GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #4 : Solving Inequalities

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

Possible Answers:

[4, \infty )

[-\infty , -4]

(-\infty , -4)

(-\infty , 4)

(-\infty , -4]

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 #2 : 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 #3 : Solving Inequalities

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

Possible Answers:

x\geq 2, x\leq -2

all real numbers

all non-negative real numbers

all positive 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 #3 : 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 #2 : 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 #2 : 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 #4 : Solving Inequalities

How many integers  satisfy the following inequality:

Possible Answers:

Five

Two

One

Three

Four

Correct answer:

Two

Explanation:

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

Example Question #118 : Algebra

What is the lowest value the integer  can take?

Possible Answers:

Correct answer:

Explanation:

The lowest value n can take is 6.

Example Question #11 : Solving Inequalities

What value of  will make the following expression negative:

 

Possible Answers:

Correct answer:

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 #120 : Algebra

Solve for 

Possible Answers:

Correct answer:

Explanation:

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

 

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