GMAT Math : Solving inequalities

Example Questions

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Example Question #1 : Solving Inequalities

How many integers  can complete this inequality?

Explanation:

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.

Example Question #2 : Inequalities

Solve .

Explanation:

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 #2 : Solving Inequalities

Solve .

Explanation:

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 #2 : Solving Inequalities

Solve .

Explanation:

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 : Inequalities

Solve .

Explanation:

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 #3 : Solving Inequalities

Find the domain of .

all non-negative real numbers

all positive real numbers

all real numbers

Explanation:

We want to see what values of x satisfy the equation.  is under a radical, so it must be positive.

Example Question #4 : Solving Inequalities

Solve the inequality:

Explanation:

When multiplying or dividing by a negative number on both sides of an inequality, the direction of the inequality changes.

Example Question #3 : Solving Inequalities

Find the solution set for :

Explanation:

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 #4 : Solving Inequalities

Which of the following is equivalent to ?

Explanation:

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 #5 : Solving Inequalities

How many integers  satisfy the following inequality:

One

Five

Three

Four

Two