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SAT Math : Absolute Value

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Example Questions

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 #4 : 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:

$6\frac{1}{2}$

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

$2\frac{1}{2}$

$1\frac{1}{2}$

$5 \frac{5}{6}$

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 #5 : Absolute Value

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

Evaluate g(16) .

Possible Answers:

$3\textup{,}092$

$5\textup{,}100$

$5\textup{,}092$

$\textup{The expression is undefined.}$

$3\textup{,}100$

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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Example Question #884 : Arithmetic

Solve:

$| \frac{x}{2}+\pi | =4\pi$

Possible Answers:

$6\pi, -10\pi$

6, 10

$6\pi, 10\pi$

Correct answer:

$6\pi, -10\pi$

Explanation:

To solve this problem, we want to set what's inside the absolute value signs equal to the positive and negative value on the right side of the equation. That's because the value inside the absolute value symbols could be equivalent to $4\pi$ or $-4\pi$ , and the equation would still hold true.

So let's set $\frac{x}{2}+\pi$ equal to $4\pi$ and $-4\pi$ separately and solve for our unknown.

First:

$\frac{x}{2}+\pi = 4\pi$

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

$x=6\pi$

Second:

$\frac{x}{2}+\pi = -4\pi$

$\frac{x}{2} = -5\pi$

$x=-10\pi$

Therefore, our answers are $6\pi$ and $-10\pi$ .

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Example Question #11 : Absolute Value

Evaluate the expression if x=5 and y=7 .

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

Possible Answers:

Correct answer:

Explanation:

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

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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Example Question #12 : Absolute Value

Simplify the following:

$\left | 56-107\right |$

Possible Answers:

163

Correct answer:

Explanation:

Simplify the following:

$\left | 56-107\right |$

Begin with basic subtraction:

$\left | 56-107\right |=\left | -51\right |$

Next, remember what we do with absolute value signs; we change negative values to positive values, and positive values remain positive.

$\left | -51\right |=51$

So our answer is positive 51

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Example Question #14 : How To Find Absolute Value

Solve for all possible values of x.

$\left | 84x+14\right |=7x-35$

Possible Answers:

$\frac{-7}{11}, \frac{3}{13}$

$\frac{7}{11}$

$\frac{-3}{11}$

$\frac{7}{11}, \frac{-7}{13}$

Correct answer:

$\frac{-7}{11}, \frac{3}{13}$

Explanation:

When solving for x in the presence of absolute value, there are always two answers.

To eliminate the absolute value, the equation must be re-written two ways:

84x+14=7x-35 and 84x+14=-(7x-35)

84x=7x-49 and 84x+14=-7x+35

77x=-49 and 91x+14=35

$x=\frac{-49}{77}$ and 91x = 21

$x=\frac{-7}{11}$ and $x=\frac{3}{13}$

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Example Question #1 : How To Find Absolute Value

Define an operation $\blacklozenge$ as follows:

For all real numbers a,b ,

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

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

Possible Answers:

Both and

Correct answer:

Explanation:

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

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

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

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

$= \left |11 \right |$

= 11

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SAT Math : Absolute Value

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #3 : Absolute Value

Example Question #4 : Absolute Value

Example Question #5 : Absolute Value

Example Question #243 : Integers

Example Question #12 : How To Find Absolute Value

Example Question #884 : Arithmetic

Example Question #11 : Absolute Value

Example Question #12 : Absolute Value

Example Question #14 : How To Find Absolute Value

Example Question #1 : How To Find Absolute Value

All SAT Math Resources

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