Thread: A few Problems I need urgent help with Logs, Inequalities and some trig.

1. A few Problems I need urgent help with Logs, Inequalities and some trig.

Hi again! I'm being a serial pest, I know it- I'm sorry! If you can only help with one or two of these I would be grateful! thanks so much!

I need help with this inequality its a quadratic division- I' not sure how to approach it? Do I factor or use the Q formula?

x^2- 7x +10/ x^2 + 8x + 15 is < or = to 0

Have I solved this correctly?

log x + log(x -15)=2

log x =2- log(x -15)

x =2-x -15
x+x=2-15
2x= -13
x=-13/2

Ive done this wrong haven't I?

And the Trig:

sin2x + sin5x=0

sin2x= -sin5x

2x=-5x???

Not doing this one right either!

And finally:

x^3 + x^2 + x +1 find all zeros real and complex - I just need to know how to tackle said problem, not the answer.

Thank you for any help at all!

2. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Originally Posted by Rhys101
I need help with this inequality its a quadratic division- I' not sure how to approach it? Do I factor or use the Q formula?

x^2- 7x +10/ x^2 + 8x + 15 is < or = to 0
I doubt this is the inequality you were given to solve. In any case, it has not solutions. Also, note that a / b <= 0 iff a and b have opposite signs and b ≠ 0.

Originally Posted by Rhys101
log x + log(x -15)=2
Use the fact that log(x) + log(y) = log(x * y).

Originally Posted by Rhys101
sin2x + sin5x=0
Use sum-to-product identities.

Originally Posted by Rhys101
x^3 + x^2 + x +1 find all zeros real and complex
Guess one root x₁ (an integer); use long division to divide the original polynomial by x - x₁ (the remainder is 0 by the polynomial remainder theorem). You get a quadratic equation that you can solve using the quadratic formula.

3. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Thank you I will have a look at what youtold me! however the first one was a question on my sample exam! So that is weird? trick question?

So for the log one I get

log x^2-15x=2
do I go to log x^2-15x-2=0 and somhow get rid of the log? and then factor? Thanks for your help!

4. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Originally Posted by Rhys101
Thank you I will have a look at what youtold me! however the first one was a question on my sample exam! So that is weird? trick question?
More likely you stated the problem incorrectly. Was it supposed to be $\displaystyle \frac{x^2- 7x +10}{x^2 + 8x + 15}\leq0?$ Because if so, you should have used parentheses around the numerator and denominator. Otherwise we have to read your mind to know what you meant.

If you did write the problem correctly, then there really are no solutions. But you would have to verify this in your work.

5. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Originally Posted by Rhys101
log x^2-15x=2
do I go to log x^2-15x-2=0 and somhow get rid of the log?
Again, learn how to use parentheses correctly. That should be $\displaystyle \log\left(x^2-15x\right)=2.$ To answer your question, the next step is to convert the equation to exponential form. Then solve the resulting quadratic.

In general, if $\displaystyle \log_ba = n,$ then $\displaystyle b^n=a.$

6. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Awesome, thank you Reckoner- yes I did write the first equation incorrectly there should have been parantheses around them- thanks for pointing that out- I need to learn how to use the way you guys can type this stuff out. Thanks again!

Also with no base specified, do I assume a base of ten? Or make up my own?

7. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Originally Posted by Rhys101
Also with no base specified, do I assume a base of ten? Or make up my own?
My guess that the base is specified, but possible several pages back. For example, there may be a note at the beginning of the book chapter or exam saying that, unless noted otherwise, all logarithms are to the base 2.

8. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Umm, sorry its off the same sample exam... there were not any such specifications so does that mean I have to assume a base of 10?

If so that gives me 10^2= x^2-15x (hope that came out ok) so am I looking at x^2-15x-100? because Im finding that hard to factor? or should I just use the quadratic equation?

Thanks yet again!

Rhys.

9. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

In relation to the 1st question expression = (x-5)(x-2)/(x+5)(x+3) Just by looking can you see that if x is between 2 and 5 inclusive the denominator is positive and the numerator is zero or negative. Hence we have the answer to the question

10. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Originally Posted by Rhys101
Umm, sorry its off the same sample exam... there were not any such specifications so does that mean I have to assume a base of 10?
In your class, has your instructor been using log to mean log base 10? If so, then yes, use a base of 10.

10 is commonly used, but some courses or texts might assume a base of 2 (particularly computer science texts), and some even use log to mean log base $\displaystyle e$ (but we usually see that written as ln).

Originally Posted by Rhys101
If so that gives me 10^2= x^2-15x (hope that came out ok) so am I looking at x^2-15x-100? because Im finding that hard to factor? or should I just use the quadratic equation?
You should be able to factor that. But yes, you can always fall back on the quadratic formula.

We're looking for two numbers whose product is -100 and whose sum is -15. Since the product must be negative, they must have opposite signs. And since the sum is negative, the larger number (in absolute value) must be negative and the smaller number positive.

11. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Originally Posted by Rhys101
Umm, sorry its off the same sample exam
Since this is a sample exam, you don't need to submit a definite answer. The point is to know how to solve such problems.

Originally Posted by Rhys101
... so doe that mean I have to assume a base of 10?
If you silently assume that the base is 10 and say that log(x) + log(x - 15) = 2 is equivalent to x = 20, this statement would not be true in general. On the other hand, if you explicitly assume that the base is 10 and say that log₁₀(x) + log₁₀(x - 15) = 2 is equivalent to x = 20, you would both produce a true statement and demonstrate to the teacher that you can solve such problems, even if you don't know the exact base intended by the question.

Unless you have an explicit convention, you should not silently assume anything. The crew of Air Canada Flight 143 assumed that to convert the volume of jet fuel in liters into weight in kilograms, the weight had to be multiplied by 1.77. In reality, 1.77 converted liters to pounds, while 0.803 was needed to convert liters to kilograms. As a result, they ended up running out of fuel in the middle of the flight.

12. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Hello, Rhys101!

$\displaystyle \text{Solve for }x\!:\;\;\frac{x^2- 7x +10}{x^2 + 8x + 15} \:\le\:0$
Factor: .$\displaystyle \frac{(x-2)(x-5)}{(x+3)(x+5)}$

The "critical values" are: .$\displaystyle x \;=\;\text{-}5,\;\text{-}3,\;2,\;5$

The four values divide the number line into five intervals.

. . $\displaystyle \begin{array}{ccccccccc} --- & \bullet & --- & \bullet & --- & \bullet & --- & \bullet & --- \\ & \text{-}5 && \text{-}3 && 2 && 5 \end{array}$

Test a value in each interval and see if it satisfies the inequality.

$\displaystyle x = \text{-}6\!:\;\;\frac{(\text{-}8)(\text{-}1)}{(\text{-}3)(\text{-}1)} \:=\:[+]\;\text{ . . . no}$

$\displaystyle x = \text{-}4\!:\;\;\frac{(\text{-}6)(\text{-}9)}{(\text{-}1)(1)} \:=\:[-]\;\text{ . . . yes}$

$\displaystyle x = 0\!:\;\;\frac{(\text{-}2)(\text{-}5)}{(3)(5)} \:=\: [+]\;\text{ . . . no}$

$\displaystyle x = 3\!:\;\;\frac{(1)(\text{-}2)}{(5)(7)} \:=\:[-]\;\text{ . . . yes}$

$\displaystyle x = 6\!:\;\;\frac{(4)(1)}{(9)(11)} \:=\:[+]\;\text{ . . . no}$

Therefore: .$\displaystyle (\text{-}5 ,\,\text{-}3)\,\cup\,(2,\,5)$

$\displaystyle \text{Given: }\:P(x) \:=\:x^3 + x^2 + x +1$

$\displaystyle \text{Find all real and complex zeros.}$

We see that $\displaystyle x = -1$ is a zero of the polynomial.
. . Hence, $\displaystyle (x+1)$ is a factor. .**

Dividing, we get: .$\displaystyle P(x) \:=\:(x+1)(x^2+1) \:=\:0$

. . $\displaystyle \begin{array}{ccccccc}x+1\:=\:0 & \Rightarrow & \boxed{x \:=\:\text{-}1} \\ x^2+1 \:=\:0 & \Rightarrow & x^2 \:=\:\text{-}1 & \Rightarrow & \boxed{x \:=\:\pm i} \end{array}$

** . Or we can factor "by grouping".

$\displaystyle x^3 + x^2 + x + 1 \;=\;x^2(x+1) + (x+1) \;=\;(x+1)(x^2+1)$

13. Re: A few Problems I need urgent help with Logs, Inequalities and some trig.

Thank you very much to everyone who answered! You have all helped me out a lot! Thank you!