Algebra - ACT Math
Card 0 of 11421
11/(x – 7) + 4/(7 – x) = ?
11/(x – 7) + 4/(7 – x) = ?
We must find a common denominator and here they changed the first fraction by removing a negative from the numerator and denominator, leaving –11/(7 – x). We add the numerators and keep the same denominator to find the answer.
We must find a common denominator and here they changed the first fraction by removing a negative from the numerator and denominator, leaving –11/(7 – x). We add the numerators and keep the same denominator to find the answer.
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Which of the following is a factor of the polynomial x_2 – 6_x + 5?
Which of the following is a factor of the polynomial x_2 – 6_x + 5?
Factor the polynomial by choosing values that when FOIL'ed will add to equal the middle coefficient, 3, and multiply to equal the constant, 1.
x_2 – 6_x + 5 = (x – 1)(x – 5)
Because only (x – 5) is one of the choices listed, we choose it.
Factor the polynomial by choosing values that when FOIL'ed will add to equal the middle coefficient, 3, and multiply to equal the constant, 1.
x_2 – 6_x + 5 = (x – 1)(x – 5)
Because only (x – 5) is one of the choices listed, we choose it.
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Given that x = 2 and y = 3, how much less is the value of 3_x_2 – 2_y_ than the value of 3_y_2 – 2_x_?
Given that x = 2 and y = 3, how much less is the value of 3_x_2 – 2_y_ than the value of 3_y_2 – 2_x_?
First, we solve each expression by plugging in the given values for x and y:
3(22) – 2(3) = 12 – 6 = 6
3(32) – 2(2) = 27 – 4 = 23
Then we find the difference between the first and second expressions’ values:
23 – 6 = 17
First, we solve each expression by plugging in the given values for x and y:
3(22) – 2(3) = 12 – 6 = 6
3(32) – 2(2) = 27 – 4 = 23
Then we find the difference between the first and second expressions’ values:
23 – 6 = 17
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Evaluate 4x2 + 6x – 17, when x = 3.
Evaluate 4x2 + 6x – 17, when x = 3.
Plug in 3 for x, giving you 36 + 18 – 17, which equals 37.
Plug in 3 for x, giving you 36 + 18 – 17, which equals 37.
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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?
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?
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
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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If (xy/2) – 3_w_ = –9, what is the value of w in terms of x and y?
If (xy/2) – 3_w_ = –9, what is the value of w in terms of x and y?
–3_w_ = –9 – (xy/2)
w = 3 + (xy/6)
–3_w_ = –9 – (xy/2)
w = 3 + (xy/6)
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Evaluate 5_x_2 + 16_x_ + 7 when x = 7
Evaluate 5_x_2 + 16_x_ + 7 when x = 7
Plug in 7 for x and you get 5(49) + 16(7) + 7 = 364
Plug in 7 for x and you get 5(49) + 16(7) + 7 = 364
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Solve for x: (x2 – x) / (x – 1) = 1
Solve for x: (x2 – x) / (x – 1) = 1
Begin by multiplying both sides by (x – 1):
x2 – x = x – 1
Solve as a quadratic equation: x2 – 2x + 1 = 0
Factor the left: (x – 1)(x – 1) = 0
Therefore, x = 1.
However, notice that in the original equation, a value of 1 for x would place a 0 in the denominator. Therefore, there is no solution.
Begin by multiplying both sides by (x – 1):
x2 – x = x – 1
Solve as a quadratic equation: x2 – 2x + 1 = 0
Factor the left: (x – 1)(x – 1) = 0
Therefore, x = 1.
However, notice that in the original equation, a value of 1 for x would place a 0 in the denominator. Therefore, there is no solution.
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Solve 3x2 + 10x = –3
Solve 3x2 + 10x = –3
Generally, quadratic equations have two answers.
First, the equations must be put in standard form: 3x2 + 10x + 3 = 0
Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.
Third, check the answer by plugging the answers back into the original equation.
Generally, quadratic equations have two answers.
First, the equations must be put in standard form: 3x2 + 10x + 3 = 0
Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.
Third, check the answer by plugging the answers back into the original equation.
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What is the value of m where:
What is the value of m where:
If n=4, then 64(4/12)=64(1/3)=4. Then, 4=m4(1+m)/(m+4). If 2 is substituted for m, then 4=24(1+2)/(2+4)=241/2=2√4=22=4.
If n=4, then 64(4/12)=64(1/3)=4. Then, 4=m4(1+m)/(m+4). If 2 is substituted for m, then 4=24(1+2)/(2+4)=241/2=2√4=22=4.
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The population of a bird species is modeled by the following equation:
,
where 
represents the number of years from the present. How many years will it take the population to reach 130 birds (rounded to the nearest tenth)?
The population of a bird species is modeled by the following equation:
where
Plugging in 130 for P, the equation becomes 130 = (11/8)x + 102. Solving for x, we obtain x = 20.4 years.
Plugging in 130 for P, the equation becomes 130 = (11/8)x + 102. Solving for x, we obtain x = 20.4 years.
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If bx + c = e – ax, then what is x?
If bx + c = e – ax, then what is x?
To solve for x:
bx + c = e – ax
bx + ax = e – c
x(b+a) = e-c
x = (e-c) / (b+a)
To solve for x:
bx + c = e – ax
bx + ax = e – c
x(b+a) = e-c
x = (e-c) / (b+a)
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A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?
A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?
This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.
This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.
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Solve the following system of equations:
–2x + 3y = 10
2x + 5y = 6
Solve the following system of equations:
–2x + 3y = 10
2x + 5y = 6
Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)
Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)
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The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?
The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?
Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.
Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.
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Given f(x) = x2 – 9. What are the zeroes of the function?
Given f(x) = x2 – 9. What are the zeroes of the function?
The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.
The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.
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Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?
Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?
In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.
In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.
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What is the 
of the following equation: 
?
What is the
The y-intercept is the constant at the end of the equation. Thus for our equation the y-intercept is 7
The y-intercept is the constant at the end of the equation. Thus for our equation the y-intercept is 7
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What is the equation for a circle of radius 12, centered at the intersection of the two lines:
y_1 = 4_x + 3
and
y_2 = 5_x + 44?
What is the equation for a circle of radius 12, centered at the intersection of the two lines:
y_1 = 4_x + 3
and
y_2 = 5_x + 44?
To begin, let us determine the point of intersection of these two lines by setting the equations equal to each other:
4_x_ + 3 = 5_x_ + 44; 3 = x + 44; –41 = x
To find the y-coordinate, substitute into one of the equations. Let's use _y_1:
y = 4 * –41 + 3 = –164 + 3 = –161
The center of our circle is therefore: (–41, –161).
Now, recall that the general form for a circle with center at (_x_0, _y_0) is:
(x - _x_0)2 + (y - _y_0)2 = _r_2
For our data, this means that our equation is:
(x + 41)2 + (y + 161)2 = 122 or (x + 41)2 + (y + 161)2 = 144
To begin, let us determine the point of intersection of these two lines by setting the equations equal to each other:
4_x_ + 3 = 5_x_ + 44; 3 = x + 44; –41 = x
To find the y-coordinate, substitute into one of the equations. Let's use _y_1:
y = 4 * –41 + 3 = –164 + 3 = –161
The center of our circle is therefore: (–41, –161).
Now, recall that the general form for a circle with center at (_x_0, _y_0) is:
(x - _x_0)2 + (y - _y_0)2 = _r_2
For our data, this means that our equation is:
(x + 41)2 + (y + 161)2 = 122 or (x + 41)2 + (y + 161)2 = 144
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What is the equation for a circle of radius 9, centered at the intersection of the following two lines?
What is the equation for a circle of radius 9, centered at the intersection of the following two lines?
To begin, let us determine the point of intersection of these two lines by setting the equations equal to each other:
To find the y-coordinate, substitute into one of the equations. Let's use 
:
The center of our circle is therefore 
.
Now, recall that the general form for a circle with center at 
is
For our data, this means that our equation is:
To begin, let us determine the point of intersection of these two lines by setting the equations equal to each other:
To find the y-coordinate, substitute into one of the equations. Let's use
The center of our circle is therefore
Now, recall that the general form for a circle with center at
For our data, this means that our equation is:
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