If x < -9, the numerator is negative.
But then 2x - 3 would also be negative since x < -9 also means that x < 3/2.

So negative over negative gives a positive, which makes the expression > 0.
So x < - 9 also satisfies the inequality.


Now, if x > 3/2, 2x - 3 > 0 and thus the denominator is positive. And numerator is also positive since x > 3/2 also means x > -9.

So the expression is also positive and so x > 3/2 satisfies the inequality.


But what about the in-between values?
i.e -9 < x < 3/2

The numerator would be positive. But the denominator would be negative. Overall the expression is negative and thus it doesn't satisfy the inequality.


Therefore, x > 3/2 or x ≤ -9

For f(lnx) ≤ ⅔, we use our result from before and replace x by lnx.


So lnx ≤ -9 or lnx > 3/2


For lnx > 3/2,

x > e^(3/2)


For lnx ≤ -9,

x ≤ e^(-9)

But remember that x cannot be 0 or negative as that would make lnx undefined. So x > 0. Putting these two results together,

0 < x ≤ e^(-9)

Or

0 < x ≤ 1/e^9

(depends on how you want to represent it)

So,

x > e^(3/2) or 0 < x ≤ 1/e^9

As for f|x + ½| ≤ ⅔,


|x + ½| ≤ -9 or |x + ½| > 3/2


|x + ½| ≤ -9 is not possible as absolute value is always 0 or positive. So this inequality is rejected.


As for|x + ½| > 3/2,

x + ½ > 3/2 or x + ½ < -3/2

x > 1 or x < -2