Absolute Value - SAT Subject Test in Math I

Card 0 of 7

Question

Define an operation $\vee$ on the set of real numbers as follows:

For any two real numbers

$a \vee b =| | a + 2b | + |2a + b| |$

Evaluate the expression

$4 \vee\left ( -4 \right )$

Answer

Substitute in the expression:

$a \vee b =| | a + 2b | + |2a + b| |$

$4 \vee\left ( -4 \right ) =| | 4 + 2 (-4) | + |2 (4) + (-4)| |$

$4 \vee\left ( -4 \right ) =| | 4 + (-8) | + |8+ (-4)| |$

$4 \vee\left ( -4 \right ) =| | -4 | + |4| |$

$4 \vee\left ( -4 \right ) =|4+4|$

$4 \vee\left ( -4 \right ) =|8|$

$4 \vee\left ( -4 \right ) =8$

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Question

Solve for .

$\left | 2x+3\right |=7$

Answer

To solve for x we need to make two separate equations. Since it has absolute value bars around it we know that the inside can equal either 7 or -7 before the asolute value is applied.

$\left | 2x+3\right |=7$

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Question

Simplify the following expression:

$\mid -5x\mid -4+6x-\mid -3\mid$

Answer

$\mid -5x\mid -4+6x-\mid -3\mid$

To simplify, we must first simplify the absolute values.

Now, combine like terms:

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Question

The absolute value of a negative can be positive or negative. True or false?

Answer

The absolute value of a number is the points away from zero on a number line.

Since this is a countable value, you cannot count a negative number.

This makes all absolute values positive and also make the statement above false.

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Question

Consider the quadratic equation

$x^{2} +14 = 9x$

Which of the following absolute value equations has the same solution set?

Answer

Rewrite the quadratic equation in standard form by subtracting from both sides:

$x^{2} +14 = 9x$

$x^{2} +14 - 9x = 9x - 9x$

$x^{2}- 9x +14 = 0$

Factor this as

$(x+ \square )(x+ \square ) = 0$

where the squares represent two integers with sum and product 14. Through some trial and error, we find that and work:

By the Zero Product Principle, one of these factors must be equal to 0.

If then ;

if then .

The given equation has solution set $\left { 2, 7\right }$ , so we are looking for an absolute value equation with this set as well.

This equation can take the form

This can be rewritten as the compound equation

Adding to both sides of each equation, the solution set is

and

Setting these numbers equal in value to the desired solutions, we get the linear system

Adding and solving for :

$\underline{a+ b = 7}$

$2a \div 2 = 9 \div 2$

$a = 4 \frac{1}{2}$

Backsolving to find :

$4 \frac{1}{2} + b = 7$

$4 \frac{1}{2} + b - 4 \frac{1}{2} = 7 - 4 \frac{1}{2}$

$b =2 \frac{1}{2}$

The desired absolute value equation is $\left |x- 4 \frac{1}{2} \right |= 2 \frac{1}{2}$ .

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Question

What is the value of: $\left | -(2^{11}) \right |$ ?

Answer

Step 1: Evaluate $2^{11}$ ...

$2^{11}=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2=2048$

Step 2: Apply the minus sign inside the absolute value to the answer in Step 1...

$2048\rightarrow-2048$

Step 3: Define absolute value...

The absolute value of any value $\pm a$ is always positive, unless there is an extra negation outside (sometimes)..

Step 4: Evaluate...

$\left | (-2048)\right |=2048$

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Question

Solve: $-3\left |2x+1 \right | = -3$

Answer

Divide both sides by negative three.

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

$\left |2x+1 \right |=1$

Since the lone absolute value is not equal to a negative, we can continue with the problem. Split the equation into its positive and negative components.

Evaluate the first equation by subtracting one on both sides, and then dividing by two on both sides.

Evaluate the second equation by dividing a negative one on both sides.

$\frac{-(2x+1) }{-1}= \frac{1}{-1}$

Subtract one on both sides.

Divide by 2 on both sides.

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

The answers are:

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