### Assignments:

Unfinished Assignment Study Questions for Lesson 16

### Lesson Objectives:

- Solve rational equations.

Solve rational equations by multiplying both sides of the equation by the LCD, or least common denominator. This will get rid of all of the fractions.

1/(5x+15)-1/(x^2-9) = 5/(x-3). So if we start by factoring all of our denominators, we have 1/(5(x+3))-1/((x-3)(x+3)) = 5/(x-3). Now we need to multiply both sides by our LCD. 5(x+3)(x-3). So now we can start canceling. The x+3's and the 5's cancel on the first fraction, and then the (x-3)(x+3)'s' cancel on our second fraction, and the (x-3)'s cancel in our third fraction.

So that we're left with x-3-5 = 5*5(x+3), so that we have 25(x+3), which is 25x+75 on the right and x-8 on the left. So if we subtract x from both sides, we get 24x on the right, and if we subtract 75 from both sides, this gives us -83.

This means that x = -83/24 is a possible solution. You have to plug back into our x-values in our equation to see if it is a solution.

So, 1/(5(-83/24)+15)-1/((-83/24)^2-9) = 5(-83/24-3). This gives us -24/31 on the left, and -24/31 on the right. -83/24 checks, so it is a solution.

The Principle of Powers:

If n is positive and a = b, then a^n = b^n.

Solve the following equation for x.

We're given sqrt(x)+sqrt(4+x) = 3. So using the Principle of Powers, we can go ahead and square both sides. This will give us sqrt(x)^2+2(sqrt(x))(sqrt(4+x))+sqrt(4+x)^2 = 9.

This is x+2sqrt(x(4+x))+4+x = 9. So x+2sqrt(4x+x^2)+4+x = 9. Now let's combine our x's and subtract 4 on both sides, so that we get 2x + 2sqrt(4x+x^2) = 5. And then subtract 2x from both sides, and we get 2sqrt(4x+x^2) = 5-2x. And then by the principle of powers, we can square both sides again and get 4(4x+x^2) = 25-2(5)(2x)+4x^2. This gives us 16x+4x^2 = 25-20x+4x^2. And now the 4x^2's cancel on both sides, and we can add 20x to both sides, so that we get 36x = 25.

x = 25/36. So now we need to plug 25/36 back in, to make sure it satisfies our equation. So we have sqrt(25/36)+sqrt(4+25/36) = 3. So that gives us 5/6+sqrt(169/36) = 3, or 5/6+13/6 = 3. That's 18/6 = 3, or 3 = 3, which is a true statement.

So that means that x = 25/36 is a solution.