GMAT Math : Algebra

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Example Questions

Example Question #431 : Algebra

Solve for .

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

Possible Answers:

x=2,3

x=-1, -5

x=-2,-3

x=1, 5

Correct answer:

x=-1, -5

Explanation:

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

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

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

0=x^2+6x+5

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

x=-1, -5

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Example Question #432 : Algebra

Solve for :

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

Possible Answers:

x=28,48

x=0,-3,-4

x=-3,-4

x=3,4

x=0,3,4

Correct answer:

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:

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

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

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.

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Example Question #431 : Algebra

Solve the following by factoring:

2x^2-6x-8=0

Possible Answers:

x=4,-1

x=-4,1

x=2,-6

x=6,-2

Correct answer:

x=4,-1

Explanation:

To solve, divide out a and then factor.

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

x^2-3x-4=0

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

x=4,-1

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Example Question #432 : Algebra

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

Possible Answers:

3x+9

$\frac{3}{x-3}$

3x+3

x+3

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

Correct answer:

3x+9

Explanation:

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

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

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Example Question #1 : Absolute Value

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

Possible Answers:

x=5

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

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

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

x=5 or $\dpi{100} -\frac{1}{3}$

Correct answer:

x=5 or $\dpi{100} -\frac{1}{3}$

Explanation:

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

We must solve them both to find two possible solutions.

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

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

So x=5 or $\dpi{100} -\frac{1}{3}$ .

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Example Question #2 : Absolute Value

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

Possible Answers:

$1 < x < 4$

$-2 \leq x\leq 5$

$x \leq 1, x\geq 4$

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

$x < 1, x > 4$

Correct answer:

$x \leq 1, x\geq 4$

Explanation:

It's actually easier to solve for the complement first. Let's solve $\left | 2x-5 \right |<3$ . 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 $x\leq 1 \textsc{ or } x\geq 4$ .

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Example Question #2 : Absolute Value

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

Possible Answers:

(B- 8, 0)

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

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

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

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

Correct answer:

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

Explanation:

Set f (x) = 0 and solve for :

f (x) = 0

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

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

Rewrite as a compound equation and solve each part separately:

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

4x + B = -8

4x + B -B = -B -8

4x = -B -8

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

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

4x + B = 8

4x + B -B = -B + 8

4x = -B +8

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

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

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Example Question #3 : Absolute Value

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

Possible Answers:

No such number exists.

Correct answer:

Explanation:

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

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, |N| = - N , and we can rewrite that equation as

N = -N- 10

N +N= -N- 10 +N

2N = -10

$2N \div 2 = -10\div 2$

N = -5

This is the only number that fits the criterion.

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Example Question #2 : Absolute Value

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

Possible Answers:

$\frac{1}{n}$

$\frac{n}{2}$

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

n^2

n+1

Correct answer:

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

Explanation:

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

0< n^2<1 ;

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

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

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

0<n+1<1 .

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

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

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

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Example Question #5 : Absolute Value

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

Possible Answers:

No such number exists.

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

$- 10 \textrm{ and } 20$

$- 10 \textrm{ and } 10$

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

Correct answer:

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

Explanation:

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

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

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

N = 2N - 20

Solve:

N -2N= 2N - 20-2N

-N= - 20

N = 20

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

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

N = -2N - 20

N + 2N= -2N - 20 + 2N

3N=- 20

$3N \div 3=- 20\div 3$

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

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

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #431 : Algebra

Example Question #432 : Algebra

Example Question #431 : Algebra

Example Question #432 : Algebra

Example Question #1 : Absolute Value

Example Question #2 : Absolute Value

Example Question #2 : Absolute Value

Example Question #3 : Absolute Value

Example Question #2 : Absolute Value

Example Question #5 : Absolute Value

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