Algebra - ACT Math

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.

This question relies on both the vertical-line test and the definition of a function. We need to use the vertical-line test to determine which of the graphs is not a function (i.e. the graph that has more than one output for a given input). The vertical-line test states that a graph represents a function when a vertical line can be drawn at every point in the graph and only intersect it at one point; thus, if a vertical line is drawn in a graph and it intersects that graph at more than one point, then the graph is not a function. The circle is the only answer choice that fails the vertical-line test, and so it is not a function.

There are four shifts of the graph y = f(x):

y = f(x) + c shifts the graph c units upwards.

y = f(x) – c shifts the graph c units downwards.

y = f(x + c) shifts the graph c units to the left.

y = f(x – c) shifts the graph c units to the right.

The graph has x-intercepts at x = 0 and x = 8. This indicates that 0 and 8 are roots of the function.

The function must take the form y = x(x - 8) in order for these roots to be true.

The parabola opens downward, indicating a negative leading coefficient. Expand the equation to get our answer.

y = -x(x - 8)

y = -x2 + 8x

y = 8x - x2

Therefore, the answer must be y = 8x - x2

One of the properties of taking an absolute value of a function is that the values are all made positive. The values themselves do not change; only their signs do. In this graph, none of the y-values are negative, so none of them would change. Thus the two graphs should be identical.

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.

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

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

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

–3_w_ = –9 – (xy/2)

w = 3 + (xy/6)

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

2x2 + ab – 2b – ax2 = 2x2 – 2b – ax2 + ab

Rearrange terms

= (2x2 – 2b) + (- ax2 + ab) Group

= 2(x2 – b) + a(-x2 + b) Factor each group

= 2(x2 – b) – a(x2 – b) Factor out -a

= (x2 – b) (2 – a) Factor out x2 – b

The polynomial factors into the following expression:

Therefore, the answer is

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.

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.

If n=4, then 64(4/12)=64(1/3)=4. Then, 4=mŸ4(1+m)/(m+4). If 2 is substituted for m, then 4=2Ÿ4(1+2)/(2+4)=2Ÿ41/2=2Ÿ√4=2Ÿ2=4.

To begin a problem like this, you must first convert everything to and alone. This way, you can begin to cancel and combine to its most simplified form.

Since and , we insert those identities into the equation as follows.

From here we combine the numerator and denominators of each fraction together to easily see what we can combine and cancel.

Since there is a in the numerator and the denominator, we can cancel them as they divide to equal 1. All we have left is , the answer.

The basic formula for a circle in the coordinate plane is , where is the center of the circle with radius .

We know that , since that is the only way a diameter can pass through the circle and intercept an x-coordinate of at both ends. , on the other hand, may be seen as halfway between one y-coordinate and the other y-coordinate. Averaging the two, we get:

, so becomes our . Since the diameter is units long, we know the radius is half that, so .

Thus, we have .

Plugging in 130 for P, the equation becomes 130 = (11/8)x + 102. Solving for x, we obtain x = 20.4 years.

To solve for x:

bx + c = e – ax

bx + ax = e – c

x(b+a) = e-c

x = (e-c) / (b+a)