Example 1

Replace T(d) with 45Multiply both sides by the denominator

Solve for d

The maximum depth the diver can go without decompression is approximately 70.8 feet.

(A Rational equation: an equation with a variable in the denominator.)

Example 2

11 aa

111111

aaaaaa

Step 1 – get a common denominator for all fractions; factor the denominators and take one of everything:

11 aa

Step 2 – multiply the top of each fraction with what’s not included in the original:

11

12

11

5

11

11 22

aa

aaa

aa

a

aa

aaa

Step 3 – once the denominators are the same, you only have to worry about the numerators: 12511 22 aaaaaaa

Step 4 – simplify and solve: 225

125122323

222

aaaaaaaa

aaaaaa

1 & 3

013

032

2252

2

aa

aa

aa

aaaa Step 5 – check the original equation for problems in the denominator, and throw out any extraneous solutions.

Example 3Partial fraction: one of the fractions that was added or subtracted to result in a given rational expression. Decomposing fractions is a skill used in calculus and other advanced math classes.

11

1

11

1

11

78

111111

yy

yB

yy

yA

yy

y

yyyyyy

Ex. 4Mark the zeros of the equation with vertical lines on a number line.The zeros are the numbers that would cause the numerator to equal zero.The excluded values are the numbers that would cause the denominator to equal zero.

x = 2 & x = 1

x = 3 & x = 4

The number line is split into 5 different intervals. Test any convenient value within each interval to determine if it is a solution. You don’t even need to

complete the calculations; as long as the value is negative, it is less than zero, so it is a solution.

It is a single # line because there is no “y” in the inequality.

Example 4 (cont’d)

Example 5

012

9

12

10

12

8

a

a

aa

0121212

aaa

The excluded value is:

zero

Rewrite:

The numerical common denominator of 3, 6, & 4 is

12

They all need an “a” as well.Change to = to solve.

Test values in each interval.

0

12

33

12

25

12

42

a

a

aa

HW: Page 247