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ACT Math : Algebraic Fractions

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Example Questions

Example Question #51 : Algebraic Fractions

Which of the following provides the complete solution set for |2-z|>0 ?

Possible Answers:

z=2

z<2

$\dpi{100} z|z\neq 2$

z>2

No solutions

Correct answer:

$\dpi{100} z|z\neq 2$

Explanation:

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

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Example Question #1 : How To Find Excluded Values

Find the excluded values in the following algebraic fraction.

$\frac{2x^{2}+3x-12}{2x^{2}+9x-5}$

Possible Answers:

$-\frac{1}{3}, 4$

$\frac{1}{3}, -4$

$\frac{1}{2}, -5$

$-\frac{1}{2}, 5$

Correct answer:

$\frac{1}{2}, -5$

Explanation:

In order to find excluded values, find the values that make the denominator equal zero.

To do this, you must factor the denominator:

(2x-1)(x+5)

Now, set each part equal to zero and solve for .

2x-1=0

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

x+5=0

x=-5

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Example Question #1 : How To Find Excluded Values

Find the excluded values for for the following algebraic fraction.

$\frac{x^{2}+4x-12}{x^{2}+5x+6}$

Possible Answers:

3, 2

3, -2

-3, 2

-3, -2

Correct answer:

-3, -2

Explanation:

In order to find the excluded values for x, find the values that make the denominator zero.

In order to find the zeroes, factor the denominator:

(x+3)(x+2)

Now, set each part equal to zero.

(x+3)=0

(x+2)=0

x=-3

x=-2

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Example Question #1 : How To Find Excluded Values

Find the excluded values for for the following algebraic fraction.

$\frac{x+3}{4-x^{2}}$

Possible Answers:

2, -2, 0

2, -2

Correct answer:

2, -2

Explanation:

To find the excluded values, find the values that make the denominator zero.

$4-x^{2}=0$

$4=x^{2}$

x=2, -2

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Example Question #1 : How To Find Excluded Values

Find the excluded values for for the following algebraic fraction.

$\frac{x-4}{x(x^{2}-9)}$

Possible Answers:

3, 0

3, -3, 0

3, -3

-3, 0

Correct answer:

3, -3, 0

Explanation:

To find the excluded values, find the values that make the denominator equal zero.

To do that, set the denominator equal to zero and solve for .

Begin by taking each multiple in the denominator and setting it equal to zero.

x=0

$x^{2}-9=0$

$x^{2}=9$

x=3, -3

x=3, -3, 0

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Example Question #2 : How To Find Excluded Values

Find the excluded values for in the following algebraic fraction.

$\frac{x^{3}+8x-9}{x^{2}+7x+12}$

Possible Answers:

4, -3

-4, -3

4, 3

-4, 3

Correct answer:

-4, -3

Explanation:

To find the excluded values, find the values that make the denominator equal zero.

To do that, begin by factoring the denominator.

(x+4)(x+3)

Now, set each part equal to zero.

x+4=0

x+3=0

Now, solve for .

x=-4

x=-3

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Example Question #1 : How To Find Excluded Values

Find the excluded values for in the following algebraic fraction.

$\frac{x^{2}+9}{x^{2}-16}$

Possible Answers:

4, -4

4, -4, 0

Correct answer:

4, -4

Explanation:

To find the excluded values, find the -values that make the denominator equal zero.

To do this, set the denominator equal to zero:

$x^{2}-16=0$

Next, solve for x.

$x^{2}=16$

x=4, -4

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Example Question #1 : How To Find Excluded Values

Find the excluded values for in the following algebraic fraction.

$\frac{x^{2}+14x-21}{x^{3}+27}$

Possible Answers:

3, -3, 0

3, -3

Correct answer:

Explanation:

To find the excluded values, find the -values that make the denominator equal zero.

1) set the denominator equal to zero.

$x^{3}+27=0$

2) solve for .

$x^{3}=-27$

x=-3

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Example Question #3 : How To Find Excluded Values

Find the excluded values of the following algebraic fraction

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

Possible Answers:

$x\neq4,7$

The numerator cancels all the binomials in the denomniator so ther are no excluded values.

$x\neq-11$

$x\neq4$

$x\neq7$

Correct answer:

$x\neq4,7$

Explanation:

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

(x-7)(x-4)

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

$x\neq4,7$

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Example Question #1 : How To Find Excluded Values

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

Possible Answers:

II only

II and III

III only

I, II, and III

I only

Correct answer:

III only

Explanation:

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,

x^2-4=0

The expression x^2-4 is a difference of squares that can be factored as

(x-2)(x+2)=0

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.

x^2+5x+6=0

The left side factors as follows

(x+3)(x+2)=0

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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ACT Math : Algebraic Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #51 : Algebraic Fractions

Example Question #1 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

Example Question #2 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

Example Question #3 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

All ACT Math Resources

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