Firstly, we should try to simplify the equation, to see solutions for $\displaystyle x$. We get $\displaystyle (x-3)(x-2)$. The best west way to simplify quadratic equations is to find the possible factors for the last term $\displaystyle c$ in the general quadratic equation $\displaystyle ax^{2}+bx+c$ and those two factors must add up to $\displaystyle b$. Here for example, $\displaystyle -2$ and $\displaystyle -3$ add up to $\displaystyle -5$ and their products is $\displaystyle 6$.

So we have to solutions for the equation and we need to know what $\displaystyle x$ we are looking for.

Statement 1 tells us that $\displaystyle x$ is positive, however, the two possible solutions are positive and therefore, statement 1 doesn't help us find the correct solution for $\displaystyle x$.

Statement 2 tells us that $\displaystyle x$ is smaller than 3. Only one of our solutions is smaller than 3. Therefore statement 2 alone is sufficient.

First, we should try to simplify the quadratic equation, and we get $\displaystyle (x+3)(x-1)=0$. This allows to see the two solutions for our equation.

Statement 1 tells us that $\displaystyle x$ is between $\displaystyle -4$ and $\displaystyle 2$. But both possible solutions are in this interval. Therefore statement 1 alone is not sufficient.

Statement 2 tells us that $\displaystyle x$ is an integer, which we already knew by reducing the equation. Therefore, this statements doesn't help us find a single value for $\displaystyle x$

Statements 1 and 2 together are still insufficient, since none can help us find a single value for $\displaystyle x$.

Firstly, we should try to find a simplified equation to better see the possible values for $\displaystyle x$. We get $\displaystyle (x-1)(x-5)$. We can see that $\displaystyle x$ can either be $\displaystyle 1$ or $\displaystyle 5$.

Statement 1 tells us that the square of $\displaystyle x$ is $\displaystyle 25$. It follows that $\displaystyle x$ must be $\displaystyle 5$, therefore, this statement is sufficient.

Statement 2 tells us that the absolute value of $\displaystyle x$ is $\displaystyle 5$, therefore $\displaystyle x$ must be $\displaystyle 5$ and therefore the statement is also sufficient alone.

Note that it is possible to answer with either statement only because $\displaystyle x$ can either be $\displaystyle 5$ or $\displaystyle 1$. If $\displaystyle x$ could have been $\displaystyle -5$ or $\displaystyle 5$ than statements 1 and 2 would have been insufficient.

To begin with this problem, we should try to solve  $\displaystyle \left | x-4\right |=6$. It gives us two sets of equations for us to find values for  $\displaystyle x$; $\displaystyle x-4=6$ and $\displaystyle x-4=-6$. Solving gives us two possible values for  $\displaystyle x$, $\displaystyle -10$ and $\displaystyle 2$. Let's see how can the statements help us determine the value of  $\displaystyle x$.

Statement 1 gives us an other unknown for the value of  $\displaystyle x$. Therefore, this statement is insufficient.

Statement 2 gives an absolute value for this unknown $\displaystyle A$. But we don't know what other values is $\displaystyle A$ equal to.

Taken together these statements, allow us to see that  $\displaystyle x$ must be $\displaystyle -10$ and therefore are sufficient to answer the question.

To approach this problem, we should firstly set up two possible equations for the value of  $\displaystyle x$; either $\displaystyle x-8=A$ or $\displaystyle x-8=-A$. 

Statement 1 tells us that $\displaystyle A$ is in fact zero. Than the equations return a single value for  $\displaystyle x$, therefore statement 1 alone is sufficient. 

Statement 2 tells us that $\displaystyle A=x-8$, if plug in this value for $\displaystyle A$, we get that:

$\displaystyle |x-8|=x-8$

Because there is the absolute value we get two equations:

$\displaystyle x-8=x-8$ 

$\displaystyle x=x+0$

$\displaystyle 0=0$

or 

$\displaystyle x-8=-(x-8)$

$\displaystyle x-8=-x+8$

$\displaystyle 2x=16$

$\displaystyle x=8$

$\displaystyle x$ must be 8. Plugging in the value for $\displaystyle A$ in our first equation gives us no solution because both sides are of equal value and we end up with $\displaystyle 0=0$.

Therefore, statement 2 alone is sufficient

To conclude, each statement alone is sufficient.

To answer this question, we must find a single value for $\displaystyle x$.

Statement 1 gives us an equation with two possible solutions for $\displaystyle x$. Therefore, statement 1 alone is not sufficient, since $\displaystyle x$ can either be $\displaystyle -3$ or $\displaystyle 3$

Statemnt 2 alone is also insufficient, because it gives us the same possible values for $\displaystyle x$ as the equation in statement 1.

When two statements give us the same the information the answer is either both statements together are sufficient or statements 1 and 2 together are not sufficient. Here neither statement allowed us to answer, it follows that statements 1 and 2 together are not sufficient.

To find a value for $\displaystyle x$, we should be able to get a value for $\displaystyle A$.

Statement 1 has two unknowns therefore we need another different equation with $\displaystyle A,B$ to be able to find values for these unknowns.

Statement 2 alone is also insufficient because just as statement 1 has two variables and therefore we need more information to solve it.

Taking together these equations, by adding both sides we get $\displaystyle 2A=4$  and from there we can find a single value for $\displaystyle x$.

Both statements together are sufficient.

Firstly we should try to see what are the possible values for $\displaystyle x$, by solving the equations given by the absolute value:

either  $\displaystyle \frac{1}{x-1}=3$ or  $\displaystyle \frac{1}{x-1}=-3$.

This allows us to find two values for $\displaystyle x$ which are $\displaystyle \frac{2}{3}$ and $\displaystyle \frac{4}{3}$, let's see how the statements can help us determine a single value for $\displaystyle x$.

Statement 1 tells us that $\displaystyle x$ must be greater than one. Only one of our solutions for $\displaystyle x$ is greater than one. Therefore, statement 1 alone is sufficient.

Statement 2 tells us that $\displaystyle x$ is not an integer, however both solutions are not integer values and therefore statement 2 doesn't help us find a single solution.

Statement 1 alone is sufficient.

$\displaystyle \dpi{100} \small z+x=y$

Therefore, $\displaystyle \dpi{100} \small x=y-z$

(1) If $\displaystyle \dpi{100} \small z=7$, then $\displaystyle \dpi{100} \small y-z=y-7$, and the value of $\displaystyle \dpi{100} \small y-7$ can vary. $\displaystyle \dpi{100} \small \rightarrow$ NOT sufficient

(2) Subtracting both $\displaystyle \dpi{100} \small z$ and 7 from each side of $\displaystyle \dpi{100} \small y+7=z$ gives $\displaystyle \dpi{100} \small y-z=-7$.

The value of $\displaystyle \dpi{100} \small y-z$ can be determined. $\displaystyle \dpi{100} \small \rightarrow$ SUFFICIENT

We are looking for one value of x since the quesiton specifies we only want a positive solution.

Statement 1 isn't sufficient because there are an infinite number of integers greater than 1. 

Statement 2 tells us that x = 2 or x = –2, and we know that we only want the positive answer. Then Statement 2 is sufficient.