solution set of the logarithm inequality

Solution set: Log base 1/3 (x^2+ 4x +3) < = -1

attempt:
x^2 + 4x + 3 <= (1/3)^-1
x^2 + 4x + 3 < = 3
x^2 + 4x < = 0
x ( x + 4) < = 0
x < = 0, x < = -4


i wonder why from the book i have read, the solution set is I.N. (-4, 0) or - 4 < x < 0

please guide me on this problem

thanks

At the point where you have x(x+4)<=0, sketch the graph of y = x(x+4).
The range of values for which y<=0 should be obvious.

Hi rcs!

Quote:

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Originally Posted by rcs

Solution set: Log base 1/3 (x^2+ 4x +3) < = -1

attempt:
x^2 + 4x + 3 <= (1/3)^-1

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This is incorrect.
When the base of the log is between 0 and 1 the inequality swaps around.

Quote:

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x^2 + 4x + 3 < = 3
x^2 + 4x < = 0
x ( x + 4) < = 0
x < = 0, x < = -4

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The last step is incorrect.
If x would be smaller than -4, say -5, then both factors are negative, resulting in a positive number.
Note that a product is negative, if and only if the factors have opposite signs.

Quote:

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i wonder why from the book i have read, the solution set is I.N. (-4, 0) or - 4 < x < 0

please guide me on this problem

thanks

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Btw, are you aware that you cannot take the log of zero or a negative number?
That is, you have the additional constraint that x^2+ 4x +3 > 0.
It means that when you have solved your inequality, you need to verify that this constraint is satisfied.

Quote:

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Originally Posted by ILikeSerena

Hi rcs!



This is incorrect.
When the base of the log is between 0 and 1 the inequality swaps around.



The last step is incorrect.
If x would be smaller than -4, say -5, then both factors are negative, resulting in a positive number.
Note that a product is negative, if and only if the factors have opposite signs.



Btw, are you aware that you cannot take the log of zero or a negative number?
That is, you have the additional constraint that x^2+ 4x +3 > 0.
It means that when you have solved your inequality, you need to verify that this constraint is satisfied.

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so if you were to solve it

how would it be done?

can you show it to me please...

Ah, well, I don't like giving full solutions.
But perhaps you can redo the calculation taking my remarks into account?

Quote:

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Originally Posted by ILikeSerena

Ah, well, I don't like giving full solutions.
But perhaps you can redo the calculation taking my remarks into account?

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if i would redo it how does the equation look like?

i just wonder why the symbol of inequality from <= changed to < only? how come?

Quote:

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Originally Posted by rcs

i just wonder why the symbol of inequality from <= changed to < only? how come?

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In this particular case it did not change from <= to <.
What you give as a solution is incorrect.
The proper solution is the complement of the interval (-4,0).
That is: x <= -4 or x >= 0.