For all a ≠ 0 and b ≠ 0,\(\frac{a + b}{b(a + b) − 2a(a + b)}\)= ?

Solution
If tan x = ^{3}⁄_{4} and 0◦ ≤ x◦ ≤ 90◦, then cos x = ?

Solution
The tangent of any acute angle is the opposite side divided by the length of the adjacent side \(\left ( \frac{opp}{adj} \right )\).
Cosine is equal to the length of adjacent side divided by the hypotenuse\(\left ( \frac{adj}{hyp} \right )\).
Since tan x = ^{3}⁄_{4}\(\left ( \frac{opp}{adj} \right )\), then the adjacent side must be 4; eliminate answer choices B, C, and D.
Now, figure out the measure of the hypotenuse by using the pythagorean theorem:
3^{2} + 4^{2} = (hyp)^{2}
9 + 16 = (hyp)2
25 = (hyp)^{2}
5 = the hypotenuse.
So, cos x = ^{4}⁄_{5}.
In the figure below, triangle PQR is an isosceles right triangle. What is the ratio of the hypotenuse to the length of PQ?

Solution
You are given that the triangle is an isosceles right triangle, which means that it is a 45◦−45◦−90◦ triangle.
One of the characteristics of this type of triangle is that the hypotenuse is √2 times the measure of each of the legs.
So the ratio of the hypotenuse to PQ is √2 : 1.
Which of the following represents the values of x that are solutions for the inequality (x − 1)(4 − x) < 0?

Solution
The first step in answering this question is to determine what we will call your critical numbers.
Solve for x as if there were an equal sign instead of an inequality sign:
(x − 1)(4 − x) = 0
x − 1 = 0 and 4 − x = 0
x = 1 and x = 4.
Only answer choice B has both of these numbers, so answer choice B is correct.
To make sure, choose a number that is greater than 4, like 5, and see if it is a solution to the inequality (x − 1)(4 − x) < 0: (4)(−1) = −4 < 0; x > 4 is correct.
Choose a number that is less than 1, like 0:
(−1)(4) = −4 < 0; x < 1 is also correct.
Let x = 3y − 4z + 7. What happens to the value of x if the value of y decreases by 2 and the value of z is increased by 1?

Solution
To solve this problem, replace y with y − 2, and z with z + 1:
x = 3(y − 2) − 4(z + 1) + 7
= 3y − 6 − 4z − 4 + 7
= 3y − 4z − 3
So, you went from x = 3y−4z+7 to x = 3y−4z−3.
Subtract the two to see the difference:
(3y − 4z + 7) − (3y − 4z − 3)
3y − 3y = 0
−4z − (−4z) = 0
7 − (−3) = 10
Since the difference is a positive 10, that means that the original value of x was 10 greater than the new value of x.
Thus, the value of x decreased by 10.
What is the smallest possible value for a where y = sin 2a reaches its maximum?

Solution
To solve this problem, you must first recall that the maximum value of the sine function is 1, and that the smallest value for a at this maximum value will be ^{π}⁄_{2}.
So, if 2a = ^{π}⁄_{2},then a = ^{π}⁄_{4}.
What are the solutions for the equation 3x^{2}−5x+2 = 0?

Solution
The first step in answering this question is to factor the equation 3x^{2} − 5x + 2 = 0:
(3x − 2)(x − 1) = 0
To find the two solutions to this equation, solve for zero:
3x − 2 = 0
3x = 2
x = ^{2}⁄_{3} (solution 1)
x − 1 = 0
x = 1 (solution 2)
The solutions to the equation given are x = 1 and
x = ^{2}⁄_{3}.
Anne made apple jelly and applesauce out of a bushel of apples. If the number of jars of jelly, j, is 3 less than twice the number of jars of applesauce, a, which expression shows the relationship of jars of jelly, j, to the jars of applesauce, a?

Solution
You are given that number of jars of jelly, j, is three less than twice the number of jars of applesauce, a.
Put this into equation form: j = 2a − 3.
Since none of the answers match this one, rearrange the equation: j + 3 = 2a.
If two lines in the standard (x,y) coordinate plane are perpendicular and the slope of one of the lines is −^{5}⁄_{7}, what is the slope of the other line?

Solution
Perpendicular lines have slopes that are the negative reciprocal of each other.
Therefore, if one of the lines has a slope of −^{5}⁄_{7}, the slope of the other line must be ^{7}⁄_{5}.
Three vertices of a rectangle in the standard (x,y) coordinate plane have the coordinates (−2,3), (4,3) and (4,2).
What are the coordinates of the fourth vertex?

Solution
Graph the rectangle on a coordinate graph.
As you can see from the graph, the fourth vertex must be (−2, 2).