Domain of a Logarithmic Function

[xxMsJojoxx]

For this question, I got up to the step of x and 5/2. Why do we switch sign to 'less than' though?

After -2x>-5, the next step should be x>5/2?

I don't understand why we have to switch signs?


Thank you!

 

[pka]

View attachment 22095
For this question, I got up to the step of x and 5/2. Why do we switch sign to 'less than' though?
After -2x>-5, the next step should be x>5/2?
I don't understand why we have to switch signs?

If \(-x\le a\) then \(x\ge -a\) likewise if \(-y\ge a\) then \(y\le -a\).
Thus if \(-2x>-5\) then \(-x> -\frac{5}{2}\) so that \(x<\frac{5}{2}\)

 

[JeffM]

[MATH]a < b \implies \exists \ c \text { such that } c > 0 \text { and } a + c = b \implies c = b - a.[/MATH]
Will you accept that as a theorem?

[MATH]\text {Given:} a < b \implies \exists \ c \text { such that } c > 0 \text { and } c = b - a\\ \text {ASSUME, for purposes of contradiction that } - a < - b\\ \therefore \exists \ d \text { such that } d > 0 \text { and } - a + d = - b \implies\\ (-1)(-a + d) = (-1)(b) \implies a - d = b \implies - d = b - a \implies \\ - d = c > 0 \implies d < 0, \text { which contradicts } d \ge 0, \implies d \not > 0.[/MATH][MATH]\text {ASSUME, for purposes of contradiction that } - a = - b \implies \\ (-1)(-a) = (-1)((-b) \implies a = b, \text { which contradicts } a < b, \implies \\ -a \not = - b.[/MATH][MATH]- a \not \le - b \implies - a > - b.[/MATH]
What is this saying in practical terms.

[MATH]- 4 < - 2 \implies 4 > 2.[/MATH]
When you multiply or divide both sides of an inequality by a negative number, the inequality reverses.

 

[Harry_the_cat]

When solving inequalities, if you multiply or divide both sides by a negative the sign changes.
Consider you case where 5-2x>0. Your answer of x>5/2 is obviously incorrect if you check it. If x >5/2, then x=3 should be a valid answer, but if you pop x=3 into your original inequality, you get 5-2x3>0 which is NOT true.

 

[Steven G]

3< 5 ok. I will multiply both sides by -2 (same as dividing by -1/2). Then we get -6< -10 using your method. Is -6 really less than -10?

-4<5 so -4(-6) < 5(-6), ie 24<-30?? Not true.

Suppose x<y, then y-x>0. Of course if I multiply a positive number by -1 it will become negative, ie less than 0.

So -1(y-x) = -y + x < 0 so -y <-x or -x>-y. The sign changes, as Harry_the_cat stated, if you multiply both sides by -1. Not to multiply by say -7, you can first multiply by -1 and of course change the sign and then multiply by 7. Of course you can just multiply by -7 and change the sign.