1. Listed below are several properties of real numbers. Give an example of each of the properties.

i. Commutative Property of Addition

a + b = b + a i______________________

…

2. Solve.

1. = 12x=_________
2. 

= 0 x=_________

1. Recall, |a| = 7 means that a is either 7 or it is –7. Likewise, |a + 5 | = 7 means (a + 5) is equal to 7 or (a + 5) is equal to –7. This implies that a = 2 or a = -12. Solve for x:

1. |x + 3| = 8x=_________

1. Determine whether the points given are solutions to the following equations, (i.e., does the point given lie on the graph of the equation?).

1. y = 2x – 9(1, 2)Yes / No

1. Find the length of the segment joining the given points, i.e. find the distance, d, between the following points. Then find the midpoint, M, of the segment. Recall, d=and M=

.

1. (1, 2) & (4, 5).

1. The point-slope form of a linear equation is y – y₁ = m(x –x₁). Using the point-slope form, write an equation of the line that passes through the given point P(x₁, y₁) and has the given slope, m. Simplify the equation and rewrite it in slope-intercept form, y = mx + b.

1. P(1, 2) , m=½

1. Using the point-slope form, write an equation of the line that passes through the given points (x₁, y₁) & (x₂, y₂). Simplify.

1. (2, 3) & (4, 6) ______________

Inequalities with two variables.

1. Graph the following inequalities.

1. 3x + 2y ³ 12

1. Given an equation of a parabola of the form, f(x) = y = a(x – h)² + k, we have that (h, k) is the vertex …

Give the vertex of the following parabolas.

1. y = x² – 6x – 1 (_____,_____)

1. Graph the following parabolas.

1. y = x² – 6x + 12

1. Solve for the indicated variable.

1. ax + by = c for xx=______________
2. C = 2p r for rr=______________
3. y = mx + b for bb=______________
4. V = 2p r² + 2p rhfor hh=______________

 Exponents & Radicals

Some properties of exponents and radicals.

i. aⁿ × a^m = a^n+m …

4. =

Example #4:

Simplify.

====.

Example #5:

====

1. Simplify.

1. =______________
2. =______________
3. 

= ______________

Rationalizing

When working with radicals, we never leave a radical in the denominator of a fraction. …

1. Solve.

1. = 5x = _____________
2. - 4 = 5x = _____________
3. 5x – 30

+ 45 = 0 x = _____________

Polynomials

Perfect Square Trinomials (a + b)² = a² + 2ab + b²

Difference of Squares a² – b² = (a + b)(a – b) …

1. Add or Subtract the following polynomials by combining like terms.

1. (x³ + 3x² – 4x + 4) + (4x³ – 9x² – 5x + 3)

1. Multiply the following polynomials.

1. (3x³ – x² + 2x – 4)(x² – 5x + 3) ____________________________________

1. Expand (3x + 7y)².____________________________________
2. Factor completely:

1. 2x² – x – 6 ____________________________________

1. Factor completely. (Hint: for some try using special formulas.)

1. 49x² – 25y²____________________________________
2. (m² + 18m + 81) – (n² + 4n + 4)____________________________________

1. Simplify.

1. –
1. _________________

 Solving Rational Equations.

When solving rational equations, one must check answers for extraneous solutions. In other words, one must check to make sure the answers obtained are in the domain of the rational expressions (the denominator of the rational expressions cannot equal zero).

1. Solve for x.

1. =
1. x =______

1. Seven times a number is the same as twelve more than three times the number. Find the number.
2. Find two consecutive integers whose sum is –29.
3. The sum of the reciprocals of two consecutive integers is .
4. Jonathan hiked up 10.5km on a trail on Longs Peak. It took him ½ hour less on the return trip since when coming down the mountain, he increased his speed by ½km/h. How long did it take him to hike up and down the mountain trail? (Hint: do not convert hrs to min. as units of time)
5. y varies directly as x and y = 48 when x = 3. What is y when x = 12?
6. If a chemist starts with 15cm³ of a 20% saline solution, how many cubic centimeters of water must she add to obtain a 10% saline solution?

1. Find the area of square A._____________________
2. Find the area of rectangle C & D._____________________

i = i² = ()² = -1 i³ = ()³ = ()²() = -1()= -i

Zero Product Property

ab = 0 means a = 0 or b = 0.

I.e. if you have two numbers whose product is zero, one of them has to equal 0.

So, if given (x – 3)(x – 2) = 0, we have that a = x – 3 and b = x – 2 and either a or b equaling zero would make the product zero. Therefore, either

a = x – 3 = 0 or b = x – 2 = 0 which implies that either

x = 3 or x = 2.

This only works if the product ab = 0! If ab = 4, there are several numbers that can be multiplied to get 4 (2 and 2, 1 and 4, ½ and 8 to name a few). Therefore, it is vital that one does not make the following mistake:

x² – 4x = 4

x(x – 4) = 4 means

x = 4 or x – 4 = 4 WRONG, WRONG, WRONG!

...

Factoring using the "dream method"

Try it!

Factor 6x² + 11x – 10 using both the trial and error method and the dream method.

Working with matrices.

The dimensions of a matrix are always given as "# rows x # columns." For, example, the matrix is a 3 x 2 matrix, while is a 2 x 3 matrix. …

In order to well understand the other geometries, it is best to first grasp a strong understanding of Euclidean geometry.

1. Three points are collinear.__________________
2. Three points determine a plane.__________________
3. Three non-collinear points determine a plane.__________________
4. If two coplanar points determine a line, the line lies in the plane.__________________
5. If one point is in a plane, a line passing through that point is also in the plane._________________
6. Given a point and a line, one can draw a line through the point parallel to the line._____________