Sign of an inequality I have the following
$\log(0,46)^{k+1}<\log(0,018)$
I solved this by writing
$k<\frac{\log(0,018)}{\log(0,46)}-1$
so $k < 4,17$. The result should be $k > 4,17$, why is that? Where am I getting wrong?
 A: The log of $.46$ is negative, so when you divided, you should have changed the direction of the inequality.
A: Since $\log(0.46)$ is negative (because $0.46 \leq e$, or $0.46 \leq 10$, depending on what base we're using for the $\log$), when we divide through we must change the direction of the inequality. The rest looks fine.
The root of this matter is that $x \geq 0$ implies $-x \leq 0$. We can invent any example with concrete numbers: consider $-2 \leq 6$, and divide through by $-2$ to get $1 \geq -3$, not $1 \leq -3$. However, we could divide by $6$ and get $-\frac{1}{3} \leq 1$, still valid, because we're dividing by a positive number.
To answer your question in the comments:
When we have something like $\displaystyle \frac{2x + 3}{5x - 1} \geq 2$, we might be tempted to multiply by $5x-1$, but since you don't know its sign, it's better to avoid multiplying or dividing by a number which, for all we know, might be negative, and for example substract $2$ from both sides: $\displaystyle \frac{2x + 3}{5x - 1} - 2 = \frac{2x + 3 - 2(5x - 1)}{5x - 1} =\frac{-8x+5}{5x-1} \geq 0$, and keep the division as a last resource, when we'll have to use the next tip. To multiply or divide by an unknown quantity we have to consider every possible sign of what we're multiplying or dividing by, for example:
If $5x-1 \gt 0$, then we multiply without changing the direction: $\displaystyle 2x + 3 \geq 2(5x - 1)$, and if $5x-1 \lt 0$, then $\displaystyle 2x + 3 \leq 2(5x - 1)$; finally, we continue manipulating the expression to solve for $x$. In each case we must remember that we imposed a condition on $x$ (like $5x-1 \gt 0$ or $5x-1 \lt 0$), and take it into account at the end, even though it's not written on the equation.
The basic idea is:


*

*try to do simpler steps whenever possible; and

*we definitely have to change the direction if we're multiplying or dividing by a negative number, so if we don't know the sign, we must separate in cases.

