GRE Math : Equations / Inequalities

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Example Questions

Example Question #7 : How To Find The Solution To An Inequality With Division

What is the solution set of the inequality $\dpi{100} \small 3x+8<35$ ?

Possible Answers:

$\dpi{100} \small x<27$

$\dpi{100} \small x<35$

$\dpi{100} \small x>9$

$\dpi{100} \small x>27$

$\dpi{100} \small x<9$

Correct answer:

$\dpi{100} \small x<9$

Explanation:

We simplify this inequality similarly to how we would simplify an equation

$\dpi{100} \small 3x+8-8<35-8$

$\dpi{100} \small \frac{3x}{3}<\frac{27}{3}$

Thus $\dpi{100} \small x<9$

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Example Question #4 : Inequalities

What is a solution set of the inequality 2x+12>42 ?

Possible Answers:

x>4

x<9

x<15

$x>\frac{3}{2}$

x>15

Correct answer:

x>15

Explanation:

In order to find the solution set, we solve 2x+12>42 as we would an equation:

2x+12>42

2x>30

x>15

Therefore, the solution set is any value of x>15 .

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Example Question #3 : How To Find The Solution To An Inequality With Division

$|3x+4|\leq29$

$|2y+6|\leq18$

Quantity A:

The smallest possible value for

Quantity B:

The smallest possible value for

Which of the following is true?

Possible Answers:

A comparison cannot be detemined from the given information.

The two quantities are equal.

Quantity B is larger.

Quantity A is larger.

Correct answer:

Quantity A is larger.

Explanation:

Recall that when you have an absolute value and an inequality like

$|3x+4|\leq29$ ,

this is the same as saying that 3x+4 must be between and . You can rewrite it:

$-29 \leq 3x+4 \leq 29$

To solve this, you just apply your modifications to each and every part of the inequality.

First, subtract :

$-33 \leq 3x \leq 25$

Then, divide by :

$-11\leq x \leq \frac{25}{3}$

Next, do the same for the other equation.

$|2y+6|\leq18$

becomes...

$-18 \leq 2y+6 \leq18$

Then, subtract :

$-24 \leq 2y \leq 12$

Then, divide by :

$-12 \leq y \leq 6$

Thus, the smallest value for is , while the smallest value for is . Therefore, quantity A is greater.

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Example Question #2 : Inequalities

The cost, in cents, of manufacturing $\dpi{100} \small x$ pencils is $\dpi{100} \small 1200+20x$ , where 1200 is the number of cents required to run the factory regardless of the number of pencils made, and 20 represents the per-unit cost, in cents, of making each pencil. The pencils sell for 50 cents each. What number of pencils would need to be sold so that the revenue received is at least equal to the manufacturing cost?

Possible Answers:

$\dpi{100} \small 30$

$\dpi{100} \small 27$

$\dpi{100} \small 40$

$\dpi{100} \small 33$

$\dpi{100} \small 36$

Correct answer:

$\dpi{100} \small 40$

Explanation:

If each pencil sells at 50 cents, $\dpi{100} \small x$ pencils will sell at $\dpi{100} \small 50x$ . The smallest value of $\dpi{100} \small x$ such that

$\dpi{100} \small 50x\geq 1200+20x$

$\dpi{100} \small x\geq 40$

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Example Question #1 : How To Find The Solution To An Inequality With Subtraction

Find the slope of the inequality equation $y-7< x+2y-14$

Possible Answers:

–1

–7

Correct answer:

–1

Explanation:

The answer is:

$y-7< x+2y-14$

$-y-7< x-14$

$-1(-y< x-7)$

$y > -x+7$

From the equation we can see that the slope is –1.

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Example Question #11 : Inequalities

and are both integers.

If y-3x>21 , y>1 , and y<4 , which of the following is a possible value of ?

Possible Answers:

Correct answer:

Explanation:

Take the values of y that are possible, i.e. 2 and 3, and plug them into the first inequality. First, plug in 2. 2 – 3x > 21. Subtract 2 from both sides, and then divide by –3. Don't forget that when you divide or multiply by a negative number in an inequality you must flip the inequality sign. Thus, x < –19/3. Now plug in 3. We find, following the same steps, that when y=3, x < –6. Thus –7 is the correct answer.

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Example Question #213 : Gre Quantitative Reasoning

Quantity A:

The value(s) for which the following function is undefined:

$f(x)=\sqrt{5x-135}$

Quantity B:

Which of the following is true?

Possible Answers:

The two quantities are equal.

Quantity A is larger.

A comparison cannot be detemined from the given information.

Quantity B is larger.

Correct answer:

Quantity B is larger.

Explanation:

This question is not as hard as it seems. Remember that for real numbers, square roots cannot be taken of negative numbers. Therefore, we know that this function is undefined for:

5x-135 < 0

This is simple to solve. Merely add 135 to both sides:

5x<135

Then, divide by :

x < 27

Therefore, quantity A is less than quantity B. This means that quantity B is greater than it.

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Example Question #214 : Gre Quantitative Reasoning

|x+12|<42

Quantity A:

Quantity B:

Which of the following is true?

Possible Answers:

Quantity A is larger.

Quantity B is larger.

A comparison cannot be detemined from the given information.

The two quantities are equal.

Correct answer:

A comparison cannot be detemined from the given information.

Explanation:

Recall that when you have an absolute value and an inequality like

|x+12|<42 ,

this is the same as saying that x+12 must be between and . You can rewrite it:

-42 < x+12<42

To solve this, you merely need to subtract from all three values:

-54 < x <30

Since is between and , it could be both larger or smaller than . Therefore, you cannot determine the relationship based on the given information.

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Example Question #171 : Equations / Inequalities

Quantitative Comparison

x > 0

Column A:

Column B: $\frac{5}{x}$

Possible Answers:

The quantities are equal.

Quantity A is greater.

The relationship cannot be determined from the information provided.

Quantity B is greater.

Correct answer:

The relationship cannot be determined from the information provided.

Explanation:

For quantitative comparison questions involving a shared variable between quantities, the best approach is to test a positive integer, a negative integer, and a fraction. Half of our work is eliminated, however, because the question stipulates that x > 0. We only need to check a positive integer and a positive fraction between 0 and 1. Plugging in 2, we see that quantity A is greater than quantity B. Checking 1/2, however, we find that quantity B is greater than quantity A. Thus the relationship cannot be determined.

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Example Question #221 : Gre Quantitative Reasoning

If –1 < n < 1, all of the following could be true EXCEPT:

Possible Answers:

16n² - 1 = 0

n² < n

n² < 2n

|n² - 1| > 1

(n-1)² > n

Correct answer:

|n² - 1| > 1

Explanation:

N_part_1

N_part_2

N_part_3

N_part_4

N_part_5

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GRE Math : Equations / Inequalities

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #7 : How To Find The Solution To An Inequality With Division

Example Question #4 : Inequalities

Example Question #3 : How To Find The Solution To An Inequality With Division

Example Question #2 : Inequalities

Example Question #1 : How To Find The Solution To An Inequality With Subtraction

Example Question #11 : Inequalities

Example Question #213 : Gre Quantitative Reasoning

Example Question #214 : Gre Quantitative Reasoning

Example Question #171 : Equations / Inequalities

Example Question #221 : Gre Quantitative Reasoning

All GRE Math Resources

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