GMAT Math : Absolute Value

Study concepts, example questions & explanations for GMAT Math

varsity tutors app store varsity tutors android store

Example Questions

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:

\(\displaystyle 159\)

\(\displaystyle -159\)

\(\displaystyle 147\)

\(\displaystyle -147\)

\(\displaystyle 17\)

Correct answer:

\(\displaystyle -159\)

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 \(\displaystyle 147\).

\(\displaystyle \left | -17\right |=17\)

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:

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

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

\(\displaystyle -6-153=-6+-153=-159\)

The correct answer is \(\displaystyle -159\).

Example Question #12 : Absolute Value

Solve \(\displaystyle \left | 5x+20\right |=55\).

Possible Answers:

 \(\displaystyle x=-7\) or \(\displaystyle x=-15\)

\(\displaystyle x=-15\)

 \(\displaystyle x=7\) or \(\displaystyle x=-15\)

 \(\displaystyle x=-7\)

 \(\displaystyle x=7\)

Correct answer:

 \(\displaystyle x=7\) or \(\displaystyle x=-15\)

Explanation:

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

1.) \(\displaystyle 5x+20=55\)

2.) \(\displaystyle 5x+20=-55\)

Solving Equation 1:

\(\displaystyle 5x+20=55\)

\(\displaystyle 5x=35\)

\(\displaystyle x=7\)

Solving Equation 2:

\(\displaystyle 5x+20=-55\)

\(\displaystyle 5x=-75\)

\(\displaystyle x=-15\)

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

Example Question #11 : Understanding Absolute Value

Solve \(\displaystyle \left | 8x-16\right |=24\).

Possible Answers:

\(\displaystyle x=5\) or \(\displaystyle x=-1\)

\(\displaystyle x=5\)

\(\displaystyle x=-5\)

\(\displaystyle x=-5\) or \(\displaystyle x=-1\)

\(\displaystyle x=-1\)

Correct answer:

\(\displaystyle x=5\) or \(\displaystyle x=-1\)

Explanation:

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

1.) \(\displaystyle 8x-16=24\)

2.) \(\displaystyle 8x-16=-24\)

Solving Equation 1:

\(\displaystyle 8x-16=24\)

\(\displaystyle 8x=40\)

\(\displaystyle x=5\)

Solving Equation 2:

\(\displaystyle 8x-16=-24\)

\(\displaystyle 8x=-8\)

\(\displaystyle x=-1\)

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

Example Question #441 : Algebra

Define an operation \(\displaystyle \ast\) as follows:

For all real numbers \(\displaystyle a ,b\),

\(\displaystyle a \ast b = \left | \frac{1}{4}a - \frac{1}{7}b \right |\)

Evaluate: \(\displaystyle 7 \ast 4\)

Possible Answers:

\(\displaystyle 1\frac{1}{7}\)

\(\displaystyle 1\frac{5}{28}\)

\(\displaystyle 2\frac{9}{28}\)

\(\displaystyle 0\)

\(\displaystyle 3\frac{1}{2}\)

Correct answer:

\(\displaystyle 1\frac{5}{28}\)

Explanation:

\(\displaystyle a \ast b = \left | \frac{1}{4}a - \frac{1}{7}b \right |\)

\(\displaystyle 7 \ast 4= \left | \frac{1}{4} \cdot 7 - \frac{1}{7} \cdot 4 \right |\)

\(\displaystyle = \left | \frac{7}{4} - \frac{4}{7} \right |\)

\(\displaystyle = \left | \frac{49}{28} - \frac{16}{28} \right |\)

\(\displaystyle = \left | \frac{33}{28} \right |\)

\(\displaystyle = \left |1 \frac{5}{28} \right |\)

\(\displaystyle = 1 \frac{5}{28}\)

Example Question #12 : Understanding Absolute Value

Define an operation \(\displaystyle \ast\) as follows:

For all real numbers \(\displaystyle a ,b\),

\(\displaystyle a \ast b = |a+b| + |a-b|\)

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

Possible Answers:

\(\displaystyle 4\)

\(\displaystyle 14\)

\(\displaystyle 10\)

\(\displaystyle 0\)

\(\displaystyle 24\)

Correct answer:

\(\displaystyle 14\)

Explanation:

\(\displaystyle a \ast b = |a+b| + |a-b|\)

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

\(\displaystyle = |-12| + |-2|\)

\(\displaystyle = 12+2\)

\(\displaystyle = 14\)

Example Question #16 : Absolute Value

Define an operation \(\displaystyle \ast\) as follows:

For all real numbers \(\displaystyle a ,b\),

\(\displaystyle a \ast b = |a+b| - |a-b|\)

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

Possible Answers:

\(\displaystyle 14\)

\(\displaystyle 32\)

\(\displaystyle 0\)

\(\displaystyle 18\)

\(\displaystyle 4\)

Correct answer:

\(\displaystyle 14\)

Explanation:

\(\displaystyle a \ast b = |a+b| - |a-b|\)

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

\(\displaystyle = |-16| - |2|\)

\(\displaystyle = 16 - 2\)

\(\displaystyle = 14\)

Example Question #11 : Absolute Value

Define a function \(\displaystyle f\) to be

\(\displaystyle f(x)= |3x- 12 | + 5\)

Give the range of the function.

Possible Answers:

\(\displaystyle [-4, \infty)\)

\(\displaystyle [5, \infty)\)

\(\displaystyle [4, \infty)\)

\(\displaystyle (-\infty, \infty)\)

\(\displaystyle [-5, \infty)\)

Correct answer:

\(\displaystyle [5, \infty)\)

Explanation:

An absolute value of a number must always assume a nonnegative value, so

\(\displaystyle |3x- 12 | \ge 0\), and

\(\displaystyle |3x- 12 | + 5\ge 0 +5\)

Therefore, 

\(\displaystyle f(x)=|3x- 12 | + 5\ge 5\)

and the range of \(\displaystyle f\) is the set \(\displaystyle [5, \infty)\).

Example Question #18 : Absolute Value

Solve the following equation:

\(\displaystyle \left | -4+3(-5)-(-6)\right |+3x=\left |7-2(4) \right |\)

Possible Answers:

\(\displaystyle x=-\frac{14}{3}\)

\(\displaystyle x=8\)

\(\displaystyle x=-4\)

\(\displaystyle x=\frac{14}{3}\)

\(\displaystyle x=4\)

Correct answer:

\(\displaystyle x=-4\)

Explanation:

Before we apply the absolute value to the two terms in the equation, we simplify what's inside of them first:

\(\displaystyle \left | -4+3(-5)-(-6)\right |+3x=\left |7-2(4) \right |\)

\(\displaystyle \left | -4-15+6\right |+3x=\left |7-8 \right |\)

\(\displaystyle \left | -13\right |+3x=\left |-1 \right |\)

Now we can apply the absolute value to each term. Remember that taking the absolute value of a quantity results in solely its value, regardless of what its sign was before the absolute value was taken. This means that that absolute value of a number is always positive:

\(\displaystyle \left | -13\right |+3x=\left |-1 \right |\)

\(\displaystyle 13+3x=1\)

\(\displaystyle 3x=-12\)

\(\displaystyle x=-4\)

Example Question #19 : Absolute Value

Give the range of the function

\(\displaystyle f(x)= |x-10|+ |x-6|\)

Possible Answers:

\(\displaystyle [6, \infty)\)

\(\displaystyle [-6, \infty)\)

\(\displaystyle [10, \infty)\)

\(\displaystyle [-4, \infty)\)

\(\displaystyle [4, \infty)\)

Correct answer:

\(\displaystyle [4, \infty)\)

Explanation:

The key to answering this question is to note that this equation can be rewritten in piecewise fashion. 

If \(\displaystyle x \ge 10\), since both \(\displaystyle x-10\) and \(\displaystyle x-6\) are nonnegative, we can rewrite \(\displaystyle f\) as

\(\displaystyle f(x)= x-10 + x-6\), or

\(\displaystyle f(x) = 2x-16\).

On  \(\displaystyle [10, \infty)\), this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is \(\displaystyle [f(10), \infty)\), or, since

\(\displaystyle f(10)= |10-10|+|10-6| = 0 + 4 = 4\)

\(\displaystyle [4, \infty)\).

 

If \(\displaystyle 6 < x < 10\), since \(\displaystyle x-10\) is negative and \(\displaystyle x-6\) is positive, we can rewrite \(\displaystyle f\) as

\(\displaystyle f(x)= -x+10 + x-6\), or

\(\displaystyle f(x)=4\)

\(\displaystyle f\) is a constant function on this interval and its range is \(\displaystyle \{4 \}\).

 

If \(\displaystyle x \le 6\), since both \(\displaystyle x-10\) and \(\displaystyle x-6\) are nonpositive, we can rewrite \(\displaystyle f\) as

\(\displaystyle f(x)= -x+10 -x+6\), or

\(\displaystyle f(x) = -2x+16\).

On  \(\displaystyle [- \infty, 6)\), this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is \(\displaystyle [f(6), \infty)\), or, since 

\(\displaystyle f(6)= |6-10|+|6-6| = 4+0=4\)\(\displaystyle [4, \infty)\).

 

The union of the ranges is the range of the function - \(\displaystyle [4, \infty)\).

Example Question #20 : Absolute Value

Give the range of the function

\(\displaystyle f(x)= |x-10|- |x-6|\)

Possible Answers:

None of the other choices gives a correct answer.

\(\displaystyle [-6,10]\)

\(\displaystyle [6,10]\)

\(\displaystyle [-10,-6]\)

\(\displaystyle [-10,6]\)

Correct answer:

None of the other choices gives a correct answer.

Explanation:

The key to answering this question is to note that this equation can be rewritten in piecewise fashion. 

 

If \(\displaystyle x > 10\), since both \(\displaystyle x-10\) and \(\displaystyle x-6\) are positive, we can rewrite \(\displaystyle f\) as

\(\displaystyle f(x)= x-10 - x+6\), or

\(\displaystyle f(x)= -4\),

a constant function with range \(\displaystyle \left \{ -4\right \}\).

 

If \(\displaystyle 6 \le x \le 10\), since \(\displaystyle x-10\) is negative and \(\displaystyle x-6\) is positive, we can rewrite \(\displaystyle f\) as

\(\displaystyle f(x)=-x+10-x+6\), or

\(\displaystyle f(x)=-2x+16\)

This is decreasing, as its graph is a line with negative slope. The range is \(\displaystyle [f(10), f(6)]\),

or, since

\(\displaystyle f(10)= |10-10|- |10-6| = 0 - 4 = -4\)

and

\(\displaystyle f(6)= |6-10|- |6-6| = 4 - 0 = 4\),

\(\displaystyle [-4,4]\).

 

If \(\displaystyle x < 6\), since both \(\displaystyle x-10\) and \(\displaystyle x-6\) are negative, we can rewrite \(\displaystyle f\) as

\(\displaystyle f(x)= -x+10 +x-6\), or

\(\displaystyle f(x)= 4\),

a constant function with range \(\displaystyle \left \{ 4\right \}\).

 

The union of the ranges is the range of the function - \(\displaystyle [-4,4]\) - which is not among the choices.

Tired of practice problems?

Try live online GMAT prep today.

1-on-1 Tutoring
Live Online Class
1-on-1 + Class
Learning Tools by Varsity Tutors