**6 Complex Fractions**

The first step in simplifying complex fractions is to write the numerator as one
fraction and

the denominator as one fraction.

Example:

Example:

Here are some to try. Answers are in the solution section.

**7 Functions**

For the function defined by f(x) = 3x - x^{2}, we express in English that the
function takes a

number, multiplies it by 3 and subtracts its square.

For the function defined by g(x) = 1 - 3x^{2}, we express in English that the
function takes a

number, squares it, multiplies the result by 3 and subtracts that result from 1.

Here are some for you to try. The answers are in the solution section.

1. If f(x) = 2x - x^{2}, find f(x + h) and simplify completely.

2. If find f(x + h) and simplify completely.

3. If f(x) = 3 - 2x^{2}, find f(x + h) and simplify completely.

**8 Linear Equations**

Rewrite the equation so that there are no variables in the denominator and no
parentheses.

Move all terms with the variable for which we are solving on one side of the
equation and

move all terms without that variable on the other side. Factor out the variable
and solve.

Example: Solve 3(x + 2) = 4x + 1 for x: We distribute to get 3x + 6 = 4x + 1.
Isolating the

x's on one side, we get 3x - 4x = 1 - 6 or -x = -5. So the answer is x = 5.

Example: Solve
for y: Multiply both sides by 2x to get y = 2x(y + 1) or

y = 2xy + 2x. Isolate the terms with y to get y - 2xy = 2x or y(1- 2x) = 2x.
The solution

is

Try the following problems. The answers are in the solution section.

1. Solve

2. Solve A(B + C) = BC + A for C.

**9 Quadratic Equations**

We can solve quadratic equations by factoring or by using the quadratic equation.

Example: Solve x^{2}-x = 6 by factoring: We put all terms on one side, getting
x^{2}-x-6 = 0.

Next, factor to get (x - 3)(x + 2) = 0. If a product ab equals 0, either a = 0
or b = 0. For

our problem, we conclude x - 3 = 0 or x + 2 = 0. The answer is x = 3 or x = -2.

Notice that, if ab = 6, we cannot conclude a = 6 or b = 6. Maybe a = 2 and b = 3
or a =1/2

and b = 12. So, if you factored without moving all terms to one side to get x(x
- 1) = 6,

you cannot conclude x = 6 or x - 1 = 6.

Example: Solve x^{2} -x = 6 by using the quadratic formula: Again, we need to put
all terms

on one side, getting x^{2} - x - 6 = 0. To solve ax^{2} + bx + c = 0, the quadratic
formula tells

us that

In this example, a = 1, b = -1, and c = -6, so

.
The solution is

Here are some problems to try. The answers are in the solution section.

1. Solve the equation for x: x^{2} - 2x = 8

2. Solve the equation for x:

**10 Common Mistakes**

Remember that For example,
since

Similarly, However, if x and y are nonnegative, and

In a similar vein, since

. However,
since

Also,

Care needs to be taken when deciding if canceling is possible. We have that
but

Recall that, for inequalities, if we multiply or divide both sides by a negative
number, the

inequality sign gets changed. We know that -3 ≤ 2 but, if we multiply by -2, we
get

(-3)(-2) ≥ (2)(-2) since 6 ≥ -4.

Example: Solve 3 - 2x ≤ 9 for x: We get -2x ≤ 9 - 3 or -2x
≤ 6 or x ≥ -3. The
answer

in interval notation is [-3;∞).

Example: Solve x + 2 > 4x - 1 for x: We get x - 4x >
-1 - 2 or -3x > -3 or
x < 1. The

answer in interval notation is (-∞; 1).

Decide if the following statements are true or false. Assume that all variable
are nonnegative.

Answers are in the solution section.

**11 Solutions**

Section 1:

Section 2:

1.
and the equation of the line is y
- (-2) = -1(x - 4).

2. The points are (3, 0) and (0,-2). So
and the equation of the

line is

3. To find the slope of the line 2x + 6y = 1, we solve for y: 6y = 1
- 2x so

The slope is
and the equation of the parallel line is

Section 3:

Section 4:

x

Section 5:

Section 6:

Section 7:

Section 8:

1. Multiplying the equation
by ABC on both sides, we get BC +AC = AB.

We isolate the terms with B on the left to get BC-AB = -AC. So B(C-A) = -AC

and

2. Distributing, we get AB +AC = BC +A. We isolate the terms with C on the left
and

get AC - BC = A - AB. So C(A - B) = A - AB and

Section 9:

1. x^{2} - 2x - 8 = 0. Factoring, we get (x - 4)(x + 2) = 0 so x
- 4 = 0 or x +
2 = 0. The

answers are x = 4,-2.

2. Multiplying both sides of the equation by x, we get 2+x = x^{2} +2x. So 0 = x^{2}
+ x - 2

and 0 = (x + 2)(x - 1). The answer is x = -2, 1.

Section 10:

1. False:

2. True

3. False

4. False: (2 + 3)^{3} = 5^{3} = 125

5. False: If -2x < 4 then x > -2.

6. True

7. True

8. False

9. True

10. False

11. False: (3x)^{2} = 9x^{2}

12. True

13. True