Solve, 2^(x-1) =5.
You will need to use logarithms to solve this. So, the easiest way to do this is by multiplying each part by "ln". (This is a lowercase L, not an uppercase I.) ln (2^(x-1)) = ln(5) this will turn into (x-1) - ln(2) = ln(5), you are trying to get x-1 alone so you can solve for it, therefore using the laws of an inequality, you then divide to get x-1 alone. This turns into x-1 = ln(5)/ln(2), add one to both sides to get x alone and you have your final answer of x= ln(5)/ln(2) +1
You have 14 grams of C2H4. How many moles, molecules, and atoms of Carbon are there?
Moles: Grams/Grams per Mole. 14g/28g/mol (Because 28 is the mass of 2 carbon atoms). Dividing this will give you 0.5 moles. Now, to convert this to molecules you multiply by the constant 6.022 x 10^23 by the number of moles you got, 0.5. This will give you 3.011 x 10^23 molecules. To find the number of atoms of Carbon, you multiply 3.011 x 10^23, the number of molecules you found by the number of atoms of carbon in the equation. In this example, we have C2, two atoms of Carbon. So you multiply by 2, giving you 6.022 x 10^23 atoms. Coincidently this is the constant you used to find the number of molecules. Simple as that!
Simplify and solve for the domain of 10X/(X-3) ≥ 5X/(X+5).
First, set your equation equal to 0.... 10X/(X-3) - 5X/(X+5) ≥ 0, by subtracting 5X/(X+5), the right-hand side of your equation must equal Zero. Next, in order to solve this, you need to set your denominators equal to each other. In order to do this with X in the denominator, you will have to simply multiply (X-3) by (X+5). And I know its very tempting to go ahead and factor them, but leave them be! In order to solve for the denominator, they need to be left as (X-3)(X+5). Now, remember what you do to the denominator, you must do to the numerator. So, If you multiplied the denominator of 10X/(X-3) by (X+5) to get the denominator equal to (X-3)(X+5), you must now multiply 10X by (X+5) and same goes for 5X/(X+5), you should then multiply 5X by (X-3). You should now have an equation that looks like, 10X(X+5)/((X-3)(X+5)) - 5X(X-3)/((X-3)(X-5)), now factor the numerators! 10X multiplied by X gives you 10X^2. and 10X times 5 will give you 50X. Now, 5X multiplied by X gives you 5X^2, and 5X times -3, is -15X. So, you should now have an equation that looks something like, (10X^2+50X)/((X-3)(X+5)) - (5X^2-15X)/((X-3)(X+5)). Now that we have our numerators factored and our denominators set equal, we can combine! So you will have (10X^2+50X) - (5X^2-15X)/((X-3)(X+5). Which can be simplified by factoring to 5X^2-35X/((X-3)(X+5)) ≥ 0. (If needed the factoring of this can be explained) Now that we have our new simplified equation, in order to find the domain restrictions, we must set the denominators equal to 0. So X-3=0, and X+5=0. When we solve for this we will get that X CANNOT equal 3 or -5. Now, set your numerator equal to 0. 5X^2-35X=0 (Factor this) = 5X(X-7)=0, Set 5X=0 and (X-7)=0. You get X=0 and X=7. Sometimes it helps to draw a sign diagram to visually show this, and if we did you would see that (-∞, -5] U (0,3] U (7, ∞). The brackets show the numbers that are NOT included in our denominator and the parenthesis show the numbers that are.