GMAT Math : Algebra

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Example Questions

Example Question #3 : Absolute Value

Solve for in the absolute value equation

$1-\frac{5}{2}\left |x-5 \right | = 15$

Possible Answers:

$-\frac{53}{3}$

$-\frac{3}{5}$

None of the other answers

$\frac{53}{3}$

$\frac{3}{5}$

Correct answer:

None of the other answers

Explanation:

The correct answer is that there is no .

We start by adding to both sides giving

$-\frac{5}{2} \left|x-5\right|=14$

Then multiply both sides by .

$-5\left|x-5|=28$

Then divide both sides by

$|x-5|=-\frac{28}{5}$

Now it is impossible to go any further. The absolute value of any quantity is always positive (or sometimes ). Here we have the absolute value of something equaling a negative number. That's never possible, hence there is no that makes this a true equation.

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Example Question #4 : Absolute Value

Solve the following equation:

$32-\left | x-23\right |=13$

Possible Answers:

x={-19,19}

x={42,4}

x={10,19}

x={-22,10}

x={-23,0}

Correct answer:

x={42,4}

Explanation:

We start by isolating the expression with the absolute value:

$32-\left | x-23\right |=13$ becomes $-\left | x-23\right |=13-32=-19$

So: x-23=19 or x-23=-19

We then solve the two equations above, which gives us 42 and 4 respectively.

So the solution is x={42,4}

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Example Question #2 : Understanding Absolute Value

Solve the absolute value equation for .

|2x-5|+3=5

Possible Answers:

$x = 2,\frac{3}{2}$

$x = -\frac{7}{2}, \frac{7}{2}$

The equation has no solution

None of the other answers.

$x = \frac{7}{2}, \frac{3}{2}$

Correct answer:

$x = \frac{7}{2}, \frac{3}{2}$

Explanation:

We proceed as follows

|2x-5| +3 =5 (Start)

|2x -5| =2 (Subtract 3 from both sides)

2x-5 =2 or 2x-5 = -2 (Quantity inside the absolute value can be positive or negative)

2x = 7 or 2x = 3 (add five to both sides)

$x = \frac{7}{2}$ or $x = \frac{3}{2}$

Another way to say this is $x = \frac{7}{2},\frac{3}{2}$

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Example Question #3 : Absolute Value

$\left | 2m-\frac{1}{2}\right |<4$

Which of the following could be a value of ?

Possible Answers:

$\frac{9}{4}$

$\frac{9}{2}$

$-\frac{7}{2}$

$\frac{7}{4}$

$-\frac{7}{4}$

Correct answer:

$\frac{7}{4}$

Explanation:

To solve an inequality we need to remember what the absolute value sign says about our expression. In this case it says that

$\left | 2m-\frac{1}{2}\right |<4$

can be written as

$2m-\frac{1}{2}<4$ Of $2m-\frac{1}{2}>-4$ .

Rewriting this in one inequality we get:

$-4<2m-\frac{1}{2}<4$

From here we add one half to both sides .

$-\frac{7}{2}<2m<\frac{9}{2}$

Finally, we divide by two to isolate and solve for m.

$-\frac{7}{4}<m<\frac{9}{4}$

-1.75<m<2.25

Only $\frac{7}{4}$ is between -1.75 and 2.25

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Example Question #11 : Understanding Absolute Value

The absolute value of negative seventeen is multiplied by a number that is three fewer than twelve. The resulting number is subtracted from negative six. What number is yielded at the end of this sequence of operations?

Possible Answers:

159

147

Correct answer:

Explanation:

This is a problem where we need to use our translating skills. We are given a word problem and asked to solve it. To do so, we need to rewrite our word problem as an equation and then use arithmetic to find the answer. In these types of problems, the hardest step is usually translating correctly, so make sure to be meticulous and work step-by-step!

1)"The absolute value of negative seventeen": Recall that absolute value means that we will just change the sign to positive. Missing that will end up giving you the trap answer 147 .

$\left | -17\right |=17$

2)"is multiplied by a number which is three fewer than twelve." We need a number that is three fewer than twelve, so we need to subtract. Follow it up with multiplication and you get:

17*(12-3)=17*9=153

3)"The resulting number is subtracted from negative six." The key word here is "from"—make sure you aren't computing 153 -(-6) , which would result in another one of the trap answers!

-6-153=-6+-153=-159

The correct answer is .

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Example Question #11 : Understanding Absolute Value

Solve $\left | 5x+20\right |=55$ .

Possible Answers:

x=7 or x=-15

x=-15

x=7

x=-7

x=-7 or x=-15

Correct answer:

x=7 or x=-15

Explanation:

Since we are solving an absolute value equation, $\left | 5x+20\right |=55$ , we must solve for both potential values of the equation:

1.) 5x+20=55

2.) 5x+20=-55

Solving Equation 1:

5x+20=55

5x=35

x=7

Solving Equation 2:

5x+20=-55

5x=-75

x=-15

Therefore, for $\left | 5x+20\right |=55$ , x=7 or x=-15 .

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Example Question #12 : Understanding Absolute Value

Solve $\left | 8x-16\right |=24$ .

Possible Answers:

x=-5 or x=-1

x=-1

x=-5

x=5 or x=-1

x=5

Correct answer:

x=5 or x=-1

Explanation:

Since we are solving an absolute value equation, $\left | 8x-16\right |=24$ , we must solve for both potential values of the equation:

1.) 8x-16=24

2.) 8x-16=-24

Solving Equation 1:

8x-16=24

8x=40

x=5

Solving Equation 2:

8x-16=-24

8x=-8

x=-1

Therefore, for $\left | 8x-16\right |=24$ , x=5 or x=-1 .

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Example Question #11 : Understanding Absolute Value

Define an operation $\ast$ as follows:

For all real numbers a ,b ,

$a \ast b = \left | \frac{1}{4}a - \frac{1}{7}b \right |$

Evaluate: $7 \ast 4$

Possible Answers:

$2\frac{9}{28}$

$1\frac{5}{28}$

$1\frac{1}{7}$

$3\frac{1}{2}$

Correct answer:

$1\frac{5}{28}$

Explanation:

$a \ast b = \left | \frac{1}{4}a - \frac{1}{7}b \right |$

$7 \ast 4= \left | \frac{1}{4} \cdot 7 - \frac{1}{7} \cdot 4 \right |$

$= \left | \frac{7}{4} - \frac{4}{7} \right |$

$= \left | \frac{49}{28} - \frac{16}{28} \right |$

$= \left | \frac{33}{28} \right |$

$= \left |1 \frac{5}{28} \right |$

$= 1 \frac{5}{28}$

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Example Question #12 : Understanding Absolute Value

Define an operation $\ast$ as follows:

For all real numbers a ,b ,

$a \ast b = |a+b| + |a-b|$

Evaluate $\left (-7 \right ) \ast \left (-5 \right )$

Possible Answers:

Correct answer:

Explanation:

$a \ast b = |a+b| + |a-b|$

$\left (-7 \right ) \ast \left (-5 \right )= |-7+\left (-5 \right )| + |-7-\left (-5 \right )|$

= |-12| + |-2|

= 12+2

= 14

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Example Question #445 : Algebra

Define an operation $\ast$ as follows:

For all real numbers a ,b ,

$a \ast b = |a+b| - |a-b|$

Evaluate $\left (-7 \right ) \ast \left (-9 \right )$

Possible Answers:

Correct answer:

Explanation:

$a \ast b = |a+b| - |a-b|$

$\left (-7 \right ) \ast \left (-9 \right )= |\left (-7 \right )+\left (-9 \right )| - |\left (-7 \right )-\left (-9 \right )|$

= |-16| - |2|

= 16 - 2

= 14

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #3 : Absolute Value

Example Question #4 : Absolute Value

Example Question #2 : Understanding Absolute Value

Example Question #3 : Absolute Value

Example Question #11 : Understanding Absolute Value

Example Question #11 : Understanding Absolute Value

Example Question #12 : Understanding Absolute Value

Example Question #11 : Understanding Absolute Value

Example Question #12 : Understanding Absolute Value

Example Question #445 : Algebra

All GMAT Math Resources

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