GMAT Math : Algebra

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Example Questions

Example Question #31 : Solving Inequalities

Solve for :

|12 - 5x | < 160

Possible Answers:

$\left ( - \infty , 12 \frac{11}{12}\right ) \cup \left ( 13\frac{3}{4} , \infty \right )$

$\varnothing$

$\left (12 \frac{11}{12}, 13\frac{3}{4} \right )$

$\left ( - \infty , -29 \frac{3}{5} \right ) \cup \left ( 34 \frac{ 2}{5}, \infty \right )$

$\left ( -29 \frac{3}{5}, 34 \frac{ 2}{5} \right )$

Correct answer:

$\left ( -29 \frac{3}{5}, 34 \frac{ 2}{5} \right )$

Explanation:

|12 - 5x | < 160

can be rewritten as the inequality

-160<12 - 5x < 160

-160- 12 <12 - 5x - 12 < 160 - 12

-172 < - 5x < 148

$\frac{-172}{-5} > \frac{- 5x }{-5} > \frac{148}{-5}$ (note the change in direction of the inequality symbols)

$34 \frac{ 2}{5} > x > -29 \frac{3}{5}$

This is the set $\left ( -29 \frac{3}{5}, 34 \frac{ 2}{5} \right )$ .

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Example Question #31 : Solving Inequalities

Solve the following inequality:

$-3+9x+14\leq 6x+1-2x$

Possible Answers:

$x\leq-2$

$x\geq2$

$x\geq \frac{4}{3}$

$x\leq-\frac{4}{3}$

$x\leq1$

Correct answer:

$x\leq-2$

Explanation:

Like any other equation, we solve the inequality by first grouping like terms. Grouping the terms on the left side of the equation and the constants on the right side of the equation, we have:

$-3+9x+14\leq 6x+1-2x$

$9x-6x+2x\leq 1+3-14$

$5x\leq -10$

$x=\frac{-10}{5}$

$x\leq-2$

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Example Question #141 : Algebra

There is water tank already $\frac{4}{7}$ full. If Jose adds 5 gallons of water to the water tank, the tank will be $\frac{13}{14}$ full. How many gallons of water would the water tank hold if it were full?

Possible Answers:

$25$

$14$

$15$

$5$

$20$

Correct answer:

$14$

Explanation:

In this case, we need to solve for the volume of the water tank, so we set the full volume of the water tank as $x$ . According to the question, $\frac{4}{7}$ -full can be replaced as $\frac{4}{7}x$ . $\frac{13}{14}$ -full would be $\frac{13}{14}x$ . Therefore, we can write out the equation as:

$\frac{4}{7}x+5=\frac{13}{14}x$ .

Then we can solve the equation and find the answer is 14 gallons.

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Example Question #142 : Algebra

There exists a set = {1, 2, 3, 4}. Which of the following defines a function of ?

Possible Answers:

two are functions

$f=\left \{ (2,3),(1,4),(2,1),(3,2),(4,4) \right \}$

$h=\left \{ (2,1),(3,4),(1,4),(2,1),(4,4) \right \}$

$g=\left \{ (3,1),(4,2),(1,1) \right \}$

none are functions

Correct answer:

$h=\left \{ (2,1),(3,4),(1,4),(2,1),(4,4) \right \}$

Explanation:

Let's look at $f,g,and\ h$ and see if any of them are functions.

1. = {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)}: This cannot be a function of because two of the ordered pairs, (2, 3) and (2, 1) have the same number (2) as the first coordinate.

2. = {(3, 1), (4, 2), (1, 1)}: This cannot be a function of because it contains no ordered pair with first coordinate 2. Because the set = {1, 2, 3, 4}, we need an ordered pair of the form (2, ) .

3. = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}: This is a function. Even though two of the ordered pairs have the same number (2) as the first coordinate, is still a function of because (2, 1) is simply repeated twice, so the two ordered pairs with first coordinate 2 are equal.

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Example Question #1 : Understanding Functions

Let be a function that assigns $x^{2}$ to each real number . Which of the following is NOT an appropriate way to define ?

Possible Answers:

all are appropriate ways to define

$x\rightarrow x^{2}$

$f(x)=x^{2}$

$y=x^{2}$

$f(y)=x^{2}$

Correct answer:

$f(y)=x^{2}$

Explanation:

This is a definition question. The only choice that does not equal the others is $f(y)=x^{2}$ . This describes a function that assigns $x^{2}$ to some number , instead of assigning $x^{2}$ to its own square root, .

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Example Question #143 : Algebra

If $f(x)=x^{2}$ , find $\frac{f(x+h)-f(x)}{h}$ .

Possible Answers:

$x^{2}+4x+4$

2x+h

$x^{2}$

$x^{2}+2xh+h^{2}$

Correct answer:

2x+h

Explanation:

We are given f(x) and h, so the only missing piece is f(x + h).

$f(x+h)=(x+h)^{2}=x^{2}+2xh+h^{2}$

Then $\frac{f(x+h)-f(x)}{h}= \frac{x^{2}+2xh+h^{2}-x^{2}}{h} = \frac{2xh+h^{2}}{h}=2x+h$

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Example Question #4 : Understanding Functions

Give the range of the function:

$f (x) = \left\{\begin{matrix} 2x - 1 & & x < -2\\ -3x& & -2\leq x\leq 2\ \\5 & & \; x > 2 \end{matrix}\right.$

Possible Answers:

$(-\infty , 6]$

$(-\infty , -6] \cup [-5, 6]$

$[-6, \infty )$

$(-\infty , 5]$

$[-5, \infty)$

Correct answer:

$(-\infty , 6]$

Explanation:

We look at the range of the function on each of the three parts of the domain. The overall range is the union of these three intervals.

On $(-\infty , -2)$ , f (x) takes the values:

x < -2

2x - 1 < 2 (-2) - 1

2x - 1 < -5

$f\left (x \right )< -5$

or $(-\infty , -5)$

On [-2, 2] , f (x) takes the values:

$-2 \leq x\leq 2$

$-3(-2) \geq -3x\geq -3 (2)$

$6 \geq -3x\geq -6$

$-6 \leq f(x)\leq 6$ ,

or [-6, 6]

On $(2, \infty)$ , f (x) takes only value 5.

The range of f (x) is therefore $(-\infty , -5) \cup [-6, 6] \cup \left \{ 5\right \}$ , which simplifies to $(-\infty , 6]$ .

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Example Question #6 : Understanding Functions

A sequence begins as follows:

8, 12, ...

It is formed the same way that the Fibonacci sequence is formed. What are the next two numbers in the sequence?

Possible Answers:

8, 4

20, 32

20, 35

18, 27

16, 20

Correct answer:

20, 32

Explanation:

Each term of the Fibonacci sequence is formed by adding the previous two terms. Therefore, do the same to form this sequence:

8 + 12= 20

12+20=32

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Example Question #1 : Understanding Functions

Give the inverse of f (x) = 5x - 7

Possible Answers:

$f^{-1}(x)= -\frac{1}{5}x +7$

$f^{-1}(x)=- \frac{1}{5}x + \frac{7}{5}$

$f^{-1}(x)= \frac{1}{5}x +7$

$f^{-1}(x)= \frac{1}{5}x - \frac{7}{5}$

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

Correct answer:

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

Explanation:

The easiest way to find the inverse of f (x) is to replace f (x) in the definition with , switch with , and solve for in the new equation.

y = 5x - 7

x = 5y - 7

x + 7 = 5y - 7 + 7

x + 7 = 5y

$\frac{1}{5} \cdot\left ( x + 7 \right ) = \frac{1}{5} \cdot 5y$

$\frac{1}{5} x + \frac{7}{5} = y$

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

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Example Question #144 : Algebra

Define $f (x) = x^{3} - 8$ . Give $f^{-1} (x)$

Possible Answers:

$f^{-1}(x) = \sqrt[3]{x +2}$

$f^{-1}(x) = 8+ \sqrt[3]{x }$

$f^{-1}(x) = \sqrt[3]{x +8}$

$f^{-1}(x) = 2- \sqrt[3]{x }$

$f^{-1}(x) = 2+ \sqrt[3]{x }$

Correct answer:

$f^{-1}(x) = \sqrt[3]{x +8}$

Explanation:

The easiest way to find the inverse of f (x) is to replace f (x) in the definition with , switch with , and solve for in the new equation.

$y = x^{3} - 8$

$x= y^{3} - 8$

$x +8= y^{3} - 8 +8$

$x +8= y^{3}$

$\sqrt[3]{x +8}= \sqrt[3]{y^{3}}$

$y = \sqrt[3]{x +8}$

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #31 : Solving Inequalities

Example Question #31 : Solving Inequalities

Example Question #141 : Algebra

Example Question #142 : Algebra

Example Question #1 : Understanding Functions

Example Question #143 : Algebra

Example Question #4 : Understanding Functions

Example Question #6 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #144 : Algebra

All GMAT Math Resources

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