In the following figure, what is the value of yin terms of x?

Solution
The question gives you two of the three interior angles of the triangle: x° and 100°.
The remaining interior angle is supplementary with y°, so it’s (180 – y)°.
Thus, you can make the following equation:
x + 100 + (180 – y) = 180
Solve for yin terms of x:
x + 100 + 180 – y = 180
x + 100 – y = 0
x + 100 = y
If the height of an equilateral triangle is 9, what is its area?

Solution
An equilateral triangle divides into two 306090 triangles, whose sides have a ratio of x : x√3 : 2x.
The height of 9 corresponds to the x√3, so
x√3 = 9
\(x = \frac{9}{\sqrt{3}}\)
You can simplify this value as I show you in Chapter 4:
\(\frac{9}{\sqrt{3}} = \frac{9}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}\)
So the base of the triangle is twice this value, which is 6√3 Plug the lengths of the base and the height into the formula for the area of a triangle:
\(A = \frac{1}{2}bh = \frac{1}{2}\left ( 6\sqrt{3} \right )\left ( 9 \right ) = 27\sqrt{3}\)
If \(\frac{3n}{2} = \frac{4n + 3}{3}\), then n =

Solution
Crossmultiply to get rid of the fractions:
\(\frac{3n}{2} = \frac{4n + 3}{3}\)
3(3n) = 2(4n + 3)
Now simplify and solve for n:
9n = 8n + 6
n = 6
What is the value of x if x^{2} – 5x – 14 = 0 and x > 0?

Solution
Factor the left side of the equation:
x^{2} – 5x – 14 = 0
(x + 2)(x – 7) = 0
Next, split the equation into two separate equations and solve each for x:
x + 2 = 0 ; x – 7 = 0
x = –2 ; x = 7
Because x > 0, the value of x is 7.
If 3x + 5y = 4, which of the following is equivalent to the expression (6x + 10y)(100x + 100y)?

Solution
To begin, factor a 2 out of (6x + 10y):
(6x + 10y)(100x + 100y) = 2(3x + 5y)(100x + 100y)
Now substitute 4 for 3x + 5y and distribute:
2(4)(100x + 100y) = 8(100x + 100y) = 800x + 800y
What is the slope of a line that includes the points (–4, 1) and (10, –6)?

Solution
Plug the values (–4, 1) and (10, –6) into the twopoint slope formula:
\(Slope = \frac{y_{2}  y_{1}}{x_{2}  x_{1}} = \frac{6  1}{10  \left ( 4 \right )} = \frac{7}{14} = \frac{1}{2}\)
How many different positive integers are factors of both 28 and 42?

Solution
List the factors of both 28 and 42:
Factors of 28: 1 2 4 7 14 28
Factors of 42: 1 2 3 6 7 14 21 42
Now you can see that 1, 2, 7, and 14 are factors of both numbers, so the correct answer is Choice (D).
Which of the following is equivalent to –8(x – 2) < 3x – 6?

Solution
Distribute the left side and combine like terms:
–8(x – 2) < 3x – 6 –8x + 16 < 3x – 6 –11x + 16 < –6 –11x < –22 To solve for x,divide both sides by –11 and reverse the inequality: x > 2
What is the value of k if \(\sqrt{10k + 3} = 5\)?

Solution
Start by squaring both sides of the equation to undo the square root:
\(\sqrt{10k + 3} = 5\)
10k + 3 = 25
Now solve for k:
10k = 22
k = 2.2
Tara’s three bowling scores in a tournament were 167, 178, and 186. What was her average score for the tournament?

Solution
Tara’s scores for 3 games were 167, 178, and 186.
To find the average, simply place these numbers into the formula for the mean:
\(Mean = \frac{Sum \,\; of \;\, Values}{Number \,\; of \,\; Values}\) = \(\frac{167 + 178 + 186}{3} = \frac{531}{3} = 177\)