GMAT Math : Algebra

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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 x=1 . When x=1 , $(x-1)^{2}=0$ .

Then we know that the inequality is only satisfied when x+4<0 , and $x\neq 1$ . Therefore x<-4 , 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 .

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Example Question #4 : Solving Inequalities

Solve x(x-1)(x+1)>0 .

Possible Answers:

$x=-1\ or\ x>1$

-1<x<1

x>1

$-1<x<0\ or\ x>1$

$x<-1\ or\ x>1$

Correct answer:

$-1<x<0\ or\ x>1$

Explanation:

The roots we need to look at are $x=0, x=1,\ and\ x=-1.$

Try x=-2

-2(-2-1)(-2+1)=-4<0 , so

x<-1 does not satisfy the inequality.

-1<x<0 :

Try $x=-\frac{1}{2}$

$-\frac{1}{2}\cdot (-\frac{1}{2}-1)(-\frac{1}{2}+1)=\frac{3}{8}>0$

so -1<x<0 does satisfy the inequality.

0<x<1 :

Try $x=\frac{1}{2}$

$\frac{1}{2}\left (\frac{1}{2}-1 \right )\left ( \frac{1}{2}+1 \right )=-\frac{3}{8}<0$

so 0<x<1 does not satisfy the inequality.

x>1 :

Try x=2 .

2(2-1)(2+1)=6>0

so x>1 satisfies the inequality.

Therefore the answer is -1<x<0 and x>1 .

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Example Question #5 : Solving Inequalities

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

Possible Answers:

all real numbers

all non-negative real numbers

$x>2\ or\ x<-2$

all positive real numbers

$x\geq 2, x\leq -2$

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.

$x^{2}-4\geq 0$

$x^{2}\geq 4$

$x\geq 2, x\leq -2$

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Example Question #3 : Solving Inequalities

Solve the inequality:

$-3x+7\geq13$

Possible Answers:

$x\leq2$

$x\geq-2$

$x\leq-2$

$x\geq2$

Correct answer:

$x\leq-2$

Explanation:

$-3x+7\geq13$

$-3x\geq6$

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

$x\leq-2$

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Example Question #4 : Solving Inequalities

Find the solution set for :

4 < -x + 7 < 15

Possible Answers:

$x \in (-\infty , 3) \cup (8, \infty )$

$x \in (3,8)$

$x \in (-\infty , -3) \cup (8, \infty )$

$x \in (-3,8)$

$x \in (-8,3)$

Correct answer:

$x \in (-8,3)$

Explanation:

4 < -x + 7 < 15

Subtract 7:

4-7 < -x + 7 -7 < 15-7

-3 < -x < 8

Divide by -1. Don't forget to switch the direction of the inequality signs since we're dividing by a negative number:

$-3 \cdot (-1) > -x \cdot (-1) > 8\cdot (-1)$

Simplify:

3 > x > -8 or in interval form, (-8,3) .

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Example Question #5 : Solving Inequalities

Which of the following is equivalent to 2x-5 > 5x+4 ?

Possible Answers:

x < -9

x < -3

x < 2

x > 3

$x > -\frac{1}{9}$

Correct answer:

x < -3

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:

2x-5>5x+4

-5>3x+4

-9>3x

Now by dividing by 3 we get our solution.

-3>x or x<-3

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Example Question #5 : Solving Inequalities

How many integers satisfy the following inequality:

4< 4m-5<13

Possible Answers:

Two

One

Four

Five

Three

Correct answer:

Two

Explanation:

4< 4m-5<13

9<4m<18

$\frac{9}{4}<m<\frac{9}{2}$

2.25<m<4.5

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

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Example Question #118 : Algebra

What is the lowest value the integer can take?

12<3n-5<27

Possible Answers:

Correct answer:

Explanation:

12<3n-5<27

17<3n<32

$\frac{17}{3}<n<\frac{32}{3}$

The lowest value n can take is 6.

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Example Question #11 : Solving Inequalities

What value of will make the following expression negative:

$0.4(50-x)\times10+0.6x\times5$

Possible Answers:

140

200

220

160

180

Correct answer:

220

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.

$0.4(50-x)\times10+0.6x\times5 < 0$

$(20-0.4x)\times 10 +0.6x \times 5<0$

Now distribute the 10 to the binomial.

$200-4x+0.6x \times 5<0$

Now multiply 0.6 by 5

200-4x+3x<0

200-x<0

-x<-200

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

x>200

220 will make the expression negative.

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Example Question #120 : Algebra

Solve for .

$3x+12\leq 48$

Possible Answers:

$x\leq -12$

$x\leq12$

$x\geq 12$

$x\leq20$

Correct answer:

$x\leq12$

Explanation:

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

$3x+12\leq48$

$3x\leq36$

$x\leq12$

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #4 : Solving Inequalities

Example Question #4 : Solving Inequalities

Example Question #5 : Solving Inequalities

Example Question #3 : Solving Inequalities

Example Question #4 : Solving Inequalities

Example Question #5 : Solving Inequalities

Example Question #5 : Solving Inequalities

Example Question #118 : Algebra

Example Question #11 : Solving Inequalities

Example Question #120 : Algebra

All GMAT Math Resources

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