Absolute Value - ACT Math

Card 0 of 20

Question

Find the absolute value of the following when x = 2,

$\left | \frac{x^3-12}{2}\right |$

Answer

$2^{3}=8$

and $\frac{-4}{2}=-2$

It is important to know that the absolute value of something is always positive so the absolute value of is

2 is your answer.

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Question

What are the values of a and b, if any, where –a|b + 7| > 0?

Answer

The absolute value will always yield a positive, as long it is not zero. Therefore, b cannot equal –7. For the value to be positive, a must be a negative number.

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Question

What is the absolute value of 19 – 36(3) + 2(4 – 87)?

Answer

19 – 36(3) + 2(4 – 87) =

19 – 108 + 2(–83) =

19 – 108 – 166 = –255

Absolute value is the non-negative value of the expression

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Question

Solve for z where | z + 1 | < 3

Answer

Absolute value problems generally have two answers:

z + 1 < 3 or z + 1 > –3 and subtracting 1 from each side gives z < 2 or z > –4 which bcomes –4 < z < 2

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Question

Solve $|z + 2| \leq -2$

Answer

Absolute values measure the distrance from the origin and is always positive, thus it can never be less than or equal to a negative number (unless a negative number is multiplied outside the absolute value). So the correct answer is no solutions.

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Question

$\left | 7 - 2\right | - \left | 4 - 8\right | =?$

Answer

Absolute value is the key here. Absolute value means the number's distance from zero. So we must account for that. Therefore $\left | 4-8\right |$ $= \left | -4\right | = 4$ .

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Question

Evaluate the expression if and .

$\left | 4x+3y \right | - \left | 4x-3y \right |$

Answer

$\left | 4x+3y \right | - \left | 4x-3y \right |$

To solve, we replace each variable with the given value.

$\left | 4 (5) +3 ( 7) \right | - \left | 4 (5) -3 ( 7) \right |\ .$

$\left | 20 +21 \right | - \left | 20 -21 \right |$

Simplify. Remember that terms inside of the absolute value are always positive.

$\left | 41 \right | - \left | -1 \right |=41-1 = 40$

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Question

Evaluate for :

$\left | 2x - 18 \right | + \left | 3x - 7 \right |$

Answer

$\left | 2x -18 \right | + \left | 3x - 7 \right |$

$= \left | 2 \cdot 2 - 18 \right | + \left | 3 \cdot 2 - 7 \right |$

$= \left | 4 - 18 \right | + \left | 6 - 7 \right |$

$= \left | -14 \right | + \left | -1 \right |$

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Question

Evaluate for :

$\left | 4x - 1.4 \right | + \left | x^{2} - 1 \right |$

Answer

Substitute 0.6 for :

$\left | 4x - 1.4 \right | + \left | x^{2} - 1 \right |$

$= \left | 4 \cdot 0.6 - 1.4 \right | + \left | 0.6^{2} - 1 \right |$

$= \left | 2.4 - 1.4 \right | + \left | 0.36 - 1 \right |$

$= \left | 1 \right | + \left | -0.64 \right |$

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Question

Evaluate for :

$\left |0.5 x - 0.7 \right | - \left |0.6 x - 0.4 \right |$

Answer

Substitute .

$\left |0.5 x - 0.7 \right | - \left |0.6 x - 0.4 \right |$

$= \left |0.5 \cdot 0.6 - 0.7 \right | - \left |0.6 \cdot 0.6 - 0.4 \right |$

$= \left |0.3- 0.7 \right | - \left |0.36 - 0.4 \right |$

$= \left |-0.4 \right | - \left | - 0.04 \right |$

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Question

Which of the following sentences is represented by the equation

Answer

is the absolute value of , which in turn is the sum of a number and seven and a number. Therefore, can be written as "the absolute value of the sum of a number and seven". Since it is equal to , it is three less than the number, so the equation that corresponds to the sentence is

"The absolute value of the sum of a number and seven is three less than the number."

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Question

Define an operation $\blacklozenge$ as follows:

For all real numbers ,

$a \blacklozenge b = \left | 2a- 2b + 5 \right |$

Evaluate $(-4) \blacklozenge (-7)$

Answer

$a \blacklozenge b = \left | 2a- 2b + 5 \right |$

$(-4) \blacklozenge (-7) = \left | 2 (-4)- 2(-7) + 5 \right |$

$= \left |-8- (-14) + 5 \right |$

$= \left |-8+14 + 5 \right |$

$= \left |11 \right |$

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Question

Define an operation $\triangleright$ as follows:

For all real numbers ,

$a \triangleright b = \left | a- \frac{1}{2}b\right | + \left | \frac{1}{2} a+b\right |$

Evaluate $\frac{1}{3} \triangleright 4$ .

Answer

$a \triangleright b = \left | a- \frac{1}{2}b\right | + \left | \frac{1}{2} a+b\right |$

$\frac{1}{3} \triangleright 4 = \left | \frac{1}{3} - \frac{1}{2} \cdot 4\right | + \left | \frac{1}{2} \cdot \frac{1}{3} +4\right |$

$= \left | \frac{1}{3} - 2\right | + \left | \frac{1}{6} +4\right |$

$= \left | -1 \frac{2}{3} \right | + \left | 4 \frac{1}{6} \right |$

$= 1 \frac{2}{3} + 4 \frac{1}{6}$

$= 5 \frac{5}{6}$

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Question

Define an operation $\blacktriangledown$ as follows:

For all real numbers ,

$a \blacktriangledown b= \frac{a+1}{\left | a\right |+ \left | b\right |}$

Evaluate: $\frac{4}{5} \blacktriangledown \left (-\frac{4}{5} \right )$ .

Answer

$a \blacktriangledown b= \frac{a+1}{\left | a\right |+ \left | b\right |}$ , or, equivalently,

$a \blacktriangledown b=\left ( a+1 \right ) \div ( | a |+ | b | )$

$\frac{4}{5} \blacktriangledown \left (-\frac{4}{5} \right )=\left ( \frac{4}{5}+1 \right ) \div \left (\left| \frac{4}{5} \right |+ \left |- \frac{4}{5}\right | \right )$

$= \frac{9}{5} \div \left ( \frac{4}{5} +\frac{4}{5} \right )$

$= \frac{9}{5} \div \frac{8}{5}$

$= \frac{9}{5} \times \frac{5}{8}$

$= \frac{9}{8}$

$=1 \frac{1}{8}$

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Question

Define $f(x) = |3x - |x^{2}- 7|; |$

Evaluate .

Answer

$f(x) = |3x - |x^{2}- 7|; |$

$f(2) = |3 \cdot 2 - |2^{2}- 7|; |$

$= |3 \cdot 2 - |4- 7|; |$

$= |3 \cdot 2 - |-3|; |$

$= |3 \cdot 2 -3 |$

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Question

Define $g(x)= \left | \left | 1,000- \sqrt{x}\right | - x^{3} \right |$ .

Evaluate .

Answer

$g(x)= \left | \left | 1,000- \sqrt{x}\right | - x^{3} \right |$

$g(16)= \left | \left | 1,000- \sqrt{16}\right | - 16^{3} \right |$

$= \left | \left | 1,000- 4 \right | - 16^{3} \right |$

$= \left | \left | 996 \right | - 16^{3} \right |$

$= \left | 996 - 16^{3} \right |$

$= \left | 996 - 4,096 \right |$

$= \left | -3,100 \right |$

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Question

Define $p(x) = \frac{\left |x+2 \right |-1}{\left |x+1 \right |-2}$ .

Evaluate $p \left (-1 \frac{1}{5} \right )$ .

Answer

$p(x) = \frac{\left |x+2 \right |-1}{\left |x+1 \right |-2}$ , or, equivalently,

$p(x) =\left ( \left |x+2 \right |-1 \right ) \div \left ( \left |x+1 \right |-2 \right )$

$p\left (-1 \frac{1}{5} \right )=\left ( \left |-1 \frac{1}{5}+2 \right |-1 \right ) \div \left ( \left |-1 \frac{1}{5}+1 \right |-2 \right )$

$=\left ( \left | \frac{4}{5} \right |-1 \right ) \div \left ( \left |- \frac{1}{5} \right |-2 \right )$

$=\left ( \frac{4}{5} -1 \right ) \div \left ( \frac{1}{5} -2 \right )$

$= - \frac{1}{5} \div \left ( - \frac{9}{5} \right )$

$= \frac{1}{5} \div \frac{9}{5}$

$= \frac{1}{5} \times \frac{5}{9}$

$= \frac{1} {9}$

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Question

What is the minimum value for if ?

Answer

When solving an absolute value equation, you should remember that you can have either a positive or a negative value in the absolute value. So, for instance:

means that can be either or .

Thus, for this question, you know that can mean:

Then, you just solve each and get:

Thus, is the minimum possible value for .

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Question

What is the largest possible value for if ?

Answer

When solving an absolute value equation, you should remember that you can have either a positive or a negative value in the absolute value. So, for instance:

means that can be either or .

Thus, for this question, you know that . Start by dividing by on both sides. This will give you:

Now, from this, we know:

Solve each equation for . The first is:

The second is . You can tell that this is going to end up being negative. You do not even need to finish. The larger value will be the positive one, .

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Question

Simplify .

Answer

Begin by simplifying the contents of the absolute value:

Remember that the absolute value of a negative number is a positive value. Thus:

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