Domain: Multiple domains

The domain of the equation is the set of domain values (of the variable in the equation) for which all algebraic expressions in the equation are well defined,

From this, of course - we exclude numbers for which arithmetic operations are not defined,

In the expression on the left side of the given equation:

$9(4x-\frac{5}{x})=20(3x-\frac{6}{x+1})$

There is a multiplication operation between fractions whose denominators contain algebraic expressions that include the variable of the equation,

These fractions are considered defined as long as the expressions in their denominators are not equal to zero (since division by zero is not possible),

Therefore, the domain of definition of the variable in the equation will be obtained from the requirement that these expressions (in the denominators of the fractions) do not equal zero, as shown below:

For the fraction inside of the parentheses in the expression on the left side we obtain the following:

$\boxed{ x\neq0}$

For the fraction inside of the parentheses in the expression on the right side we obtain the following:

$x+1\neq0 \\$Proceed to solve the second inequality above (in the same way as solving an equation):

$x+1\neq0 \\ \boxed{x\neq-1}$

Therefore, the correct answer is answer A.

Note:

It should be noted that the above inequality is a point inequality and not a trend inequality (meaning it negates equality: $(\neq)$and does not require a trend: (<,>,\leq,\geq) ) which is solved exactly like solving an equation. This is unlike solving a trend inequality where different solution rules apply depending on the type of expressions in the inequality, for example: solving a first-degree inequality with one variable (which has only first-degree and lower algebraic expressions), is solved almost identically to solving an equation. However any division or multiplication operation of both sides by a negative number requires that the trend be revered.

The domain of the equation is the set of domain values (of the variable in the equation) for which all algebraic expressions in the equation are well defined,

From this, of course - we exclude numbers for which arithmetic operations are not defined,

In the expression on the left side of the given equation:

$\frac{7}{x+5}=\frac{6}{13x}$

There is a multiplication operation between fractions whose denominators contain algebraic expressions that include the variable of the equation.

These fractions are considered to be defined as long as the expression in their denominators is not equal to zero (given that division by zero is not possible),

Therefore, the domain of definition of the variable in the equation will be obtained from the requirement that these expressions (in the denominators of the fractions) do not equal zero, as follows:

For the fraction in the expression on the left side we obtain:

$x+5\neq0 \\$For the fraction in the expression on the right side we obtain:

$13x\neq0$

$$We will solve these inequalities (in the same way as solving an equation):

$x+5\neq0 \\ \boxed{x\neq-5}$

$13x\neq0 \hspace{8pt}\text{/:13} \\ \boxed{x\neq0}$$$

Therefore, the correct answer is answer A.

Note:

It should be noted that the above inequality is a point inequality and not a directional inequality (meaning it negates equality: $(\neq)$and does not require direction: (<,>,\leq,\geq) ) which is solved exactly like solving an equation. This is unlike solving a directional inequality where different solution rules apply depending on the type of expressions in the inequality. For example: solving a first-degree inequality with one variable (which only has first-degree algebraic expressions and below), is solved almost identically to solving an equation. However, any division or multiplication of both sides by a negative number requires reversing the direction.

To solve this problem, we need to identify the values of $y$ for which the denominator of the expression becomes zero, as these values are not part of the domain.

First, let's simplify the denominator of the given equation:

Original equation: $\frac{\sqrt{15} + \frac{34}{z}}{4y - 12 + \frac{8}{2}} = 5$

Simplifying the terms: $34:z \text{ remains as it is for simplification purposes, and } \frac{8}{2} = 4$

Thus, the denominator becomes: $(4y - 12 + 4) = 4y - 8$

We need to ensure the denominator is not zero to avoid undefined expressions: $4y - 8 \neq 0$

Simplify and solve for $y$: $4y - 8 \neq 0 \implies 4y \neq 8 \implies y \neq 2$

Therefore, the equation is undefined for $y = 2$, and the answer is that the field of application excludes $y = 2$.

Given the possible choices for the problem, the correct choice is: $y\operatorname{\ne}2$

The solution to this problem is $y \neq 2$.

In order to solve the equation, start by removing the denominators.

To do this, we'll multiply the denominators:

$(2x+1)^2\cdot(2x+1)+(x+2)^2\cdot(x+2)=4.5x(2x+1)(x+2)$

Open the parentheses on the left side, making use of the distributive property:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=4.5x(2x+1)(x+2)$

Continue to open the parentheses on the right side of the equation:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=4.5x(2x^2+5x+2)$

Simplify further:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=9x^3+22.5x+9x$

Go back and simplify the parentheses on the left side of the equation:

$8x^3+8x^2+2x+4x^2+4x+1+x^3+4x^2+4x+2x^2+8x+8=9x^3+22.5x+9x$

Combine like terms:

$9x^3+18x^2+18x+9=9x^3+22.5x+9x$

Notice that all terms can be divided by 9 as shown below:

$x^3+2x^2+2x+1=x^3+2.5x+x$

Move all numbers to one side:

$x^3-x^3+2x^2-2.5x^2+2x-x+9=0$

We obtain the following:

$0.5x^2-x-1=0$

In order to remove the one-half coefficient, multiply the entire equation by 2

$x^2-2x-2=0$

Apply the square root formula, as shown below-

$\frac{2±\sqrt{12}}{2}$

Apply the properties of square roots in order to simplify the square root of 12:

$\frac{2±2\sqrt{3}}{2}$Divide both the numerator and denominator by 2 as follows:

$1±\sqrt{3}$