GMAT Math : Algebra

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Example Questions

Example Question #11 : Solving Inequalities

What value of satisfies both of the following inequalities?

$9-3m\leq5$

$2m+7\leq13-m$

Possible Answers:

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{8}{3}$

$\frac{7}{3}$

$\frac{5}{4}$

Correct answer:

$\frac{3}{2}$

Explanation:

Solve for in both inequalities:

(1)

$9-3m\leq5$

Subtract 9 from each side of the inequality:

$-3m\leq-4$

Then, divide by . Remember to switch the direction of the sign from "less than or equal to" to "greater than or equal to."

$m\geq\frac{4}{3}$

(2)

$2m+7\leq13-m$

Add to both sides of the inequality:

$3m+7\leq13$

Subtract 7 from each side of the inequality:

$3m\leq6$

Divide each side of the inequality by 3:

$m\leq2$

From solving both inequalities, we find such that:

$\frac{4}{3}\leq m\leq 2$

$\frac{4}{3}\approx 1.33$

Only $\frac{3}{2}$ is in that interval $(\frac{3}{2}=1.5)$ .

$\frac{5}{4}$ and $\frac{2}{3}$ are less than 1.33, so they are not in the interval.

$\frac{7}{3}$ and $\frac{8}{3}$ are more than 2, so they are not in the interval.

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Example Question #2 : How To Find The Solution To An Inequality With Division

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

$8-7x\leq11x+4$

Possible Answers:

$-\frac{1}{3}$

$-\frac{2}{9}$

$-\frac{2}{3}$

$\frac{4}{9}$

$\frac{1}{9}$

Correct answer:

$\frac{4}{9}$

Explanation:

The inequality that is presented in the problem is:

$8-7x\leq11x+4$

Start by moving your variables to one side of the inequality and all other numbers to the other side:

$-7x-11x\leq4-8$

$-18x\leq-4$

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!

$x\geq\frac{-4}{-18}$

Reduce:

$x\geq\frac{2}{9}$

The only answer choice with a value greater than $\frac{2}{9}$ is $\frac{4}{9}$ .

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Example Question #123 : Algebra

6m-7>11

n<2n-9

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

Possible Answers:

m*n<3

-m*n>0

m+n>12

m+n<-12

$\frac{m}{n}<0$

Correct answer:

m+n>12

Explanation:

First let's solve each of the inequalities:

6m-7>11

6m>18

m>3

n<2n-9

-n<-9

Don't forget to flip the direction of the sign when dividing by a negative number:

n>9

m+n>12 is the correct answer. is an integer greater than 3 and is greater than 9. Therefore, the sum of and is greater than 12.

m+n<-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.

$\frac{m}{n}<0$ 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.

m*n<3 is not true. is greater than 3 and is greater than 9. The product of and cannot be less than 3.

-m*n>0 is not true. and are positive. Therefore, the product of and is negative and cannot be greater than 0.

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Example Question #13 : Solving Inequalities

3-x>8-2x

y-2>9

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

Possible Answers:

5<m<11

m<11

m<5

m=5

m>11

Correct answer:

m>11

Explanation:

First, we solve both inequalities:

3-x>8-2x

3-x+2x>8

x>5

y-2>9

y>11

If satisfies both inequalities, then is greater than 5 AND is greater than 11. Therefore is greater than 11.

m>11 is the correct answer.

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Example Question #1205 : Gmat Quantitative Reasoning

Solve the following inequality:

7x-4<13x+26

Possible Answers:

x<3

x>-5

x<-5

x<-3

x>3

Correct answer:

x>-5

Explanation:

We start by simplifying our inequality like any other equation:

7x-4<13x+26

-6x<30

Now we must remember that when we divide by a negative, the inequality is flipped, so we obtain:

x>-5

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Example Question #1203 : Gmat Quantitative Reasoning

Solve the following inequality:

$\small 4x-7\geq13x+11$

Possible Answers:

$\small \small \small x\leq2$

$\small \small \small \small x\geq 2$

$\small \small \small \small \small x\geq -2$

$\small \small x\leq-2$

$\small \small \small \small \small x\geq -4$

Correct answer:

$\small \small x\leq-2$

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

$4x-7\geq13x+11$

Start by moving the and the 13x over:

$4x-7({\color{Blue} -13x})({\color{Red} +7})\geq13x+11({\color{Red} +7})({\color{Blue} -13x})$

Simplify to get the following:

$-9x\geq18$

Then, we will divide both sides of the equation by . Remember to switch the direction of the inequality sign!

$\frac{-9x}{-9}\geq\frac{18}{-9}$

$\small x\leq\frac{18}{-9}$

So,

$\small \small x\leq-2$

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Example Question #127 : Algebra

x = 2y- 6 and x = 10 - 5z . If the greatest number of the three, then what are the possible values of ?

Possible Answers:

x>6

$x>1 \frac{2}{3}$

$1\frac{2}{3} < x<6$

$x<1 \frac{2}{3}$

x<6

Correct answer:

x>6

Explanation:

The question is equivalent to asking when x > y and x > z are both true statements.

We can solve for in the first equation:

x = 2y- 6

2y = x + 6

$y = \frac{1}{2}(x+6)$

$y = \frac{1}{2}x+3$

Now use substitution to solve the inequality

x > y

$x > \frac{1}{2}x+3$

$x - \frac{1}{2}x > \frac{1}{2}x+3 - \frac{1}{2}x$

$\frac{1}{2}x > 3$

$2 \cdot \frac{1}{2}x >2 \cdot 3$

x > 6

Therefore, x > y if and only if x > 6 .

Similarly, we can solve for in the first equation:

x = 10 - 5z

x + 5z - x = 10 - 5z + 5z - x

5z = 10-x

$z = \frac{1}{5}(10-x)$

$z = 2 -\frac{1}{5}x$

Now use substitution to solve the inequality

x > z

$x >2 -\frac{1}{5}x$

$x+ \frac{1}{5}x >2 -\frac{1}{5}x + \frac{1}{5}x$

$\frac{6}{5}x >2$

$\frac{5}{6} \cdot \frac{6}{5}x >\frac{5}{6} \cdot 2$

$x > \frac{5}{3} = 1\frac{2}{3}$

Therefore, x > z if and only if $x > 1\frac{2}{3}$ .

We combine these results, and conclude that both x > y and x > z hold - that is, is the greatest of the three - if and only if x > 6 .

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Example Question #21 : Solving Inequalities

Solve for :

8<-5x+18<103

Possible Answers:

$x\textup{ is undefined}$

-17<x<2

$x<-17\textup{ or }x>2$

$x<5.2\textup{ or }x>24.2$

5.2<x<24.2

Correct answer:

-17<x<2

Explanation:

The first step to solving this inequality is to subtract 18 from all three segments of the equation:

8 (- 18) < -5x + 18 ( - 18) < 103 ( - 18)

-10 < -5x < 85

We then divide the entire equation by -5. According to inequality rules, when we multiply or divide an inequality by a negative number, we must flip the inequality signs.

(-10/-5) > (-5x/-5) > (85/-5)

2 > x > -17 or in conventional layout -17 < x < 2

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Example Question #21 : Solving Inequalities

Solve: -45 < -6x+15< 15

Possible Answers:

-10 > x or x > 0

10 > x > 0

10< x or x< 0

-60 < x < 0

-10<x<0

Correct answer:

10 > x > 0

Explanation:

In order to solve this inequality, we must split it into two inequalities:

-45 < -6x + 15 and -6x + 15 < 15

We then solve each, remembering of course that, when multiplying or dividing by a negative number, we must switch the direction of the inequality sign.

-45 - 15 < -6x + 15 - 15

-60 < -6x

$\frac{-60}{-6} > \frac{-6x}{-6}$

10 > x

and

-6x + 15 < 15

-6x + 15 - 15 < 15 - 15

-6x < 0

$\frac{-6x}{-6} > \frac{0}{-6}$

x> 0

Therefore:

0< x < 10

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Example Question #128 : Algebra

Give the solution set of the inequality

$\frac{1}{x} > \frac{2}{x-4}$

Possible Answers:

$(-\infty, -4) \cup (0, 4)$

$(-4, 0) \cup (4, \infty)$

$(-\infty, -4) \cup (4, \infty)$

$(-\infty, 0) \cup (4, \infty)$

(-4, 4 )

Correct answer:

$(-\infty, -4) \cup (0, 4)$

Explanation:

To solve a rational inequality, move all expressions to the left first:

$\frac{1}{x} > \frac{2}{x-4}$

$\frac{1}{x} - \frac{2}{x-4} > \frac{2}{x-4} - \frac{2}{x-4}$

$\frac{1}{x} - \frac{2}{x-4} >0$

$\frac{1(x-4 )}{x(x-4 )} - \frac{2x}{x(x-4 )} >0$

$\frac{x-4}{x(x-4 )} - \frac{2x}{x(x-4 )} >0$

$\frac{-x-4}{x(x-4 )} >0$

$-1 \cdot \frac{-x-4}{x(x-4 )} < -1 \cdot 0$

$\frac{x+4}{x(x-4 )} < 0$

The boundary points of the solution set will be the points at which:

x+4 = 0 - that is, x = -4 ;

x = 0 ;

x-4 = 0 ; that is, x = 4 .

None of these values will be included in the solution set, since equality is not allowed by the inequality symbol.

Test the intervals

$(-\infty, -4) , (-4, 0) , (0, 4), (4, \infty)$

by choosing a value in each interval and testing the truth of the inequality.

$(-\infty, -4)$ : test x= -5

$\frac{1}{x} > \frac{2}{x-4}$

$\frac{1}{-5} > \frac{2}{-5-4}$

$-\frac{1}{5}>-\frac{2}{9}$

True; include the interval $(-\infty, -4)$

(-4, 0) : test x = -1

$\frac{1}{-1} > \frac{2}{-1-4}$

$\frac{1}{-1} > \frac{2}{-5}$

$-1 > -\frac{2}{5}$

False; exclude the interval (-4, 0)

(0, 4) : test x =1

$\frac{1}{1} > \frac{2}{1-4}$

$\frac{1}{1} > \frac{2}{-3}$

$1 > - \frac{2}{3}$

True; include the interval (0, 4) .

$(4, \infty)$ : test x = 5

$\frac{1}{5} > \frac{2}{5-4}$

$\frac{1}{5} > \frac{2}{1}$

$\frac{1}{5} > 2$

False; exclude the interval $(4, \infty)$ .

The solution set of the inequality is $(-\infty, -4) \cup (0, 4)$ .

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Solving Inequalities

Example Question #2 : How To Find The Solution To An Inequality With Division

Example Question #123 : Algebra

Example Question #13 : Solving Inequalities

Example Question #1205 : Gmat Quantitative Reasoning

Example Question #1203 : Gmat Quantitative Reasoning

Example Question #127 : Algebra

Example Question #21 : Solving Inequalities

Example Question #21 : Solving Inequalities

Example Question #128 : Algebra

All GMAT Math Resources

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