What would be the value or values of this variable that make the denominators zero?

  a)  x=3 and x=-6 will make the denominator terms zero, hence x may not have those values.

  b)  no solution

The left side can be written over a common denominator so as to make the equation be ...

  ((x+6) -3(x-3))/((x-3)(x+6)) = 9/((x -3)(x +6))

  (-2x +15)/((x -3)(x +6)) = 9/((x -3)(x +6)) . . . . simplify

  -2(x -3)/((x -3)(x +6)) = 0 . . . . . . . . . . . . . . . . subtract the right side

  -2/(x+6) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . cancel factors of (x -3)

This equation has no solution. (There is no value of x that will make -2 = 0.)

Comment on denominator zeros

After you've seen a few instances of the equation

  x + c = 0

it doesn't take long for you to recognize that the solution is ...

  x = -c

That is, when a denominator factor is (x+c), the corresponding value of x that makes this factor zero is x = -c. So, a denominator factor of x+c means the domain must exclude x = -c.

Knowing this, we can write the restrictions on a rational function "by inspection" without having to do anything except mental pattern matching.

Comment on the solution method

By rewriting the rational equation to be something of the form (a/b) = 0, we have the opportunity, as here, to cancel factors that would otherwise result in extraneous solutions. Then, we don't have to be concerned with whether or not the solutions found are extraneous.

A. -4 and 4

B. No solution.

Step-by-step explanation:

The given equation is

A.

Equate the denominators equal to 0 to find the restrictions on the variable.

Therefore, .

B.

We have,

Multiply both sides by (x-4)(x+4).

Add 4 on both sides.

Divide both sides by 5.

Here the solution is x=4 but it is the restricted value.

Therefore, the given equation has no solution.

a. 0

b. x = 1.25

Step-by-step explanation:

The given equation is:

a. The denominators are x and 4x. The values that make a denominator zero are:

b. Solving the equation:

The solution is x = 1.25

X = 3/5

Step-by-step explanation:

2/x=4/5x+2

Find the LCM of the denominator 5x and 1

2/x =4/5x + 2/1

2/x = (4 + 10x)/5x

Cross multiply the equation

2× 5x = (4+ 10x) × x

10x = 4x + 10x^2

Collect like term of the mixed number

10x - 4x = 10x^2

6x = 10x^2

Divide both side by 2x

6x/2x = {10x^2 } / 2x

3 = 5x

Divide both side by the coefficient of x

3/5 = 5x/5

X = 3/5

x=-5

Step-by-step explanation:

Denominators are:

x+5, 5, and 5x+25

5 cannot be turned to 0 since it does not have a variable.

So we have x+5 and 5x+25.

eqyal each denominator to 0 since that is what we want.

x+5=0              5x+25=0

x=-5                 5x=-25

                        x=-25/5

                        x=-5

∴x=-5 for all denominators to make 0.