Math

Question Solve the equation 2x1=8x|2 x-1|=8 x and check graphically. Select the correct choice: A. x=x= (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) or B. There is no solution.

Studdy Solution

STEP 1

Assumptions1. The given equation is x1=8x|x-1|=8x . We are asked to solve for xx
3. We are also asked to check the solution graphically

STEP 2

The absolute value function x|x| can be defined as a piecewise functionx={xif x0xif x<0|x| =\begin{cases}x & \text{if } x \geq0 \\ -x & \text{if } x <0\end{cases}

STEP 3

We can apply this definition to our equation 2x1=8x|2x-1|=8x and create two separate equations1. 2x1=8x2x-1=8x when 2x102x-1 \geq0
2. (2x1)=8x-(2x-1)=8x when 2x1<02x-1 <0

STEP 4

Let's solve the first equation 2x1=8x2x-1=8xRearrange the equation to isolate xx2x8x=12x -8x =1

STEP 5

implify the equationx=1-x =1

STEP 6

Divide by -6 to solve for xxx=16=16x = \frac{1}{-6} = -\frac{1}{6}

STEP 7

Now, let's check if 16-\frac{1}{6} is a valid solution for the first equation. Remember, we assumed 2x102x-1 \geq0 when we derived this equation. Substitute x=16x = -\frac{1}{6} into 2x12x-12(16)1=131=432(-\frac{1}{6}) -1 = -\frac{1}{3} -1 = -\frac{4}{3}Since 43<0-\frac{4}{3} <0, our assumption was incorrect and 16-\frac{1}{6} is not a valid solution for the equation 2x1=x|2x-1|=x.

STEP 8

Now, let's solve the second equation (2x1)=8x-(2x-1)=8xRearrange the equation to isolate xx2x+1=8x-2x +1 =8x

STEP 9

implify the equationx=x =

STEP 10

Divide by10 to solve for xxx=10x = \frac{}{10}

STEP 11

Now, let's check if 10\frac{}{10} is a valid solution for the second equation. Remember, we assumed x<0x- <0 when we derived this equation. Substitute x=10x = \frac{}{10} into xx-(10)=5=45(\frac{}{10}) - = \frac{}{5} - = -\frac{4}{5}Since 45<0-\frac{4}{5} <0, our assumption was correct and 10\frac{}{10} is a valid solution for the equation x=8x|x-|=8x.

STEP 12

To check the solution graphically, plot the functions y=2xy=|2x-| and y=8xy=8x on the same graph. The x-coordinate of the point where the two graphs intersect is the solution to the equation. You should find that the graphs intersect at x=10x=\frac{}{10}, confirming our solution.
The correct choice is A. x=10x=\frac{}{10}.

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