GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #131 : Algebra

\displaystyle x^{2}-5x=-6 What is \displaystyle x?

(1) \displaystyle x is positive

(2) \displaystyle x< 3

Possible Answers:

Statement 1 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Statement 2 alone is sufficient.

Each statement alone is sufficient.

Correct answer:

Statement 2 alone is sufficient.

Explanation:

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.

 

 

Example Question #1 : Linear Equations, One Unknown

\displaystyle x^{2}+2x=3. What is \displaystyle x?

(1) \displaystyle -4< x< 2

(2) \displaystyle x is an integer

Possible Answers:

Each statement alone is sufficient.

Statement 1 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Statement 2 alone is sufficient.

Correct answer:

Statements 1 and 2 together are not sufficient.

Explanation:

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.

Example Question #131 : Algebra

\displaystyle x^{2}-6x=-5. What is \displaystyle x?

(1) \displaystyle x^{2}=25

(2) \displaystyle \left | x\right |= 5

Possible Answers:

Statement 2 alone is sufficient.

Each statement alone is sufficient.

Both statements together are insufficient.

Statement 1 alone is sufficient.

Statements 1 and 2 together are not sufficient.

Correct answer:

Each statement alone is sufficient.

Explanation:

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.

Example Question #931 : Data Sufficiency Questions

What is \displaystyle x if,  \displaystyle \left | x-4\right |=6?

(1) \displaystyle x=A

(2) \displaystyle \left | A\right |=10

Possible Answers:

Each statement alone is sufficient.

Statements 1 and 2 together are not sufficient.

Both statements together are sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Correct answer:

Both statements together are sufficient.

Explanation:

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.

Example Question #11 : Dsq: Solving Linear Equations With One Unknown

\displaystyle \left | x-8\right |=A. What is  \displaystyle x?

(1) \displaystyle A=0

(2) \displaystyle A=x-8

Possible Answers:

Each statement alone is sufficient.

Statement 2 alone is sufficient.

Statement 1 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Correct answer:

Each statement alone is sufficient.

Explanation:

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.

Example Question #933 : Data Sufficiency Questions

What is \displaystyle x?

(1) \displaystyle x^{2}=9

(2) \displaystyle \left | x^{3} \right |=27

Possible Answers:

Statements 1 and 2 taken together are not sufficient.

Statement 2 alone is sufficient.

Each statement alone is suffcient.

Both statements together are sufficient.

Statement 1 alone is sufficient.

Correct answer:

Statements 1 and 2 taken together are not sufficient.

Explanation:

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.

Example Question #131 : Algebra

\displaystyle x=A, What is \displaystyle x?

(1) \displaystyle A+B= 3

(2) \displaystyle A-B=1

Possible Answers:

Statements 1 and 2 together are not sufficient.

Both statements together are sufficient.

Each statement alone is sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Correct answer:

Both statements together are sufficient.

Explanation:

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.

Example Question #931 : Data Sufficiency Questions

\displaystyle \left |\frac{1}{x-1} \right |=3 and \displaystyle x is different than \displaystyle 1. What is \displaystyle x?

(1) \displaystyle x>1

(2) \displaystyle x is not an integer. 

 

Possible Answers:

Statements 1 and 2 together are sufficient.

Each statement alone is sufficient.

Both statements together are sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Correct answer:

Statement 1 alone is sufficient.

Explanation:

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.

Example Question #1 : Equations

If \dpi{100} \small z+x=y, what is the value of \dpi{100} \small x?

(1)\dpi{100} \small z=7

(2) \dpi{100} \small y+7=z

Possible Answers:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Correct answer:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Explanation:

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

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

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

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

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

Example Question #2 : Equations

If \dpi{100} \small x> 0, what is the value of x?

Statement 1: \dpi{100} \small x> 1

Statement 2: \dpi{100} \small x^{2}-4=0

 

Possible Answers:

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Statement 2 ALONE is sufficient, but Statement 1 is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but Statement 2 is not sufficient.

Correct answer:

Statement 2 ALONE is sufficient, but Statement 1 is not sufficient.

Explanation:

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.

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