Rationals Review

Rationals Unit Test Review

Topics on the Test:

• Simplify, Multiply & Divide Rationals

• Add/Subtract Rationals (identify restrictions)

• Solve Rational Equations

• Rational Graph Characterisitics: Vertical and Horizontal Asymptotes, x and y intercepts, and Domain/Range

Simplify Rationals

Factor and cancel like factors from 'top' and 'bottom'---never cancel factors on the same 'level'

Multiple Choice

Simplify and state excluded values (that make the denom. = 0)

$$\frac{7m+42}{m^2+3m-18}$$  

• $$\frac{m+6}{m-3},\ m\ne-6,\ 3$$  

• $$\frac{7}{m-3},\ m\ne-6,\ 3$$  

• $$7\left(m+6\right),\ m\ne-6,\ 3$$  

• $$m-3,\ m\ne-6,\ 3$$  

Multiply Rationals

Factor each numerator and denominator then simplify.

Again, cancel only like factors from 'top' and 'bottom'!

Multiple Choice

Multiply by factoring and canceling like factors:

$$\frac{x^2+3x-18}{3x+18}\times\frac{x^2+x-90}{x^2+7x-30}$$  

• $$3x^2\left(x-9\right)$$  

• $$\frac{x-9}{3}$$  

• $$\frac{x+7}{60}$$  

• $$\left(x-3\right)\left(x+10\right)$$  

Divide Rationals

Flip the 2nd fraction, then factor and simplify as done with multiplication.

Multiple Choice

Divide: Flip the 2nd fraction, then factor and cancel just like multiplication.

$$\frac{x^2-4x-5}{x^2-3x+2}\div\frac{x^2-3x-10}{x^2-4}$$  

• $$\frac{x-5}{x-1}$$  

• $$\frac{x+1}{x-1}$$  

• $$\frac{1}{x-1}$$  

• $$x+1$$  

Add/Subtract Rat'ls.

Get like denominators, then combine into one fraction. Here's how =====>

Example of Adding Rationals with unlike denominators

Multiple Choice

What is the Least common denominator (LCD)?

• (n+1)(n-1)

• 2(n+1)

• 2(n-1)

• 2(n+1)(n-1)

Multiple Choice

Using the LCD of 2(n-1)(n+1), simplify into one fraction:

(multiply each numerator by the part of the LCD in the denominator that's missing)

Solve Rational Equations

Multiply each term by the LCD to get rid of the fractions, then solve.

Remember to check your solutions for restricted x-values! (Can't divide by 0!)  In the example,  $x\ne0,\ 8$  So the solutions of  $x=12,\ 2$  work!

Multiple Choice

Which x-value(s) must be restricted from the solution?

• -1, 5

• 0

• -3

• 1

Multiple Choice

Solve. Don't forget to check for extraneous/restricted x-values!

• x = -1, 5

• No Solution

• x = -4, 5

• x= -4

Multiple Choice

Solve.

• x = 4.5

• x = -4.5

• x = 3

• x = -3

Multiple Choice

Solve for r

• 5
• - 5, 4
• - 5
• 5, 4

Finding x & y intercepts

Graph and look, if you're able!

OR

From an equation:

y intercepts: make all x's =0

x intercepts: set numerator = 0

Multiple Choice

What is the y-intercept?

• (0,1)
• (0,1/9)
• (0,-1/9)
• (0,-1)

Multiple Choice

What is/are the x-intercepts?

• (-2, 0) (2, 0)

• (0, -2) (0, 2)

• (2, 0) (3, 0)

• (0, 3)

Vertical Asymptotes

Set Denominator = 0
Always begins with  $x=$  

Notice: this is the same as the restricted values!

Again, a graph might help!

Multiple Choice

$$y=\frac{x-5}{x^2+4x-12}$$  Find the vertical asymptote(s).

(factor denominator and set each factor = 0!)

• x = 3 and x = -4

• x=-6 and x = 2

• x = 5

• x = 12

Multiple Choice

Find the Vertical Asymptotes.

• x= 0, -2

• x= 2

• x= -3, 2

• x = -3,1

Horizontal Asymptotes

Compare the DEGREE of the numerator and denominator.

Bigger on Bottom--> y =0

Bigger on Top --> No HA

Same degree --> divide 1st term on top/1st term on bottom

On a graph, look for the horizontal line that runs 'between' the curves.

Multiple Choice

Find the horizontal asymptote.

(Compare the degree of num and denom. to determine)

• y = 0

• y = 1/2

• y = 2

• None

Multiple Choice

Find the horizontal asymptote.

• None
• y=-2
• y=2
• y=0

Multiple Choice

Find the horizontal asymptote.

• y=4

• y=0

• y=-2

• y=1

Domain & Range

Skip over the asymptotes values:

Domain: All Reals except the VA
Range: Generally, All Reals except the HA

Ex. In the graph shown:
Domain: All reals except  $x\ne-3$  

Range:  All reals except $x\ne-5$  

Multiple Choice

What is the domain and range?

(A.R.N. stands for 'All Real Numbers')

Hint: use the asymptotes to help--'skip' over those values

• Domain: A.R.N. except x= -3

  Range: A.R.N. except y= 1

• Domain: A.R.N. except x= 3

  Range: A.R.N. except y= -1

• Domain: A.R.N. except x= 1

  Range: A.R.N. except y= -3

• Domain: A.R.N. except x= -3

  Range: A.R.N. except y= -1

• Domain: A.R.N. except x= -1

  Range: A.R.N. except y= 3