**a. Multiply**74X910**and explain all your steps.**

Using the fraction rule, abXcd = acbd

Therefore, 74X910 = 7×94×10

= 6340

**Multiply**nn-3X9n+3**and explain all your steps.**

From the fraction rule, nn-3X9n+3 = n×9(n-3)×(n+3)

Simplifying the denominator, (n-3)×(n+3) using the difference of two squares,

(a-b) (a+b) = a2 – b2

a = n, b=3

So, (n-3)×(n+3) = n2 -9

Therefore, n×9(n-3)×(n+3) = 9nn2-9

**Evaluate your answer to part (b) when**n=7**. Did you get the same answer you got in part (a)? Why or why not?**

Substituting n with 7 gives, 9*772-9

= 6349-9

= 6340

Yes, I got the same answer I got in part a.

**Explain how you find the Least Common Denominator of**x2+5x+4 and x2-16

Least Common Denominator is the smallest denominator that is divisible by both x2+5x+4 and x2-16

Factoring x2+5x+4 gives (x+1)(x+4)

Factoring x2-16 by difference of two squares gives (x+4)(x-4)

The next step is to multiply each factor with the highest power

LCD = (x+1)(x+4)(x-4)

**Simplify the expression below and explain all your steps.**

4n2+6n+9 – 1n2-9

The first step is to factor n2 +6n + 9, which gives (n+3)2

The second step is to factor n2-9 to get (n+3)(n-3)

= 4(n+3)2 – 1(n+3)(n-3)

The Least Common Denominator is (n+3)2 (n-3)

Solving the fraction gives

= 4(n-3)n+32(n-3) – n+3n+32(n-3)

Since the denominators are equal, combining the fractions gives,

= 4n-3-(n+3)n+32(n-3)

Expanding the numerator gives 4n-12-n-3 = 3n-15

= 3n-15n+32(n-3)

**A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression. Provide three (3) examples.**

- 5x-28x+2

- 5y-4x-26y+5x+2

- 5+5x-25x-2-2 (Lial et al., 2004)

**Why is there no solution to the equation**3x-2=5x-2

The first step is to multiply both sides by x-2

So, 3x-2 (x-2)=5x-2** (x-2)**

Simplifying gives; 3=5

Since the sides are not equal, the equation has no solution

**Find a printed map and then write and solve an application problem similar to example 8.79 from the book.**

**In your own words, explain the difference between direct variation and inverse variation.**

Direct variation means that a change in one quantity leads to a corresponding change in the other quantity in a direct proportion. If one quantity increases, the other quantity increases in the same proportion. If the same quantity decreases, the other quantity decreases in the same proportion (“Use direct and inverse variation – Elementary algebra,” 2016).)

On the other hand, inverse proportion means that if one quantity increases, it causes the other quantity to decrease in value and vice versa.

**Make up an example from your life experience of inverse variation.**

Once, we were tasked with packing beverage factory products into their numerous delivery trucks every day. Together with other 3 workers, (4 workers in total) we could fill one truck in 45 minutes. When one worker went on leave one day, we, 3 workers, filled one truck in 58 minutes.

This scenario is an example of inverse variation since a decrease in the number of workers led to an increase in the amount of time taken to fill one truck.

## References

Lial, M. L., Hornsby, J., & McGinnis, T. (2004). *Beginning algebra: Margaret L. Lial, John Hornsby, Terry McGinnis*. Boston: Pearson/Addison-Wesley.

Manes, Michelle. “The Key Fraction Rule”. *Pressbooks-Dev.Oer.Hawaii.Edu*, 2021, http://pressbooks-dev.oer.hawaii.edu/math111/chapter/the-key-fraction-rule/.

*Use direct and inverse variation – Elementary algebra*. (2016, March 17). BCcampus Open Publishing – Open Textbooks Adapted and Created by BC Faculty. https://opentextbc.ca/elementaryalgebraopenstax/chapter/use-direct-and-inverse-variation/