GMAT Math : Solving inequalities

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Example Questions

Example Question #1 : Solving Inequalities

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

$7< 2x-3 <15$

Possible Answers:

$\dpi{100} \small 4$

$\dpi{100} \small 5$

$\dpi{100} \small 6$

$\dpi{100} \small 9$

$\dpi{100} \small 3$

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.

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Example Question #2 : Solving Inequalities

Solve $5 < 3x + 10 \leq 16$ .

Possible Answers:

$\left ( \frac{5}{3}, 2 \right ]$

$\left [ \frac{5}{3}, 2\right )$

$\left ( -\frac{5}{3}, 2 \right )$

$\left ( -\frac{5}{3}, 2 \right ]$

$\left [ -\frac{5}{3}, 2 \right ]$

Correct answer:

$\left ( -\frac{5}{3}, 2 \right ]$

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 $-\frac{5}{3}$ and cannot equal $-\frac{5}{3}$ , yet CAN equal 2. Therefore $-\frac{5}{3}$ should have a parentheses around it, and 2 should have a bracket: is in

$\left ( -\frac{5}{3}, 2 \right ]$

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Example Question #1 : Solving Inequalities

Solve -2x+3<7 .

Possible Answers:

$[-2, \infty ]$

$(- \infty , -2)$

$(2, \infty )$

$[-2, \infty )$

$(-2, \infty )$

Correct answer:

$(-2, \infty )$

Explanation:

-2x+3<7

Subtract 3 from both sides: -2x<4

Divide both sides by : x>-2

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.

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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 #2 : Solving Inequalities

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

Possible Answers:

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

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

x>1

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

-1<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 #3 : Solving Inequalities

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

Possible Answers:

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

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

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.

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

$x^{2}\geq 4$

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

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Example Question #3 : Solving Inequalities

Solve the inequality:

$-3x+7\geq13$

Possible Answers:

$x\geq-2$

$x\leq2$

$x\leq-2$

$x\geq2$

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 #2 : Solving Inequalities

Find the solution set for :

4 < -x + 7 < 15

Possible Answers:

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

$x \in (3,8)$

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

$x \in (-8,3)$

$x \in (-3,8)$

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 #2 : Solving Inequalities

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

Possible Answers:

x > 3

x < 2

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

x < -9

x < -3

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 #4 : Solving Inequalities

How many integers satisfy the following inequality:

4< 4m-5<13

Possible Answers:

Five

Two

One

Three

Four

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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GMAT Math : Solving inequalities

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #1 : Solving Inequalities

Example Question #4 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #3 : Solving Inequalities

Example Question #3 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #4 : Solving Inequalities

All GMAT Math Resources

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