Solve the equation check for extraneous solutions

6/6-4x/=8x+4 I used/ / for the brackets
posssible answers
x=1
x=5/2 or x=2
x=5/2
x=5/2 or x=1
Not looking for just the answer, would like to know how to get it.

or, if as in your later post, we have

6|6-4x| = 8x+4, then
if 6-4x >= 0, x=1 as above

else, if 6-4x < 0, we have
-6(6-4x) = 8x+4
-36 + 24x = 8x + 4
16x = 40
x = 5/2

so, x = 1 or 5/2

Thank you Steve

6(6-4x) = 8x+4

36 - 24x = 8x+4
32 = 32x
x = 1

To solve the equation (6/6-4x) / (8x+4) = 8x+4, follow these steps:

1. Simplify the expression on the left-hand side of the equation: 6/(6-4x) / (8x+4) = 8x+4.

2. Multiply both sides of the equation by (6-4x) to eliminate the fraction on the left-hand side:

6 = (8x+4)(6-4x).

3. Expand the expression on the right-hand side of the equation using the distributive property:

6 = 48x - 32x^2 + 24 - 16x.

4. Combine like terms on the right-hand side:

6 = -32x^2 + 32x + 24.

5. Move all terms to one side of the equation to set it equal to zero:

0 = -32x^2 + 32x + 18.

6. Rearrange the equation in descending order of the exponent:

-32x^2 + 32x + 18 = 0.

Now, to solve this quadratic equation, you can either factor, complete the square, or use the quadratic formula. In this case, factoring is not straightforward, so we can apply the quadratic formula:

7. The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a), where the quadratic equation is in the form ax^2 + bx + c = 0.

For our equation -32x^2 + 32x + 18 = 0, we have a = -32, b = 32, and c = 18.

Substituting these values into the quadratic formula gives:

x = (-32 ± √(32^2 - 4(-32)(18))) / (2(-32)).

8. Simplify the expression under the square root:

x = (-32 ± √(1024 + 2304)) / (-64).

x = (-32 ± √3328) / (-64).

x = (-32 ± √(256 * 13)) / (-64).

x = (-32 ± 16√13) / (-64).

9. Further simplify by factoring out a common factor of -16:

x = (-32 ± (-16)√13) / (-64).

x = (1/2) (2 ± √13).

10. Now, we have two possible solutions:

x = (1/2) (2 + √13) = 1 + (1/2)√13.

x = (1/2) (2 - √13) = 1 - (1/2)√13.

So, the possible solutions to the equation are x ≈ 1 + (1/2)√13 and x ≈ 1 - (1/2)√13. You can check each possible solution by substituting it back into the original equation and verifying if both sides of the equation are equal.