Rational equation

Question

(2x+2/x-1)-4=(6x/x+4)

1. Solve the rational equation using a calculator. Graph the left side of the equation as y1 and the right sides as y2. Enter your solution(s):

2.Explain how you used the graph to find the solutions using complete sentences.:

Answer 1

When solving an equation using graphing, the way we are looking at it is considering the left and right-hand-side as two different functions. The equals sign is asking "at which x-values are these two functions the same?" which amounts to looking for the graphs intersect each other! It's a pretty cool idea.

So in your situation, we consider one function

y1 = (2x+2/x-1)-4

and the other function

y2 = (6x/x+4)

Now if you graph these as separate functions, you will see two separate graphs. Then, you can find any locations where the two graphs intersect. The x-coordinates of these intersection points are the solutions to the equation!

Answer 2

(2x+2)(x-1) -4 = 6x/(x+4)

multiply by (x-1)(x+4) to eliminate fractions

(2x+2)(x+4) -4(x-1)(x+4) = 6x(x-1)

2x^2 +10x + 8 - 4x^2 -12x +16 = 6x^2- 6x

8x^2 -4x -24 =0

2x^2 -x -6 = 0

(2x+3)(x-2) =0

x = 2, -3/2

check the answers

plug into the original equation 2 and -3/2

(2x+2)/(x-1) -4 = 6x/(x+4)

try x=2

(2(2)+2)/(2-1) -4 = 6(2)/(2+4)

6/1 -4 = 12/6

2 = 2

x=2 is verified

try -3/2

(2(-3/2)+2)/(-3/2-1) -4 = 6(-3/2)/(-3/2+4)

(-3 +2)/(-5/2) -4 = -9/(4-3/2)

-1/(-5/2)-4 = -9/(5/2)

2/5- 4 = -18/5

-9/2 = - 9/5

x=-3/2 is the 2nd solution

or use a graphing calculator such as TI-83 or online Desmos

then look for the points of intersection, where the graphs of the left side and right side intersect

the left side is a rectangular hyperbola with 2 separate branches

the right side curve intersects each branch at (2,2) and at (-3/2, -3.6)

the x coordinates of the intersection points are the solutions to the original equation