GMAT Math : Solving inequalities

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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Example Question #1 : Solving Inequalities

How many integers \dpi{100} \small (x) can complete this inequality?

7< 2x-3 <15

Possible Answers:

\dpi{100} \small 3

\dpi{100} \small 5

\dpi{100} \small 6

\dpi{100} \small 4

\dpi{100} \small 9

Correct answer:

\dpi{100} \small 3

Explanation:

7< 2x-3 <15

3 is added to each side to isolate the \dpi{100} \small x term:

10< 2x <18

Then each side is divided by 2 to find the range of \dpi{100} \small x:

5< x <9

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 5 < 3x + 10 \leq 16.

Possible Answers:

Correct answer:

Explanation:

5 < 3x + 10 \leq 16

Subtract 10: -5 < 3x \leq 6

Divide by 3: -5/3 < x \leq 2

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 .

Possible Answers:

(- \infty , -2)

(-2, \infty )

(2, \infty )

[-2, \infty ]

[-2, \infty )

Correct answer:

(-2, \infty )

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 (-2, \infty )

Note: Infinity should ALWAYS have a parentheses around it, NEVER a bracket.

Example Question #4 : Solving Inequalities

Solve (x-1)^{2}(x+4)<0.

Possible Answers:

(-\infty , -4)

(-\infty , -4]

(-\infty , 4)

[-\infty , -4]

[4, \infty )

Correct answer:

(-\infty , -4)

Explanation:

(x-1)^{2} must be positive, except when .  When , (x-1)^{2}=0.

Then we know that the inequality is only satisfied when , and x\neq 1.  Therefore , which in interval notation is (-\infty , -4).

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

Solve .

Possible Answers:

Correct answer:

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 #6 : Solving Inequalities

Find the domain of y=\sqrt{x^{2}-4}.

Possible Answers:

x\geq 2, x\leq -2

all non-negative real numbers

all positive real numbers

all real numbers

Correct answer:

x\geq 2, x\leq -2

Explanation:

We want to see what values of x satisfy the equation.  x^{2}-4 is under a radical, so it must be positive.

x^{2}-4\geq 0

x^{2}\geq 4

x\geq 2, x\leq -2

Example Question #7 : Solving Inequalities

Solve the inequality:

Possible Answers:

Correct answer:

Explanation:

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 :

Possible Answers:

Correct answer:

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 #9 : Solving Inequalities

Which of the following is equivalent to ?

Possible Answers:

Correct answer:

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 #2 : Solving Inequalities

How many integers  satisfy the following inequality:

Possible Answers:

Four

Five

One

Three

Two

Correct answer:

Two

Explanation:

There are two integers between 2.25 and 4.5, which are 3 and 4.

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