GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

varsity tutors app store varsity tutors android store

Example Questions

Example Question #431 : Algebra

Solve for \displaystyle x.

\displaystyle (x-3)^2=2x^2+14

Possible Answers:

\displaystyle x=-1, -5

\displaystyle x=1, 5

\displaystyle x=2,3

\displaystyle x=-2,-3

Correct answer:

\displaystyle x=-1, -5

Explanation:

In order to solve this problem, we need to combine like terms and then factor:

\displaystyle (x-3)^2=2x^2+14

\displaystyle x^2-6x+9=2x^2+14

\displaystyle 0=x^2+6x+5

\displaystyle 0=(x+5)(x+1)

\displaystyle x=-1, -5

Example Question #21 : Solving By Factoring

Solve for \displaystyle x:

\displaystyle 4x^{3} - 28x^{2} + 48x = 0

Possible Answers:

\displaystyle x=28,48

\displaystyle x=0,3,4

\displaystyle x=3,4

\displaystyle x=-3,-4

\displaystyle x=0,-3,-4

Correct answer:

\displaystyle x=0,3,4

Explanation:

To solve this we must factor:

The first step is to recognize that all of the terms on the equation's left side contain 4x. Therefore we can pull 4x out:

\displaystyle 4x(x^{2}-7x +12) = 0

Then we factor the inside of the parentheses, realizing that only -3, and -4 add to form -7, and multiply to form 12:

\displaystyle 4x(x - 3)(x - 4) = 0

Now we can see that, for the left side of the equation to equal zero, x can only equal 0, 3, or 4.

Example Question #1511 : Gmat Quantitative Reasoning

Solve the following by factoring:

\displaystyle 2x^2-6x-8=0

Possible Answers:

\displaystyle x=4,-1

\displaystyle x=2,-6

\displaystyle x=-4,1

\displaystyle x=6,-2

Correct answer:

\displaystyle x=4,-1

Explanation:

To solve, divide out a \displaystyle 2 and then factor.

\displaystyle 2(x^2-3x-4)=0

\displaystyle x^2-3x-4=0

\displaystyle (x-4)(x+1)=0

\displaystyle x=4,-1

Example Question #21 : Solving By Factoring

Simplify \displaystyle \frac{3x^{2}-27}{x-3}.

Possible Answers:

\displaystyle \frac{3x+3}{x-3}

\displaystyle 3x+9

\displaystyle x+3

\displaystyle 3x+3

\displaystyle \frac{3}{x-3}

Correct answer:

\displaystyle 3x+9

Explanation:

\displaystyle \frac{3x^{2}-27}{x-3}=\frac{3(x^{2}-9)}{x-3}=\frac{3(x-3)(x+3)}{x-3}

\displaystyle =3(x+3)=3x+9.

Example Question #1 : Absolute Value

Solve \left | 3x - 7 \right |=8\displaystyle \left | 3x - 7 \right |=8.

Possible Answers:

\displaystyle x=5

\displaystyle x=5 or \displaystyle x=

\displaystyle x=-5 or \displaystyle x= \displaystyle \frac{1}{3}

\displaystyle x=5 or \displaystyle x= \displaystyle \frac{1}{3}

\displaystyle x=-\frac{1}{3}

Correct answer:

\displaystyle x=5 or \displaystyle x=

Explanation:

\left | 3x - 7 \right |=8\displaystyle \left | 3x - 7 \right |=8 really consists of two equations: 3x - 7 = \pm 8\displaystyle 3x - 7 = \pm 8

We must solve them both to find two possible solutions.

3x - 7 = 8 \Rightarrow 3x = 15\Rightarrow x = 5\displaystyle 3x - 7 = 8 \Rightarrow 3x = 15\Rightarrow x = 5

3x - 7 = - 8 \Rightarrow 3x = -1\Rightarrow x = -1/3\displaystyle 3x - 7 = - 8 \Rightarrow 3x = -1\Rightarrow x = -1/3

So \displaystyle x=5 or \displaystyle x= .

Example Question #2 : Absolute Value

Solve \left | 2x - 5 \right |\geq 3\displaystyle \left | 2x - 5 \right |\geq 3.

Possible Answers:

x < 1, x > 4\displaystyle x < 1, x > 4

x \leq 1, x\geq 4\displaystyle x \leq 1, x\geq 4

-2 \leq x\leq 5\displaystyle -2 \leq x\leq 5

x \leq -1, x\geq -4\displaystyle x \leq -1, x\geq -4

1 < x < 4\displaystyle 1 < x < 4

Correct answer:

x \leq 1, x\geq 4\displaystyle x \leq 1, x\geq 4

Explanation:

It's actually easier to solve for the complement first.  Let's solve \left | 2x-5 \right |<3\displaystyle \left | 2x-5 \right |< 3.  That gives -3 < 2x - 5 < 3.  Add 5 to get 2 < 2x < 8, and divide by 2 to get 1 < x < 4.  To find the real solution then, we take the opposites of the two inequality signs.  Then our answer becomes x\leq 1 \textsc{ or } x\geq 4.

Example Question #3 : Absolute Value

Give the \displaystyle x-intercept(s), if any, of the graph of the function \displaystyle f (x) = \left | 4x + B \right | - 8 in terms of \displaystyle B

Possible Answers:

\displaystyle \left ( \frac{-B -8 }{4}, 0 \right ), \left ( \frac{-B +8 }{4}, 0 \right )

\displaystyle \left ( \frac{B -8 }{4}, 0 \right ), \left ( \frac{B +8 }{4}, 0 \right )

\displaystyle \left ( B-2, 0 \right ), \left ( B+2, 0 \right )

\displaystyle (B- 8, 0)

\displaystyle \left ( -B-2, 0 \right ), \left ( -B+2, 0 \right )

Correct answer:

\displaystyle \left ( \frac{-B -8 }{4}, 0 \right ), \left ( \frac{-B +8 }{4}, 0 \right )

Explanation:

Set \displaystyle f (x) = 0 and solve for \displaystyle x:

\displaystyle f (x) = 0

\displaystyle \left | 4x + B \right | - 8 = 0

\displaystyle \left | 4x + B \right | = 8

 

Rewrite as a compound equation and solve each part separately:

\displaystyle 4x + B = -8 \textrm{ or } 4x + B = 8

 

\displaystyle 4x + B = -8

\displaystyle 4x + B -B = -B -8

\displaystyle 4x = -B -8

\displaystyle 4x \div 4 =\left ( -B -8 \right ) \div 4

\displaystyle x = \frac{-B -8 }{4}

 

 

\displaystyle 4x + B = 8

\displaystyle 4x + B -B = -B + 8

\displaystyle 4x = -B +8

\displaystyle 4x \div 4 =\left ( -B +8 \right ) \div 4

\displaystyle x = \frac{-B +8 }{4}

Example Question #4 : Absolute Value

A number is ten less than its own absolute value. What is this number?

Possible Answers:

\displaystyle 10

\displaystyle -5

\displaystyle 5

No such number exists.

\displaystyle -10

Correct answer:

\displaystyle -5

Explanation:

We can rewrite this as an equation, where \displaystyle N is the number in question:

\displaystyle N = |N| - 10

A nonnegative number is equal to its own absolute value, so if this number exists, it must be negative.

In thsi case, \displaystyle |N| = - N, and we can rewrite that equation as

\displaystyle N = -N- 10

\displaystyle N +N= -N- 10 +N

\displaystyle 2N = -10

\displaystyle 2N \div 2 = -10\div 2

\displaystyle N = -5

This is the only number that fits the criterion.

Example Question #2 : Understanding Absolute Value

If \displaystyle -1< n< 0, which of the following has the greatest absolute value?

Possible Answers:

\displaystyle \frac{1}{n^{2}}

\displaystyle n^2

\displaystyle \frac{1}{n}

\displaystyle n+1

\displaystyle \frac{n}{2}

Correct answer:

\displaystyle \frac{1}{n^{2}}

Explanation:

Since \displaystyle -1< n< 0, we know the following:  

\displaystyle 0< n^2< 1 ;

\displaystyle -1< \frac{n}{2}< 0;

\displaystyle \frac{1}{n^2}>1;

\displaystyle \frac{1}{n}< -1;

\displaystyle 0< n+1< 1.

Also, we need to compare absolute values, so the greatest one must be either \displaystyle \left | \frac{1}{n^2}\right | or \displaystyle \left | \frac{1}{n}\right |.

We also know that \displaystyle \left | n^2\right |< \left | n\right | when \displaystyle -1< n< 0.

Thus, we know for sure that \displaystyle \left | \frac{1}{n^2}\right |>\left | \frac{1}{n}\right |.

 

Example Question #3 : Understanding Absolute Value

Give all numbers that are twenty less than twice their own absolute value.

Possible Answers:

\displaystyle - 10 \textrm{ and } 20

\displaystyle - 10 \textrm{ and } 10

\displaystyle - 6\frac{2}{3} \textrm{ and } 20

\displaystyle - 6\frac{2}{3} \textrm{ and } 10

No such number exists.

Correct answer:

\displaystyle - 6\frac{2}{3} \textrm{ and } 20

Explanation:

We can rewrite this as an equation, where \displaystyle N is the number in question:

\displaystyle N = 2\cdot |N| - 20

If \displaystyle N is nonnegative, then \displaystyle | N | = N, and we can rewrite this as 

\displaystyle N = 2N - 20

Solve:

\displaystyle N -2N= 2N - 20-2N

\displaystyle -N= - 20

\displaystyle N = 20

 

If \displaystyle N is negative, then \displaystyle | N | =- N, and we can rewrite this as 

\displaystyle N = 2\left (-N \right ) - 20

\displaystyle N = -2N - 20

\displaystyle N + 2N= -2N - 20 + 2N

\displaystyle 3N=- 20

\displaystyle 3N \div 3=- 20\div 3

\displaystyle N=- 6\frac{2}{3}

 

The numbers \displaystyle - 6\frac{2}{3}, 20 have the given characteristics.

Tired of practice problems?

Try live online GMAT prep today.

1-on-1 Tutoring
Live Online Class
1-on-1 + Class
Learning Tools by Varsity Tutors