All GMAT Math Resources
Example Questions
Example Question #12 : Solving Inequalities
What value of satisfies both of the following inequalities?
Solve for in both inequalities:
(1)
Subtract 9 from each side of the inequality:
Then, divide by . Remember to switch the direction of the sign from "less than or equal to" to "greater than or equal to."
(2)
Add to both sides of the inequality:
Subtract 7 from each side of the inequality:
Divide each side of the inequality by 3:
From solving both inequalities, we find such that:
Only is in that interval .
and are less than 1.33, so they are not in the interval.
and are more than 2, so they are not in the interval.
Example Question #13 : Solving Inequalities
Which of the following could be a value of , given the following inequality?
The inequality that is presented in the problem is:
Start by moving your variables to one side of the inequality and all other numbers to the other side:
Divide both sides of the equation by . Remember to flip the direction of the inequality's sign since you are dividing by a negative number!
Reduce:
The only answer choice with a value greater than is .
Example Question #123 : Algebra
If and are two integers, which of the following inequalities would be true?
First let's solve each of the inequalities:
Don't forget to flip the direction of the sign when dividing by a negative number:
is the correct answer. is an integer greater than 3 and is greater than 9. Therefore, the sum of and is greater than 12.
is not true. and are two positive integers as is greater than 3 and is greater than 9. The sum of two positive integers cannot be a negative number.
is not true. and are two positive integers as is greater than 3 and is greater than 9. The division of two positive numbers is positive and therefore cannot be less than 0.
is not true. is greater than 3 and is greater than 9. The product of and cannot be less than 3.
is not true. and are positive. Therefore, the product of and is negative and cannot be greater than 0.
Example Question #13 : Solving Inequalities
If an integer satisfies both of the above inequalities, which of the following is true about ?
First, we solve both inequalities:
If satisfies both inequalities, then is greater than 5 AND is greater than 11. Therefore is greater than 11.
is the correct answer.
Example Question #1205 : Gmat Quantitative Reasoning
Solve the following inequality:
We start by simplifying our inequality like any other equation:
Now we must remember that when we divide by a negative, the inequality is flipped, so we obtain:
Example Question #14 : Solving Inequalities
Solve the following inequality:
Solving inequalities is very similar to solving equations, but we need to remember an important rule:
If we multiply or divide by a negative number, we must switch the direction of the inequality. So a "greater than" sign will become a "less than" sign and vice versa.
We are given
Start by moving the and the over:
Simplify to get the following:
Then, we will divide both sides of the equation by . Remember to switch the direction of the inequality sign!
So,
Example Question #127 : Algebra
and . If the greatest number of the three, then what are the possible values of ?
The question is equivalent to asking when and are both true statements.
We can solve for in the first equation:
Now use substitution to solve the inequality
Therefore, if and only if .
Similarly, we can solve for in the first equation:
Now use substitution to solve the inequality
Therefore, if and only if .
We combine these results, and conclude that both and hold - that is, is the greatest of the three - if and only if .
Example Question #21 : Solving Inequalities
Solve for :
The first step to solving this inequality is to subtract 18 from all three segments of the equation:
8 (- 18) < -5x + 18 ( - 18) < 103 ( - 18)
-10 < -5x < 85
We then divide the entire equation by -5. According to inequality rules, when we multiply or divide an inequality by a negative number, we must flip the inequality signs.
(-10/-5) > (-5x/-5) > (85/-5)
2 > x > -17 or in conventional layout -17 < x < 2
Example Question #21 : Solving Inequalities
Solve:
or
or
In order to solve this inequality, we must split it into two inequalities:
and
We then solve each, remembering of course that, when multiplying or dividing by a negative number, we must switch the direction of the inequality sign.
and
Therefore:
Example Question #128 : Algebra
Give the solution set of the inequality
To solve a rational inequality, move all expressions to the left first:
The boundary points of the solution set will be the points at which:
- that is, ;
;
; that is, .
None of these values will be included in the solution set, since equality is not allowed by the inequality symbol.
Test the intervals
by choosing a value in each interval and testing the truth of the inequality.
: test
True; include the interval
: test
False; exclude the interval
: test
True; include the interval .
: test
False; exclude the interval .
The solution set of the inequality is .