How to find excluded values - SAT Math

Card 0 of 6

Question

If the average (arithmetic mean) of , , and is , what is the average of , , and ?

Answer

If we can find the sum of $\dpi{100} \small x+2$ , $\dpi{100} \small y-6$ , and 10, we can determine their average. There is not enough information to solve for $\dpi{100} \small x$ or $\dpi{100} \small y$ individually, but we can find their sum, $\dpi{100} \small x+y$ .

Write out the average formula for the original three quantities. Remember, adding together and dividing by the number of quantities gives the average: $\frac{x + y + 9}{3} = 12$

Isolate $\dpi{100} \small x+y$ :

$x + y + 9 = 36$

$x + y = 27$

Write out the average formula for the new three quantities:

$\frac{x + 2 + y - 6 + 10}{3} = ?$

Combine the integers in the numerator:

$\frac{x + y + 6}{3} = ?$

Replace $\dpi{100} \small x+y$ with 27:

$\frac{27+ 6}{3} = \frac{33}{3} = 11$

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Question

Which of the following provides the complete solution set for ?

Answer

The absolute value will always be positive or 0, therefore all values of z will create a true statement as long as . Thus all values except for 2 will work.

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Question

Which of the following are answers to the equation below?

$\frac{x^2-4}{x^2+5x+6}=0$

I. -3

II. -2

III. 2

Answer

Given a fractional algebraic equation with variables in the numerator and denominator of one side and the other side equal to zero, we rely on a simple concept. Zero divided by anything equals zero. That means we can focus in on what values make the numerator (the top part of the fraction) zero, or in other words,

The expression is a difference of squares that can be factored as

Solving this for gives either or . That means either of these values will make our numerator equal zero. We might be tempted to conclude that both are valid answers. However, our statement earlier that zero divided by anything is zero has one caveat. We can never divide by zero itself. That means that any values that make our denominator zero must be rejected. Therefore we must also look at the denominator.

The left side factors as follows

This means that if is or , we end up dividing by zero. That means that cannot be a valid solution, leaving as the only valid answer. Therefore only #3 is correct.

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Question

Find the excluded values of the following algebraic fraction

$\frac{2x^2-5x-12}{x^2-11x+28}$

Answer

To find the excluded values of a algebraic fraction you need to find when the denominator is zero. To find when the denominator is zero you need to factor it. This denominator factors into

so this is zero when x=4,7 so our answer is

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Question

$y=\frac{x^2-5}{x^2-25}$

For what value(s) of x is the function undefined?

Answer

When the denominator of a function is equal to 0, the function is undefined at that point. We can set x2-25 equal to 0 in order to find out what values of x make that true.

We can factor to solve for x.

$x=\sqrt{5}$ is an incorrect answer because for this value of x, the function equals zero, but it is not undefined.

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Question

Find the extraneous solution for

$\sqrt{x}=3x-4$

Answer

$\sqrt{x}=3x-4$

$(\sqrt{x})^2=(3x-4)^2$

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{25\pm\sqrt{(-25)^2-4\cdot9\cdot16}}{2\cdot9}$

$x=\frac{25\pm\sqrt{625-576}}{18}$

$x=\frac{25\pm\sqrt{49}}{18}$

$x=\frac{25+7}{18}=\frac{32}{18}=\frac{16}{9}$

$x=\frac{25-7}{18}=\frac{18}{18}=1$

Now lets plug these values into our original equation.

For $x=\frac{16}{9}$

$\sqrt{x}=3x-4$

$\sqrt{\frac{16}{9}}=3\left(\frac{16}{9}\right)-4$

$\frac{4}{3}=\frac{16}{3}-4$

$\frac{4}{3}=\frac{16}{3}-\frac{12}{3}=\frac{4}{3}$

So this isn't an extraneous solution.

For

$\sqrt{x}=3x-4$

$\sqrt{1}=3(1)-4$

Since , is an extraneous solution.

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