5 math questions

1. 1/x-2 - 1/x-1 = 1/3x
1          1          1  
------ - ------- = -------
x-2      x-1        3x

(x-1) - (x-2)        1  
---------------- = -------
(x-2)(x-1)          3x

x-1 - x +2          1  
---------------- = -------
(x-2)(x-1)         3x

  1                     1  
---------------- = -------
(x-2)(x-1)          3x

3x = (x-2) (x-1)
3x = x^2 – 3x +2
x^2 – 6x +2 = 0
Using quadratic formula, x = {+6+-sq. root[(36-(4)(1)(2))} /2

x = 5.646 or 0.3542

2. In (8x + 1) = 2 In (x+2)
In (8x + 1) = In (x+2)^2
(8x + 1) = (x+2)^2
8x + 1 = x^2 + 4x +4
x^2 - 4x +3 = 0
(x -1) (x -3) = 0

x = 1 or x = 3

3. sin (2兀X^2) = 0
Let k = 2兀X^2
Sin k = 0
k = 0, 兀, or 2兀
2兀X^2 =兀
2x^2 = 1
x =1/(sq root(2)) = sq root(2)/2 
2兀X^2 =2兀
x = 1

x = 0, x =sq root(2)/2, x = 1

4. Find the equation of the line which intersects the line y=3x-5 at right angles when x=4.
Put your answer in the form y=mx+b, and select the appropriate value of m.
When x = 4, y = 3(4) – 5 = 7
The other linear equation,
y = -1/3 x + b
7 = -1/3(4) + b
7 = -4/3 + b
b = 7 +4/3 = 25/3

The equation is:  y = -1/3 x + 25/3 (b = 8 and 1/3)

5. Find the equation of the line that is the mirror image of the line y=4x - 5 in the line x=-2. Put your answer in the form y=mx+b, and select the appropriate value of b.
The 2 lines intersect at x = 2, y =4x – 5, y = 4(2) -5 = 3 point of intersection (2, 3)
The slope of the other linear equation -4
y = -4x + b
3 = -4(2) + b
b = 11

The other line y = - 4x + 11


6. 3x^2 - 2/ 3x^2 +2 = ½
(3x^2)(2x^2) - 2 +2(2x^2) = ½(2x^2)
(3x^2)(2x^2) - 2 +2(2x^2) - (x^2) = 0
(3x^2)^2 + (3x^2) - 2 = 0
(3x^2 + 2)(3x^2 – 1) = 0
3x^2 = - 2 (no real root)
3x^2 = 1
x^2 = 1/3
x = 1/sq. root(3)

x = sq. root(3)/3