GMAT Math : Understanding absolute value

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Example Questions

Example Question #1 : Absolute Value

Solve $\left | 3x - 7 \right |=8$ .

Possible Answers:

$x=-\frac{1}{3}$

x=5 or $\dpi{100} -\frac{1}{3}$

x=5 or $\frac{1}{3}$

x=-5 or $\frac{1}{3}$

x=5

Correct answer:

x=5 or $\dpi{100} -\frac{1}{3}$

Explanation:

$\left | 3x - 7 \right |=8$ really consists of two equations: $3x - 7 = \pm 8$

We must solve them both to find two possible solutions.

$3x - 7 = 8 \Rightarrow 3x = 15\Rightarrow x = 5$

$3x - 7 = - 8 \Rightarrow 3x = -1\Rightarrow x = -1/3$

So x=5 or $\dpi{100} -\frac{1}{3}$ .

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Example Question #2 : Absolute Value

Solve $\left | 2x - 5 \right |\geq 3$ .

Possible Answers:

$1 < x < 4$

$x \leq 1, x\geq 4$

$-2 \leq x\leq 5$

$x < 1, x > 4$

$x \leq -1, x\geq -4$

Correct answer:

$x \leq 1, x\geq 4$

Explanation:

It's actually easier to solve for the complement first. Let's solve $\left | 2x-5 \right |<3$ . That gives -3 < 2x - 5 < 3. Add 5 to get 2 < 2x < 8, and divide by 2 to get 1 < x < 4. To find the real solution then, we take the opposites of the two inequality signs. Then our answer becomes $x\leq 1 \textsc{ or } x\geq 4$ .

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Example Question #3 : Absolute Value

Give the -intercept(s), if any, of the graph of the function $f (x) = \left | 4x + B \right | - 8$ in terms of .

Possible Answers:

(B- 8, 0)

$\left ( \frac{-B -8 }{4}, 0 \right ), \left ( \frac{-B +8 }{4}, 0 \right )$

$\left ( \frac{B -8 }{4}, 0 \right ), \left ( \frac{B +8 }{4}, 0 \right )$

$\left ( B-2, 0 \right ), \left ( B+2, 0 \right )$

$\left ( -B-2, 0 \right ), \left ( -B+2, 0 \right )$

Correct answer:

$\left ( \frac{-B -8 }{4}, 0 \right ), \left ( \frac{-B +8 }{4}, 0 \right )$

Explanation:

Set f (x) = 0 and solve for :

f (x) = 0

$\left | 4x + B \right | - 8 = 0$

$\left | 4x + B \right | = 8$

Rewrite as a compound equation and solve each part separately:

$4x + B = -8 \textrm{ or } 4x + B = 8$

4x + B = -8

4x + B -B = -B -8

4x = -B -8

$4x \div 4 =\left ( -B -8 \right ) \div 4$

$x = \frac{-B -8 }{4}$

4x + B = 8

4x + B -B = -B + 8

4x = -B +8

$4x \div 4 =\left ( -B +8 \right ) \div 4$

$x = \frac{-B +8 }{4}$

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Example Question #3 : Absolute Value

A number is ten less than its own absolute value. What is this number?

Possible Answers:

No such number exists.

Correct answer:

Explanation:

We can rewrite this as an equation, where is the number in question:

N = |N| - 10

A nonnegative number is equal to its own absolute value, so if this number exists, it must be negative.

In thsi case, |N| = - N , and we can rewrite that equation as

N = -N- 10

N +N= -N- 10 +N

2N = -10

$2N \div 2 = -10\div 2$

N = -5

This is the only number that fits the criterion.

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Example Question #5 : Absolute Value

If -1<n<0 , which of the following has the greatest absolute value?

Possible Answers:

$\frac{n}{2}$

n^2

n+1

$\frac{1}{n^{2}}$

$\frac{1}{n}$

Correct answer:

$\frac{1}{n^{2}}$

Explanation:

Since -1<n<0 , we know the following:

0< n^2<1 ;

$-1<\frac{n}{2}<0$ ;

$\frac{1}{n^2}>1$ ;

$\frac{1}{n}<-1$ ;

0<n+1<1 .

Also, we need to compare absolute values, so the greatest one must be either $\left | \frac{1}{n^2}\right |$ or $\left | \frac{1}{n}\right |$ .

We also know that $\left | n^2\right |<\left | n\right |$ when -1<n<0 .

Thus, we know for sure that $\left | \frac{1}{n^2}\right |>\left | \frac{1}{n}\right |$ .

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Example Question #6 : Absolute Value

Give all numbers that are twenty less than twice their own absolute value.

Possible Answers:

$- 10 \textrm{ and } 10$

No such number exists.

$- 6\frac{2}{3} \textrm{ and } 20$

$- 6\frac{2}{3} \textrm{ and } 10$

$- 10 \textrm{ and } 20$

Correct answer:

$- 6\frac{2}{3} \textrm{ and } 20$

Explanation:

We can rewrite this as an equation, where is the number in question:

$N = 2\cdot |N| - 20$

If is nonnegative, then | N | = N , and we can rewrite this as

N = 2N - 20

Solve:

N -2N= 2N - 20-2N

-N= - 20

N = 20

If is negative, then | N | =- N , and we can rewrite this as

$N = 2\left (-N \right ) - 20$

N = -2N - 20

N + 2N= -2N - 20 + 2N

3N=- 20

$3N \div 3=- 20\div 3$

$N=- 6\frac{2}{3}$

The numbers $- 6\frac{2}{3}, 20$ have the given characteristics.

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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 #5 : Understanding Absolute Value

Solve the following equation:

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

Possible Answers:

x={-19,19}

x={-23,0}

x={42,4}

x={10,19}

x={-22,10}

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 #3 : Absolute Value

Solve the absolute value equation for .

|2x-5|+3=5

Possible Answers:

The equation has no solution

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

None of the other answers.

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

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

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 #4 : Absolute Value

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

Which of the following could be a value of ?

Possible Answers:

$\frac{7}{4}$

$-\frac{7}{4}$

$-\frac{7}{2}$

$\frac{9}{4}$

$\frac{9}{2}$

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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GMAT Math : Understanding absolute value

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Absolute Value

Example Question #2 : Absolute Value

Example Question #3 : Absolute Value

Example Question #3 : Absolute Value

Example Question #5 : Absolute Value

Example Question #6 : Absolute Value

Example Question #3 : Absolute Value

Example Question #5 : Understanding Absolute Value

Example Question #3 : Absolute Value

Example Question #4 : Absolute Value

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