All GMAT Math Resources
Example Questions
Example Question #1 : Solving Inequalities
How many integers can complete this inequality?
3 is added to each side to isolate the term:
Then each side is divided by 2 to find the range of :
The only integers that are between 5 and 9 are 6, 7, and 8.
The answer is 3 integers.
Example Question #2 : Solving Inequalities
Solve .
Subtract 10:
Divide by 3:
We must carefully check the endpoints. is greater than and cannot equal , yet CAN equal 2. Therefore should have a parentheses around it, and 2 should have a bracket: is in
Example Question #3 : Solving Inequalities
Solve .
Subtract 3 from both sides:
Divide both sides by :
Remember: when dividing by a negative number, reverse the inequality sign!
Now we need to decide if our numbers should have parentheses or brackets. is strictly greater than , so should have a parentheses around it. Since there is no upper limit here, is in .
Note: Infinity should ALWAYS have a parentheses around it, NEVER a bracket.
Example Question #4 : Solving Inequalities
Solve .
must be positive, except when . When , .
Then we know that the inequality is only satisfied when , and . Therefore , which in interval notation is .
Note: Infinity must always have parentheses, not brackets. has a parentheses around it instead of a bracket because is less than , not less than or equal to .
Example Question #4 : Solving Inequalities
Solve .
The roots we need to look at are
:
Try
, so
does not satisfy the inequality.
:
Try
so does satisfy the inequality.
:
Try
so does not satisfy the inequality.
:
Try .
so satisfies the inequality.
Therefore the answer is and .
Example Question #5 : Solving Inequalities
Find the domain of .
all real numbers
all non-negative real numbers
all positive real numbers
We want to see what values of x satisfy the equation. is under a radical, so it must be positive.
Example Question #6 : Solving Inequalities
Solve the inequality:
When multiplying or dividing by a negative number on both sides of an inequality, the direction of the inequality changes.
Example Question #1 : Solving Inequalities
Find the solution set for :
Subtract 7:
Divide by -1. Don't forget to switch the direction of the inequality signs since we're dividing by a negative number:
Simplify:
or in interval form, .
Example Question #5 : Solving Inequalities
Which of the following is equivalent to ?
To solve this problem we need to isolate our variable .
We do this by subtracting from both sides and subtracting from both sides as follows:
Now by dividing by 3 we get our solution.
or
Example Question #1 : Solving Inequalities
How many integers satisfy the following inequality:
Five
Three
Two
One
Four
Two
There are two integers between 2.25 and 4.5, which are 3 and 4.