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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #3 : How To Find Absolute Value

Evaluate for x = 0.6 :

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

Possible Answers:

2.36

1.36

0.64

1.64

0.36

Correct answer:

1.64

Explanation:

Substitute 0.6 for :

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

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

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

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

=1 + 0.64

= 1.64

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Example Question #4 : How To Find Absolute Value

Evaluate for x = 0.6 :

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

Possible Answers:

0.76

0.44

0.36

1.04

0.96

Correct answer:

0.36

Explanation:

Substitute x = 0.6 .

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

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

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

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

= 0.4 - 0.04 = 0.36

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Example Question #881 : Arithmetic

Which of the following sentences is represented by the equation

| 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 less than the number.

None of the other responses are correct.

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

Correct answer:

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

Explanation:

| x + 7 | is the absolute value of x+ 7 , 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 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 : Absolute Value

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

Evaluate f(2) .

Possible Answers:

None of the other responses is correct.

Correct answer:

Explanation:

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

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

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

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

$= |3 \cdot 2 -3 |$

= |6 -3 |

= | 3 |

= 3

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Example Question #3 : Absolute Value

Define an operation $\blacktriangledown$ as follows:

For all real numbers a,b ,

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

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

Possible Answers:

$\frac{5}{8}$

The expression is undefined.

None of the other responses is correct.

$1 \frac{1}{8}$

Correct answer:

$1 \frac{1}{8}$

Explanation:

$a \blacktriangledown b= \frac{a+1}{\left | a\right |+ \left | b\right |}$ , 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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Example Question #3 : Absolute Value

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

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

Possible Answers:

$-\frac{1} {9}$

$-\frac{1}{11}$

$\frac{1} {9}$

$\frac{1}{11}$

Correct answer:

$\frac{1} {9}$

Explanation:

$p(x) = \frac{\left |x+2 \right |-1}{\left |x+1 \right |-2}$ , 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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Example Question #2 : How To Find Absolute Value

Define an operation $\triangleright$ as follows:

For all real numbers a,b ,

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

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

Possible Answers:

$5 \frac{5}{6}$

$6\frac{1}{2}$

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

$1\frac{1}{2}$

$2\frac{1}{2}$

Correct answer:

$5 \frac{5}{6}$

Explanation:

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

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

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

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

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

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

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Example Question #2 : How To Find Absolute Value

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

Evaluate g(16) .

Possible Answers:

$\textup{The expression is undefined.}$

$3\textup{,}092$

$3\textup{,}100$

$5\textup{,}100$

$5\textup{,}092$

Correct answer:

$3\textup{,}100$

Explanation:

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

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

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

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

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

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

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

=3,100

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Example Question #243 : Integers

Solve

$\left | x -7\right | = 3$

Possible Answers:

x=10, 4

x=-4

No solution

x=10

x=0

Correct answer:

x=10, 4

Explanation:

Since this is an absolute value equation, we must set the left hand side equal to both the positive and negative versions of the right side. Therefore,

x-7 = 3

x-7 = -3

Solving the first equation, we see that x=10

Solving the second, we see that x=4

When we substitute each value back into the original equation $\left | x -7\right | = 3$ , we see that they both check.

$\left | 10 -7\right | = \left | 3 \right | =3$

$\left | 4 -7\right | = \left | -3 \right | =3$

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Example Question #12 : How To Find Absolute Value

Solve:

|4x -2 | = 14

Possible Answers:

x=-3

x=4

None of the given answers.

x=-3, 4

x=-4, 3

Correct answer:

x=-3, 4

Explanation:

To solve this equation, we want to set 4x-2 equal to both and and solve for .

Therefore:

4x-2 = 14

4x = 16

x=4

and

4x-2=-14

4x = -12

x=-3

We can check both of these answers by plugging them back into the inequality to see if the statement is true.

|4(4) -2| = |16 -2| = |14| = 14

and

|4(-3)-2|=|-14| = 14

Both answers check, so our final answer is x=-3, 4

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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #3 : How To Find Absolute Value

Example Question #4 : How To Find Absolute Value

Example Question #881 : Arithmetic

Example Question #2 : Absolute Value

Example Question #3 : Absolute Value

Example Question #3 : Absolute Value

Example Question #2 : How To Find Absolute Value

Example Question #2 : How To Find Absolute Value

Example Question #243 : Integers

Example Question #12 : How To Find Absolute Value

All SAT Math Resources

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