# GMAT Math : Solving inequalities

## Example Questions

### Example Question #11 : Solving Inequalities

What is the lowest value the integer  can take?

Explanation:

The lowest value n can take is 6.

### Example Question #11 : Solving Inequalities

What value of  will make the following expression negative:

Explanation:

Our first step is to simplify the expression. We need to remember our order of operations or PEMDAS.

First distribute the 0.4 to the binomial.

Now distribute the 10 to the binomial.

Now multiply 0.6 by 5

Remember to flip the sign of the inequation when multiplying or dividing by a negative number.

220 will make the expression negative.

### Example Question #13 : Solving Inequalities

Solve for

Explanation:

To solve this problem all we need to do is solve for

### Example Question #14 : Solving Inequalities

What value of  satisfies both of the following inequalities?

Explanation:

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 #1 : How To Find The Solution To An Inequality With Division

Which of the following could be a value of , given the following inequality?

Explanation:

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 #15 : Solving Inequalities

If  and  are two integers, which of the following inequalities would be true?

Explanation:

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 #16 : Solving Inequalities

If an integer  satisfies both of the above inequalities, which of the following is true about ?

Explanation:

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 #17 : Solving Inequalities

Solve the following inequality:

Explanation:

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 #18 : Solving Inequalities

Solve the following inequality:

Explanation:

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 #19 : Solving Inequalities

and . If  the greatest number of the three, then what are the possible values of  ?

Explanation:

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 .

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