All GMAT Math Resources
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
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
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
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 #5 : Solving Inequalities
The roots we need to look at are
does not satisfy the inequality.
so does satisfy the inequality.
so does not satisfy the inequality.
so satisfies the inequality.
Therefore the answer is and .
Example Question #6 : Solving Inequalities
Find the domain of .
all positive real numbers
all non-negative real numbers
all real numbers
We want to see what values of x satisfy the equation. is under a radical, so it must be positive.
Example Question #7 : 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 #8 : Solving Inequalities
Find the solution set for :
Divide by -1. Don't forget to switch the direction of the inequality signs since we're dividing by a negative number:
or in interval form, .
Example Question #9 : 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.
Example Question #10 : Solving Inequalities
How many integers satisfy the following inequality:
There are two integers between 2.25 and 4.5, which are 3 and 4.