How to find the solution to an inequality with addition - GRE Quantitative Reasoning
Card 0 of 9
What values of x make the following statement true?
|x – 3| < 9
What values of x make the following statement true?
|x – 3| < 9
Solve the inequality by adding 3 to both sides to get x < 12. Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.
Solve the inequality by adding 3 to both sides to get x < 12. Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.
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Solve for 
.
Solve for
Absolute value problems always have two sides: one positive and one negative.
First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.
Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).
We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.
Absolute value problems always have two sides: one positive and one negative.
First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.
Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).
We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.
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If –1 < w < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?
If –1 < w < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?
3_w_/2 will become greater than 1 as soon as w is greater than two thirds. It will likewise become less than –1 as soon as w is less than negative two thirds. All the other options always return values between –1 and 1.
3_w_/2 will become greater than 1 as soon as w is greater than two thirds. It will likewise become less than –1 as soon as w is less than negative two thirds. All the other options always return values between –1 and 1.
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Quantitative Comparison
Quantity A: 3_x_ + 4_y_
Quantity B: 4_x_ + 3_y_
Quantitative Comparison
Quantity A: 3_x_ + 4_y_
Quantity B: 4_x_ + 3_y_
The question does not give us any specifics about the variables x and y.
If we substitute the same numbers for x and y (say, x = 1 and y = 1), the two expressions are equal.
If we substitute different number in for x and y (say, x = 2 and y = 1), the two expressions are not equal.
If there are two possible outcomes, then we need more information to determine which quantity is greater. Don't be afraid to pick "The relationship cannot be determined from the information given" as an answer choice on the GRE!
The question does not give us any specifics about the variables x and y.
If we substitute the same numbers for x and y (say, x = 1 and y = 1), the two expressions are equal.
If we substitute different number in for x and y (say, x = 2 and y = 1), the two expressions are not equal.
If there are two possible outcomes, then we need more information to determine which quantity is greater. Don't be afraid to pick "The relationship cannot be determined from the information given" as an answer choice on the GRE!
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If 
and 
, then which of the following could be the value of 
?
If
To solve this problem, add the two equations together:
The only answer choice that satisfies this equation is 0, because 0 is less than 4.
To solve this problem, add the two equations together:
The only answer choice that satisfies this equation is 0, because 0 is less than 4.
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Let 
be an integer such that 
.
Quantity A: 
Quantity B: 
Let
Quantity A:
Quantity B:
The expression 
can be rewritten as 
.
The only integer that satisfies the inequality is 0.
Thus, Quantity A and Quantity B are equal.
The expression
The only integer that satisfies the inequality is 0.
Thus, Quantity A and Quantity B are equal.
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What values of 
make the statement 
true?
What values of
First, solve the inequality 
:
Since we are dealing with absolute value, 
must also be true; therefore:
First, solve the inequality
Since we are dealing with absolute value,
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Find all solutions of the inequality 
.
Find all solutions of the inequality
Start by subtracting 3 from each side of the inequality. That gives us 
. Divide both sides by 2 to get 
. Therefore every value for 
where 
is a solution to the original inequality.
Start by subtracting 3 from each side of the inequality. That gives us
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Find all solutions of the inequality 
.
Find all solutions of the inequality
Start by subtracting 13 from each side. This gives us 
. Then subtract 
from each side. This gives us 
. Divide both sides by 2 to get 
. Therefore all values of 
where 
will satisfy the original inequality.
Start by subtracting 13 from each side. This gives us
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