Firstly, we should try to simplify the equation, to see solutions for . We get . The best west way to simplify quadratic equations is to find the possible factors for the last term  in the general quadratic equation  and those two factors must add up to . Here for example,  and  add up to  and their products is .

So we have to solutions for the equation and we need to know what  we are looking for.

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

Statement 2 tells us that  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 . This allows to see the two solutions for our equation.

Statement 1 tells us that  is between  and . But both possible solutions are in this interval. Therefore statement 1 alone is not sufficient.

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

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

Firstly, we should try to find a simplified equation to better see the possible values for . We get . We can see that  can either be  or .

Statement 1 tells us that the square of  is . It follows that  must be , therefore, this statement is sufficient.

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

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

To begin with this problem, we should try to solve  . It gives us two sets of equations for us to find values for  ;  and . Solving gives us two possible values for  ,  and . Let's see how can the statements help us determine the value of  .

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

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

Taken together these statements, allow us to see that   must be  and therefore are sufficient to answer the question.

To approach this problem, we should firstly set up two possible equations for the value of  ; either  or . 

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

Statement 2 tells us that , if plug in this value for , we get that:

Because there is the absolute value we get two equations:

or 

 must be 8. Plugging in the value for  in our first equation gives us no solution because both sides are of equal value and we end up with .

Therefore, statement 2 alone is sufficient

To conclude, each statement alone is sufficient.

To answer this question, we must find a single value for .

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

Statemnt 2 alone is also insufficient, because it gives us the same possible values for  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 , we should be able to get a value for .

Statement 1 has two unknowns therefore we need another different equation with  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   and from there we can find a single value for .

Both statements together are sufficient.

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

either   or  .

This allows us to find two values for  which are  and , let's see how the statements can help us determine a single value for .

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

Statement 2 tells us that  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.

Therefore,

(1) If , then , and the value of  can vary.  NOT sufficient

(2) Subtracting both  and 7 from each side of  gives .

The value of  can be determined.  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.