mistake mystery

[nanase]

guys i need help in reasoning this :
(7x+12)/x > 3
7x+12 > 3x
4x > -12
x > -3

but if you sketch the graph it's obvious that my answer is wrong.
what is wrong with the steps?

 

[nanase]

may I ask how long for a thread to be displayed publicly? this one is still awaiting approval even though posted 2 hours ago

 

[JeffM]

First, this is a volunteer site, with most of the volunteers residing in North America. If you ask a question between midnight and dawn in North America, you may not get an answer for several hours.

Your final answer and the reasoning to get there are both good.

 

[pka]

guys i need help in reasoning this :
(7x+12)/x > 3
7x+12 > 3x
4x > -12
x > -3
but if you sketch the graph it's obvious that my answer is wrong.
what is wrong with the steps?

I have always advised students to compare theirs to zero, as in
\(\displaystyle \frac{7x+12}{x}-3 >0 \\ \dfrac{4x+12}{x} >0\)
From the above, we easily see that the sign of \(\displaystyle \frac{4x+12}{x}\) changes at \(\displaystyle -3~\&~0\)
At \(\displaystyle x=-2,~\frac{4x+12}{x}<0\) thus at once we know the solution \(\displaystyle x<-3\text{ or }x>0\), \(\displaystyle (-\infty,-3)\cup (0,\infty)\)

 

[mmm4444bot]

may I ask how long for a thread to be displayed publicly? this one is still awaiting approval even though posted 2 hours ago

Hello. The first time you logged in, there was an alert at the top of your screen which asked you to look at the Read Before Posting notice.

? The answer to your question is clearly listed (in three places, for anyone who follows the links).

Cheers

\(\;\)

 

[Dr.Peterson]

guys i need help in reasoning this :
(7x+12)/x > 3
7x+12 > 3x
4x > -12
x > -3

but if you sketch the graph it's obvious that my answer is wrong.
what is wrong with the steps?

In case it isn't clear to you what you did wrong, it's in your first step. In multiplying by x, and not changing the direction of the comparison, you are assuming that x is positive.

We generally avoid multiplying an inequality by anything whose sign is not known. If you want to do it that way, you need to use separate cases:

• If x > 0, then (7x+12)/x > 3 --> 7x + 12 > 3x --> 4x > -12 --> x > -3; this is always true if x > 0
• If x < 0, then (7x+12)/x > 3 --> 7x + 12 < 3x --> 4x < -12 --> x < -3
• Since x is the denominator, x can't = 0

Putting these together, either x > 0, or x < -3.

What pka did is the standard method, which feels easier or safer to most of us. But there are some problems for which cases are more or less necessary.

 

[nanase]

In case it isn't clear to you what you did wrong, it's in your first step. In multiplying by x, and not changing the direction of the comparison, you are assuming that x is positive.

We generally avoid multiplying an inequality by anything whose sign is not known. If you want to do it that way, you need to use separate cases:

• If x > 0, then (7x+12)/x > 3 --> 7x + 12 > 3x --> 4x > -12 --> x > -3; this is always true if x > 0
• If x < 0, then (7x+12)/x > 3 --> 7x + 12 < 3x --> 4x < -12 --> x < -3
• Since x is the denominator, x can't = 0

Putting these together, either x > 0, or x < -3.

What pka did is the standard method, which feels easier or safer to most of us. But there are some problems for which cases are more or less necessary.

thank you sir, from your explanation I can see what is lacking in my steps. cheers

 

[nanase]

First, this is a volunteer site, with most of the volunteers residing in North America. If you ask a question between midnight and dawn in North America, you may not get an answer for several hours.

Your final answer and the reasoning to get there are both good.

thank you for the info sir, I am sorry if my post sounded pushy or demanding. I was just trying to ask politely but still sorry if my written sentence sounds the opposite