Answer:
x = 8
Step-by-step explanation:
log₅ (10x − 1) = log₅ (9x + 7)
10x − 1 = 9x + 7
x = 8

We have 2 denominators that we need to get rid of. Whenever there are the denominators, all we have to do is multiply all whole equation with the denominators.
Our denominators are both 2 and x+1. Therefore, we multiply the whole equation by 2(x+1)
![\frac{x}{2}[2(x+1)]-\frac{2}{x+1}[2(x+1)] = 1[2(x+1)]](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B2%7D%5B2%28x%2B1%29%5D-%5Cfrac%7B2%7D%7Bx%2B1%7D%5B2%28x%2B1%29%5D%20%3D%201%5B2%28x%2B1%29%5D)
Then shorten the fractions.
![\frac{x}{2}[2(x+1)]-\frac{2}{x+1}[2(x+1)] = 1[2(x+1)]\\x(x+1)-2(2)=1(2x+2)](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B2%7D%5B2%28x%2B1%29%5D-%5Cfrac%7B2%7D%7Bx%2B1%7D%5B2%28x%2B1%29%5D%20%3D%201%5B2%28x%2B1%29%5D%5C%5Cx%28x%2B1%29-2%282%29%3D1%282x%2B2%29)
Distribute in all.

We should get like this. Because the polynomial is 2-degree, I'd suggest you to move all terms to one place. Therefore, moving 2x+2 to another side and subtract.

We are almost there. All we have to do is, solving for x by factoring. (Although there are more than just factoring but factoring this polynomial is faster.)

Thus, the answer is x = 3, -2
You haven't provided the original coordinates or the figure, therefore, I cannot give an exact answer. However, I will help you with the concept.
For rotation 180° about the origin, the signs of both the x any y coordinates are changed.
<u>This can be modeled as follows:</u>
(x,y) ......> rotation 180° about the origin ........> (-x,-y)
<u>Examples:</u>
(1,2) .......> rotation 180° about the origin ........> (-1,-2)
(2,-19) ....> rotation 180° about the origin ........> (-2,19)
(-3,-8) .....> rotation 180° about the origin ........> (3,8)
(-5,7) ......> rotation 180° about the origin ........> (5,-7)
Based on the above, all you have to do to get the coordinates of C' is change the signs of both the x and y coordinates
Hope this helps :)
11.5 is a rational number
Let

and

be the sides of the rectangle. The perimeter is given to be 500m, so we are maximizing the area function

subject to the constraint

.
From the constraint, we find

so we can write the area function independently of

:

Differentiating and setting equal to zero, we find one critical point:

which means

, so in fact the largest area is achieved with a square fence that surrounds an area of

.