Expressions - ACT Math

Card 0 of 981

Question

If , what is the value of:

$6(x+y)-(x+y)^2+\frac{1}{5}(x+y)+8$

Answer

Substituting x+y for 5, we get

6(5) - 52 + (1/5)(5) + 8

30 - 25 + 1 + 8

= 14

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Question

Evaluate 4x2 + 6x – 17, when x = 3.

Answer

Plug in 3 for x, giving you 36 + 18 – 17, which equals 37.

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Question

John has a motorcycle. He drives it to the store, which is 30 miles away. It takes him 30 minutes to drive there and 60 minutes to drive back, due to traffic. What was his average speed roundtrip in miles per hour?

Answer

The whole trip is 60 miles, and it takes 90 minutes, which is 1.5 hours.

Miles per hour is 60/1.5 = 40 mph

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Question

If (xy/2) – 3_w_ = –9, what is the value of w in terms of x and y?

Answer

–3_w_ = –9 – (xy/2)

w = 3 + (xy/6)

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Question

Evaluate 5_x_2 + 16_x_ + 7 when x = 7

Answer

Plug in 7 for x and you get 5(49) + 16(7) + 7 = 364

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Question

Simplify the following rational expression: (9x - 2)/(x2) MINUS (6x - 8)/(x2)

Answer

Since both expressions have a common denominator, x2, we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

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Question

What is $\frac{x^{2} + 11x + 28}{x+7}$ ?

Answer

$x^{2} + 11x + 28$ factors to . Thus, $\frac{(x+4)(x+7)}{x+7}$ . Canceling out like terms leads to .

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Question

Simplify the following rational expression:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

Answer

Since both fractions in the expression have a common denominator of $x^{2}$ , we can combine like terms into a single numerator over the denominator:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

$=\frac{(7x-18)+(6x-14)}{x^{2}}$

$=\frac{13x-32}{x^{2}}$

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Question

Simplify the following rational expression:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

Answer

Since both rational terms in the expression have the common denominator $2x^{2}$ , combine the numerators and simplify like terms:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

$=\frac{(5x-5)+(7x+9)}{2x^{2}}$

$=\frac{12x+4}{2x^{2}}$

$=\frac{6x+2}{x^2}$

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Question

Simplify the following expression:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

Answer

Since both terms in the expression have the common denominator $x^{3}$ , combine the fractions and simplify the numerators:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

$=\frac{(10x-9)+(11x+12)}{x^{3}}$

$=\frac{21x+3}{x^{3}}$

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Question

Compute the following: $\frac{x+2}{2-x}+\frac{x-2}{2-x}$

Answer

Notice that the denominator are the same for both terms. Since they are both the same, the fractions can be added. The denominator will not change in this problem.

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

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Question

A company rents cars for a rental fee of $37.00 per day, with an additional charge of $0.45 per mile driven. Which of the following expressions represents the cost, in dollars, of renting the car for 2 days and driving it m miles?

Answer

To determine cost we add the initial rental fee of $37, times two days giving us $74 plus the mileage rate, 0.45, times the number of miles. Giving an equation of 0.45m+74.

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Question

Combine the following rational expressions:

$\frac{3x^2-4x+5}{x+1}+\frac{4x^2-5x+9}{x+1}$

Answer

When working with complex fractions, it is important not to let them intimidate you. They follow the same rules as regular fractions!

$\frac{3x^2-4x+5}{x+1}+\frac{4x^2-5x+9}{x+1}$

In this case, our problem is made easier by the fact that we already have a common denominator. Nothing fancy is required to start. Simply add the numerators:

$\frac{(3x^2-4x+5)+(4x^2-5x+9)}{x+1}$

For our next step, we need to combine like terms. This is easier to see if we group them together.

$\frac{({\color{Blue} 3x^2+4x^2})+({\color{Red} -4x+-5x})+({\color{DarkOrange} 5+9})}{x+1}$

$\frac{({\color{Blue}7x^2})+({\color{Red} -9x})+({\color{DarkOrange} 14})}{x+1}$

Thus, our final answer is:

$\frac{7x^2 -9x+14}{x+1}$

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Question

Simplify the following expression:

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

Answer

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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Question

Simplify the following:

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

Answer

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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Question

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

Answer

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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Question

Sally is ordering snacks for her class trip. She needs 85 cookies. The cookies come in cases of 6 boxes, with 7 cookies in each box. Sally can't order a partial box. What is the smallest number of cases she should order?

Answer

We first determine how many cookies are in each box. There are 7 cookies in a box, multiplied by 6 boxes, making 42 cookies in a case. We then divide the total number of cookies she needs, 85, by the number in each case, 42, giving us 2 with a remainder. This means Sally must order 3 cases of cookies.

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Question

Select the expression that is equivalent to

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

Answer

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 is . 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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Question

Combine the following two expressions if possible.

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

Answer

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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Question

Which of the following is equivalent to $\dpi{100} \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ ? Assume that denominators are always nonzero.

Answer

We will need to simplify the expression $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We can think of this as a large fraction with a numerator of $\frac{1}{t}-\frac{1}{x}$ and a denominator of $\dpi{100} x-t$ .

In order to simplify the numerator, we will need to combine the two fractions. When adding or subtracting fractions, we must have a common denominator. $\frac{1}{t}$ has a denominator of $\dpi{100} t$ , and $\dpi{100} -\frac{1}{x}$ has a denominator of $\dpi{100} x$ . The least common denominator that these two fractions have in common is $\dpi{100} xt$ . Thus, we are going to write equivalent fractions with denominators of $\dpi{100} xt$ .

In order to convert the fraction $\dpi{100} \frac{1}{t}$ to a denominator with $\dpi{100} xt$ , we will need to multiply the top and bottom by $\dpi{100} x$ .

$\frac{1}{t}=\frac{1\cdot x}{t\cdot x}=\frac{x}{xt}$

Similarly, we will multiply the top and bottom of $\dpi{100} -\frac{1}{x}$ by $\dpi{100} t$ .

$\frac{1}{x}=\frac{1\cdot t}{x\cdot t}=\frac{t}{xt}$

We can now rewrite $\frac{1}{t}-\frac{1}{x}$ as follows:

$\frac{1}{t}-\frac{1}{x}$ = $\frac{x}{xt}-\frac{t}{xt}=\frac{x-t}{xt}$

Let's go back to the original fraction $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We will now rewrite the numerator:

$\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ = $\frac{\frac{x-t}{xt}}{x-t}$

To simplify this further, we can think of $\frac{\frac{x-t}{xt}}{x-t}$ as the same as $\frac{x-t}{xt}\div (x-t)$ . When we divide a fraction by another quantity, this is the same as multiplying the fraction by the reciprocal of that quantity. In other words, $a\div b=a\cdot \frac{1}{b}$ .

$\frac{x-t}{xt}\div (x-t)$ = $\frac{x-t}{xt}\cdot \frac{1}{x-t}$ $=\frac{x-t}{xt(x-t)}$ $= \frac{1}{xt}$

Lastly, we will use the property of exponents which states that, in general, $\frac{1}{a}=a^{-1}$ .

$\frac{1}{xt}=(xt)^{-1}$

The answer is $(xt)^{-1}$ .

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