Logarhythm

180 - 120 = 60

x + 65 + 60 = 180

x + 125 = 180

x = 55

600 ml x 0.7 = 420 ml of alcohol is in the solution. 

Like I understand that you have to find the intervals, but I'm not sure how to solve the inequalities for them...
Three cases: (at least I think)

\(y<\frac{-1}{4} \)     and.    \(-\frac{1}{4} ≤ y < \frac{3}{2}\)     and     \(y ≥ \frac{3}{2}\)

p.s. why are only some of them like "≤ and ≥" but others aren't? 

it's just very confusing to me...
 

\(-99 - \sqrt{7801}= {-b - \sqrt{b^2-4ac} \over 2a}\)

\(-99 - \sqrt{7801}= {-99 - \sqrt{99^2-4(\frac{1}{2})c} \over 2 \cdot \frac{1}{2}}\)

\( \sqrt{7801}= \sqrt{99^2-4(\frac{1}{2})c}\)

\(7801 = 9801 - 2c\)

\(2c = 2000\)

\(c = 1000\)

hope this helps :)

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(x = {-(-5) \pm \sqrt{(-5)^2-4(1)(19)} \over 2(1)}\)

\(x = {5 \pm \sqrt{25-76} \over 2}\)

\(x = {5 \pm i\sqrt{51} \over 2}\)

a = \(\frac{5 + i\sqrt{51}}{2}\)

b = \(\frac{5 - i\sqrt{51}}{2}\)

(a - 1)(b - 1)

= \((\frac{5+i\sqrt{51}}{2} - 1)(\frac{5-i\sqrt{51}}{2} - 1)\)

= \((\frac{3+i\sqrt{51}}{2} )(\frac{3-i\sqrt{51}}{2} )\)

= \(\frac{(3+i\sqrt{51})(3-i\sqrt{51})}{4}\) 

= \(\frac{3^2-(i\sqrt{51})^2}{4}\) ==> difference in squares

= \(\frac{9 + 51}{4}\)

= \(\frac{60}{4}\)

= 15

(edit: I realized I made a mistake. Hopefully this answer is correct.)

3t - 2 + 4 = t - 7t

3t + 2 = -6t

9t + 2 = 0

9t = -2

t = \(\frac{-2}{9}\)

Let's split this into 3 parts: the 42km/h part, the 95km/h part and finally the 55km/h part.

42km/h: 

Since she drove for 20 minutes (a third of an hour), and she was driving at 42 km per hour, we can just take 42 ÷ 3 = 14 km.

95km/h: 

Since we don't know how long she drove, we will save this for last. 

55km/h: 

She drove for 12 minutes (or one fifth of an hour), and she was driving 55 km per hour, we can take 55 ÷ 5 = 11 km.

Now putting them all together, she drove a total of 212 km, and we know that she drove 14 km at 42km/h, and 11 km at 55km/h. 
212 - 14 - 11 = 187 km. 

Now we know how far she drove at 95km/h, and to figure out how long she drove, let's make an equation.

\(\frac{95}{60} = \frac{187}{x}\)  now cross multiply

\(95x = 11220\)

\(x = 11220 \div 95 ≈ 118.105\) 

This means she drove for about 20 + 118 + 12 = 150 minutes. (150.105 approximately if decimal is needed)

Since an hour is 60 minutes. You're driving at the speed of: \(\frac{90 km}{60 min}\), or \(\frac{1.5km}{1min}\). 
This means you'll get 1.5 x 35 = 52.5 km from your starting point :)

If you score 190 points over 4 games, just divide 190 by 4 to get 47.5 points per game. 
The same thing with 200 points in 5 games, 200 ÷ 5 = 40 points per game. 
This means you scored the most points per game :)

First, you find the biggest number that goes into 15 and 40; in other words you find their greatest common factor.

(edit: here's the working out just in case:)
Factors of 15 are: 1, 3, 5, and 15.
Factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40. 
 

In this case, it's 5. Then you divide both the numerator and denominator with that number, giving you the end result of:

\(\frac{15\div 5}{40\div 5} = \frac{3}{8}\)