In the standard (x,y) coordinate plane, what is the slope of the line with equation 4y − 6x = 8?

Solution
To solve this problem, put the given equation in the standard form, y = mx + b, where m is the slope:
4y − 6x = 8
4y = 6x + 8
y =(^{6}⁄_{4})x + 2
y =(^{3}⁄_{2})x + 2
Therefore, m = ^{3}⁄_{2}.
The lengths of the sides of a triangle are 3 consecutive even integers. If the perimeter of the triangle is 48 inches, what is the length, in inches, of the longest side?

Solution
To solve this problem, first remember that perimeter is equal to the distance around an object.
Therefore, the perimeter of a triangle is simply the sum of the lengths of its sides.
You are given that the lengths of the sides of the triangle are 3 consecutive even integers.
Set the first length (the shortest side) equal to x, the second length equal to x + 2, and the third length (the longest side) equal to x + 4.
Now, create an equation and solve for x, as follows:
x + (x + 2) + (x + 4) = 48
3x + 6 = 48
x = 14
You now know that the length of the shortest side is 14, so the length of the longest side must be 14 + 4 = 18.
Marcia’s horse’s rectangular corral is 50 feet wide by 125 feet long. Marcia wants to increase the area by 1,850 square feet by increasing the width and length by the same amount. What will be the new dimensions (width by length), in feet?

Solution
To solve this problem, first calculate the new area of Marcia’s horse corral:
Initial area = 50 × 125 = 6,250 square feet
New area = 6,250 + 1,850 = 8,100 square feet
Now, you can try the answer choices to see which combination of width and length fits the parameters given in the question:
Answer choice F: While these dimensions indicate an increase of 5 feet for both the width and the length, the area does not equal 8,100 (55 × 130 = 7,150 square feet).
Answer choice G: These dimensions reflect an increase of 10 feet for both the width and the length, and the area equals 8,100 (60×135 = 8,100 square feet), so this is the correct answer
In the standard (x,y) coordinate plane, how many times does the graph of (x + 1)(x − 2)(x + 3)(x + 4) intersect the xaxis?

Solution
A line, parabola, and so on will intersect the xaxis when y = 0.
Therefore, you can set each of the elements given in the question to 0 and solve for x, as follows:
(x + 1) = 0, x = −1
(x − 2) = 0, x = 2
(x + 3) = 0, x = −3
(x + 4) = 0, x = −4
The graph will cross the xaxis at −1, 2, −3, and −4.
A car leaves a parking lot and travels directly north for 6 miles. It then turns and travels 8 miles east. How many miles is the car from the parking lot?

Solution
To solve this problem, it might be helpful to draw a picture like the one shown next:
You can see that the route traveled creates a right triangle. Use the Pythagorean Theorem to calculate the distance from the parking lot, as follows:
6^{2} + 8^{2} = x^{2}
36 + 64 = x^{2}
100 = x^{2}
10 = x
Jenny ran 3^{1}⁄_{3} miles on Saturday and 2^{4}⁄_{5} miles on Sunday. The total distance, in miles, Jenny ran during those 2 days is within which of the following ranges?

Solution
To solve this problem, first find the common denominator, as follows:
\(3\frac{1}{2}+2\frac{4}{5}=3\frac{5}{15}+2\frac{12}{15}\)
Next, add the fractions:
\(3\frac{5}{15}+2\frac{12}{15}\)
=\(5\frac{17}{15}\)
=6\(5\frac{17}{15}\)
Now look at the fraction part of this mixed number and work through the answer choices. You know that \(\frac{2}{15}\) is less than
^{1}⁄_{2} (\(\frac{7.5}{15}\)), so eliminate answer choice A. You also know that \(\frac{2}{15}\) is
less than ^{1}⁄_{3} (\(\frac{5}{15}\)), so eliminate answer choice B.
Because 6\(\frac{2}{15}\) is greater than 6 but less than 6^{1}⁄_{3},Jenny’s total distance is between 6 and 6^{1}⁄_{3}miles.
If (x + r) 2 = x^{2} + 22x + r^{2} for all real numbers x, then r =?

Solution
To solve this problem, expand (x + r)^{2} using the FOIL method:
(x + r)^{2}
= (x + r)(x + r)
= x^{2} + 2xr + r^{2}
Now, substitute this value for (x + r)^{2} in the given equation:
x^{2} + 2xr + r^{2} = x^{2} + 22x + r^{2}
You can see that 2xr = 22x, so solve for r, as follows:
2xr = 22x
r = 22x/2x
r = 11
A rope 55 feet long is cut into two pieces. If one piece is 23 feet longer than the other, what is the length, in feet, of the shorter piece?

Solution
To solve this problem, set up an equation to determine the length of the shorter piece, substituting x for the unknown, shorter length:
x + (x + 23) = 55
2x + 23 = 55
2x = 32
x = 16
The problem states that one piece is 23 feet longer than the other piece, which does not necessarily mean that the shorter piece will be 23 feet long, so answer choice A is incorrect.
If you simply subtracted 23 from 55 you would get answer choice D, which is incorrect.
This calculation does not account for both lengths of rope. If you divided the original length of the rope (55) by 2, subtracted 23 and rounded up, you would get answer choice E.
This is not the correct calculation to perform.
Two whole numbers have a greatest common factor of 8 and a least common multiple of 48. Which of the following pairs of whole numbers will satisfy the given conditions?

Solution
You are given that both numbers have a factor of 8 and that they both factor into 48 evenly (48 is the least common multiple).
Therefore, the following is true:
48 = 8 × a × b
Because 48 = 8 × 6, a × b must equal 6; a could equal 2 and b could equal 3, which means that one of the given numbers has a factor of 2 and the other has a factor of 3.
Both numbers have a common factor of 8, so one number could be 8 × 2 = 16 and the other number could be 8 × 3 = 24.
If W = XYZ, then which of the following is an expression for Z in terms of W, X, and Y?

Solution
Don’t let the fact that there are no numbers in this math problem confuse you! Simply remember that W, X, Y, and Z each represent some number, and perform the correct mathematical operations to isolate Z on one side of the equation, as follows:
W = XYZ
\(\frac{W}{XY}\) = Z