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ACT Math : Rational Expressions

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Example Questions

Example Question #1 : Expressions

Simplify the following:

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

Possible Answers:

2x^2-4x+4

$\frac{2(x+3)(x-2)}{(x+3)(x-2)}$

$\frac{(x+3)^2+(x-2)^2}{(x+3)(x-2)}$

Correct answer:

$\frac{(x+3)^2+(x-2)^2}{(x+3)(x-2)}$

Explanation:

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

$\frac{(x+3)(x+3)}{(x-2)(x+3)}+\frac{(x-2)(x-2)}{(x+3)(x-2)}$

$\frac{(x+3)^2+(x-2)^2}{(x+3)(x-2)}$

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Example Question #1 : How To Add Rational Expressions With Different Denominators

Simplify the following $\frac{3x}{x-3} + \frac{7}{x-4}$

Possible Answers:

$\frac{3x^{2}-5x-21}{x^{2}-12}$

$\frac{3x^{2} - 5x -21}{x^{2}-7x+12}$

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

$\frac{3x^{2} -12x +21}{x^{2}-7x+12}$

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

Correct answer:

$\frac{3x^{2} - 5x -21}{x^{2}-7x+12}$

Explanation:

Find the least common denominator between x-3 and x-4, which is (x-3)(x-4). Therefore, you have $\frac{3x(x-4)}{(x-3)(x-4)} + \frac{7(x-3)}{(x-4)(x-3)}$ . Multiplying the terms out equals $\frac{3x^{2}-12x+ 7x-21}{x^{2}-7x+12}$ . Combining like terms results in $\frac{3x^{2}-5x-21}{x^{2}-7x+12}$ .

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Example Question #3 : Rational Expressions

Simplify the following expression:

$\frac{5}{7}+\frac{11}{14}$

Possible Answers:

$\frac{16}{7}$

$\frac{16}{21}$

$\frac{3}{2}$

$\frac{8}{7}$

Correct answer:

$\frac{3}{2}$

Explanation:

In order to add fractions, we must first make sure they have the same denominator.

So, we multiply $\frac{5}{7}$ by $\frac{2}{2}$ and get the following:

$\frac{5}{7}\left(\frac{2}{2}\right)+\frac{11}{14}$

$\frac{10}{14}+\frac{11}{14}$

Then, we add across the numerators and simplify:

$\frac{10}{14}+\frac{11}{14}=\frac{21}{14}=\frac{3}{2}$

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Example Question #1 : Rational Expressions

Combine the following two expressions if possible.

$\frac{x+3}{x+7} + \frac{x^2}{4-x}$

Possible Answers:

$\frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

$\frac{2x^3 -5x^2 + 2x -30}{-x^2-3x+28}$

$\frac{(x-3)^3}{-x^2-3x+28}$

$\frac{-x^3 +4x^2 + 8x -4}{x^2-3x-28}$

$\frac{x^3 +3x^2 -7x - 22}{x^2-3x-28}$

Correct answer:

$\frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

Explanation:

For binomial expressions, it is often faster to simply FOIL them together to find a common trinomial than it is to look for individual least common denominators. Let's do that here:

$\frac{x+3}{x+7} + \frac{x^2}{4-x} = \frac{x+3}{x+7} * \frac{(4-x)}{(4-x)} + \frac{x^2}{4-x} * \frac{(x+7)}{(x+7)}$

FOIL and simplify.

$\frac{-x^2 + x + 12}{-x^2 -3x+28} + \frac{x^3 +7x^2}{-x^2 -3x+28}$

Combine numerators.

$\frac{-x^2 + x + 12}{-x^2 -3x+28} + \frac{x^3 +7x^2}{-x^2 -3x+28} = \frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

Thus, our answer is $\frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

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Example Question #2 : Rational Expressions

Select the expression that is equivalent to

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

Possible Answers:

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

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

$\frac{x^3 - x^2 -9}{3x^2}$

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

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

Correct answer:

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

Explanation:

To add the two fractions, a common denominator must be found. With one-term denominators, it is easier to simply find the least common denominator between them and multiply each side to obtain it.

In this case, the least common denominator between and x^2 is 3x^2 . So the first fraction needs to be multiplied by and the second by :

$\frac{x^2 + 3x - 2}{3x} + \frac{x-3}{x^2} = \frac{x^2 + 3x - 2}{3x}* \left(\frac{x}{x}\right) + \frac{x-3}{x^2} * \left(\frac{3}{3}\right)$

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

Now, we can add straight across, remembering to combine terms where we can.

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

So, our simplified answer is $\frac{x^3 + 3x^2 +x -9}{3x^2}$

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Example Question #2 : Expressions

2x-12=24

3y-15=60

Find the product of and .

Possible Answers:

260

450

144

325

120

Correct answer:

450

Explanation:

Solve the first equation for .

2x-12=24

2x=36

x=18

Solve the second equation for .

3y-15=60

3y=75

y=25

The final step is to multiply and .

xy = (18)(25)=450

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Example Question #1 : Expressions

The following table shows the temperature of a cup of coffee at different times

Time 1:09 1:11 1:13 1:15 1:17

Temperature (ºF) 187.1 184.4 181.7 179.0 176.3

If this trend continues, what will the temperature of the coffee at minute 1:25?

Possible Answers:

168.3°F

162.9°F

160.2°F

171.0°F

165.5°F

Correct answer:

165.5°F

Explanation:

The table shows that for every two minutes, the temperature of the coffee lowers 2.7ºF. At 1:25, 16 minutes, or eight 2-minute intervals have passed, and the temperature of the coffee has lowered by 8*2.7ºF, reaching a temperature of 165.5ºF.

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Example Question #1 : Rational Expressions

Amy buys concert tickets for herself and her friends. She initially buys them at $40/ticket. Weeks later, her other friends ask her to buy them tickets, but the prices have increased to $54. Amy buys 7 tickets total and spends $350. How many tickets has she paid $40 on?

Possible Answers:

Correct answer:

Explanation:

Amy has bought 7 tickets, x of them at $40/ticket, and the remaining 7-x at $54/ticket. She spends at total of

$\\ \\ \$350 = 40x + (54)(7-x) = 40x + 378 – 54x = 378 – 14x\rightarrow \\ 14x = 378-350 = 28 \rightarrow x =2.$

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Example Question #1 : How To Evaluate Rational Expressions

What is

$\frac{3}{\sqrt{6}}$ ?

Possible Answers:

$\frac{\sqrt{12}}{3}$

$\frac{\sqrt{6}}{3}$

$\frac{\sqrt{6}}{2}$

$\frac{\sqrt{3}}{3}$

$\frac{\sqrt{3}}{2}$

Correct answer:

$\frac{\sqrt{6}}{2}$

Explanation:

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

$\frac{3}{\sqrt{6}}$ $*\frac{\sqrt{6}}{\sqrt{6}}=$

$\frac{3*\sqrt{6}}{6}=$

To simplify completely, factor out a three from the numerator and denominator resulting in the final solution.

$\frac{\sqrt{6}}{2}$

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Example Question #1 : How To Subtract Rational Expressions With Different Denominators

Simplify:

$\frac{4}{x-3}-\frac{12}{x+1}$

Possible Answers:

$\frac{4}{x-1}$

$\frac{-8}{x^2-2}$

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

$\frac{-8}{x-2}$

$\frac{40-8x}{x^2-2x-3}$

Correct answer:

$\frac{40-8x}{x^2-2x-3}$

Explanation:

The common denominator of these two fractions simply is the product of the two denominators, namely:

(x-3)(x+1)

Thus, you will need to multiply each fraction's numerator and denominator by the opposite fraction's denominator:

$\frac{4}{x-3}-\frac{12}{x+1}=\frac{4}{x-3}*\frac{x+1}{x+1}-\frac{12}{x+1}*\frac{x-3}{x-3}$

Let's first simplify the numerator:

$\frac{4(x+1)-12(x-3)}{(x-3)(x+1)}$

$\frac{4x+4-12x+36}{(x-3)(x+1)}$

$\frac{-8x+40}{(x-3)(x+1)}$ , which is the simplest form you will need for this question.

However, the correct answer has the denominator multiplied out. Merely FOIL (x-3)(x+1) :

$\frac{40-8x}{x^2-2x-3}$

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ACT Math : Rational Expressions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Expressions

Example Question #1 : How To Add Rational Expressions With Different Denominators

Example Question #3 : Rational Expressions

Example Question #1 : Rational Expressions

Example Question #2 : Rational Expressions

Example Question #2 : Expressions

Example Question #1 : Expressions

Example Question #1 : Rational Expressions

Example Question #1 : How To Evaluate Rational Expressions

Example Question #1 : How To Subtract Rational Expressions With Different Denominators

All ACT Math Resources

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