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SSAT Upper Level Math : How to find absolute value

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Example Questions

Example Question #1 : Absolute Value

Evaluate for x = 2 $\displaystyle x = 2$:

Possible Answers:

$\displaystyle 27$

$\displaystyle 23$

$\displaystyle 21$

$\displaystyle 13$

$\displaystyle 15$

Correct answer:

$\displaystyle 15$

Explanation:

$\displaystyle =14 + 1 = 15$

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Example Question #1 : How To Find Absolute Value

Evaluate for x = 0.6 $\displaystyle x = 0.6$ :

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

Possible Answers:

1.64 $\displaystyle 1.64$

2.36 $\displaystyle 2.36$

0.64 $\displaystyle 0.64$

0.36 $\displaystyle 0.36$

1.36 $\displaystyle 1.36$

Correct answer:

1.64 $\displaystyle 1.64$

Explanation:

Substitute 0.6 for $\displaystyle x$ :

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

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

$\displaystyle =1 + 0.64$

= 1.64 $\displaystyle = 1.64$

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Example Question #3 : Absolute Value

Evaluate for x = 0.6 $\displaystyle x = 0.6$:

Possible Answers:

0.36 $\displaystyle 0.36$

0.44 $\displaystyle 0.44$

0.76 $\displaystyle 0.76$

0.96 $\displaystyle 0.96$

1.04 $\displaystyle 1.04$

Correct answer:

0.36 $\displaystyle 0.36$

Explanation:

Substitute x = 0.6 $\displaystyle x = 0.6$.

$\displaystyle = 0.4 - 0.04 = 0.36$

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Example Question #1 : How To Find Absolute Value

Which of the following sentences is represented by the equation

$\displaystyle | x + 7 | = x - 3$

Possible Answers:

The sum of three and the absolute value of the sum of a number is three greater than the number.

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

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

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

None of the other responses are correct.

Correct answer:

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

Explanation:

$\displaystyle | x + 7 |$ is the absolute value of x+ 7 $\displaystyle x+ 7$, which in turn is the sum of a number and seven and a number. Therefore, $\displaystyle | x + 7 |$ can be written as "the absolute value of the sum of a number and seven". Since it is equal to x - 3 $\displaystyle x - 3$, 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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Example Question #2 : How To Find Absolute Value

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

Evaluate f(2) $\displaystyle f(2)$.

Possible Answers:

$\displaystyle 5$

$\displaystyle 9$

$\displaystyle 17$

$\displaystyle 3$

None of the other responses is correct.

Correct answer:

$\displaystyle 3$

Explanation:

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

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

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

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

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

$\displaystyle = |6 -3 |$

= | 3 | $\displaystyle = | 3 |$

= 3 $\displaystyle = 3$

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Example Question #1 : How To Find Absolute Value

Define an operation $\blacktriangledown$ $\displaystyle \blacktriangledown$ as follows:

For all real numbers a,b $\displaystyle a,b$,

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

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

Possible Answers:

$\displaystyle 0$

The expression is undefined.

None of the other responses is correct.

$\frac{5}{8}$ $\displaystyle \frac{5}{8}$

$1 \frac{1}{8}$ $\displaystyle 1 \frac{1}{8}$

Correct answer:

$1 \frac{1}{8}$ $\displaystyle 1 \frac{1}{8}$

Explanation:

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

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

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

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

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

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

$= \frac{9}{8}$ $\displaystyle = \frac{9}{8}$

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

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Example Question #11 : How To Find Absolute Value

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

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

Possible Answers:

$\displaystyle 1$

$\frac{1}{11}$ $\displaystyle \frac{1}{11}$

$\frac{1} {9}$ $\displaystyle \frac{1} {9}$

$-\frac{1}{11}$ $\displaystyle -\frac{1}{11}$

$-\frac{1} {9}$ $\displaystyle -\frac{1} {9}$

Correct answer:

$\frac{1} {9}$ $\displaystyle \frac{1} {9}$

Explanation:

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

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

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

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

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

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

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

$= \frac{1} {9}$ $\displaystyle = \frac{1} {9}$

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Example Question #691 : Arithmetic

Define an operation $\triangleright$ $\displaystyle \triangleright$ as follows:

For all real numbers a,b $\displaystyle a,b$,

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

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

Possible Answers:

$2\frac{1}{2}$ $\displaystyle 2\frac{1}{2}$

$1\frac{1}{2}$ $\displaystyle 1\frac{1}{2}$

$6\frac{1}{2}$ $\displaystyle 6\frac{1}{2}$

$5 \frac{5}{6}$ $\displaystyle 5 \frac{5}{6}$

$\text{None of the other responses are correct.}$ $\displaystyle \text{None of the other responses are correct.}$

Correct answer:

$5 \frac{5}{6}$ $\displaystyle 5 \frac{5}{6}$

Explanation:

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

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

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

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

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

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

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Example Question #12 : How To Find Absolute Value

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

Evaluate g(16) $\displaystyle g(16)$.

Possible Answers:

$3\textup{,}100$ $\displaystyle 3\textup{,}100$

$5\textup{,}100$ $\displaystyle 5\textup{,}100$

$3\textup{,}092$ $\displaystyle 3\textup{,}092$

$5\textup{,}092$ $\displaystyle 5\textup{,}092$

$\textup{The expression is undefined.}$ $\displaystyle \textup{The expression is undefined.}$

Correct answer:

$3\textup{,}100$ $\displaystyle 3\textup{,}100$

Explanation:

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

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

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

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

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

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

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

=3,100 $\displaystyle =3,100$

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Example Question #2 : Absolute Value

Define an operation $\blacklozenge$ $\displaystyle \blacklozenge$ as follows:

For all real numbers a,b $\displaystyle a,b$,

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

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

Possible Answers:

Both $\displaystyle 11$ and $\displaystyle -11$

$\displaystyle 27$

$\displaystyle 17$

$\displaystyle 11$

$\displaystyle -11$

Correct answer:

$\displaystyle 11$

Explanation:

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

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

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

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

$= \left |11 \right |$ $\displaystyle = \left |11 \right |$

= 11 $\displaystyle = 11$

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SSAT Upper Level Math : How to find absolute value

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : Absolute Value

Example Question #1 : How To Find Absolute Value

Example Question #3 : Absolute Value

Example Question #1 : How To Find Absolute Value

Example Question #2 : How To Find Absolute Value

Example Question #1 : How To Find Absolute Value

Example Question #11 : How To Find Absolute Value

Example Question #691 : Arithmetic

Example Question #12 : How To Find Absolute Value

Example Question #2 : Absolute Value

All SSAT Upper Level Math Resources

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