Manipulating Algebraic Expressions

See attached file for full problem description.

1. Write each intervals in absolute value notation.

a) xE(0,9)

The midpoint of the interval (0+9)/2. Since 9-4.5=4.5, each point in the interval is 4.5 units or less from the number 4.5. Thus the interval can be described as:
I x-4.5I less than or equal to 4.5.

b) xE(-2,20)

The midpoint of the interval (-2+20)/2. Since 20-9=11, each point in the interval is 11 units or less from the number 9. Thus the interval can be described as:
I x-9I less than or equal to 11.

2. Write a description of each of the following intervals.

a) (-4,4)
This set is known as -4 less than or equal to x less than or equal to 4

b) (0,2)
This set is known as 0 less than or equal to x less than or equal to 2

c) (-beta,10)
This set is known as -beta less than or equal to x less than or equal to 10

3. Solve the following inequalities:

a)x(x-4) greater than or equal to 0

The zeros of the function are 4 and 0. Note that the zeros are included in the solutions because we are looking for values greater than or equal to 0.

Based on the zeros of the function, the three test intervals are:

(-beta,4),(4,0),(0,beta)

Next we choose a test value in each interval. The following table will summarize the calculations:

Interval (-beta,4) (4,0) (0,-beta)
Test Value
x
(x-4)
x(x-4)

c) x3-4x2+x+6< or equal to 0

First we must factor the polynomial. Using the Factor Theoerm and factors of the constant???????

4. Solve the following inequality. Express your solution in interval notation.

x3-10x+3>or equal to 0
???