# 15 Basic Algebra Problems

1. Reduce if possible: (p^2 + 5p + 6)/(p^2 + 2p - 3)

2. Multiply: (x^2 - 3x + 2)/(x^2 - 4x + 4) times (x^2 + x - 6)/(x^2 - 1)

3. For what values of p is #1 undefined?

4. For what values of x is #2 undefined?

5. Find the LCD of the following: 1/(x + 2) and 1/(x^2 + 2x).

6. Find the LCD of the following: 1/(x^2 + x - 2) and 1/(x^2 - 3x + 2).

7. Add: 1/x + 1/x^2 + 1/x^3.

8. Do the math: 1/(a^5 * b) - 5/(ab^3)

9. Solve for x: 1/x + 1/(x-3) = (x-2)/(x-3)

10. Solve for t : (t+4)/2 = (t+4)/t

11. If Joe can paint the fence in 4.2 hrs and Bob can do it in 5.6 hrs, how long would it take if they work together?

12. A pool has a pipe that can fill it in 3.2 hrs, a smaller one that can do it in 6.5 hrs and a little bitty one that takes 10.2 hrs. How long will it take to fill the pool if all three work together?

13. You have 22 gals of 55% solution. How much pure solution do you need to add to get some 80% solution?

14. You want to end up with 12 gals of 30% solution. How much 20% solution and how much 70% solution should you mix to get the 12 gals of 30%?

15. You have 100 gals of Ooh-La-La pink paint that is 20% red and 80% white. You want some Cutie-Pie pink paint that is 25% red/75% white. How much Sissy pink (which is 40% red/60% white) should you mix with the Ooh-La-La to get the Cutie-Pie? (round to nearest hundredth)

#### Solution Preview

1. Reduce if possible: (p^2 + 5p + 6)/(p^2 + 2p - 3)

the numerator can be factored into

(p^2 + 5p + 6) = (2 + p) (3 + p)

the denominator can then be factored into

(p^2 + 2p - 3) = (p - 1) (3 + p)

(2 + p) (3 + p) / ( (p - 1) (3 + p) ) = (2 + p) / ( p - 1)

2. Multiply: (x^2 - 3x + 2)/(x^2 - 4x + 4) times (x^2 + x - 6)/(x^2 - 1)

Simplify: (x^2 - 3x + 2) / (x^2 - 4x + 4) = (x - 1) / (x - 2)

Numerator: (x^2 - 3x + 2) = (x - 2) (x - 1)

Denominator: / (x^2 - 4x + 4) = (x - 2 )^2

Factor the numerator of : (x^2 + x - 6) / (x^2 - 1)

(x^2 + x - 6) = (x - 2) (x + 3)

Your problem is now much simpler:

((x - 1) / (x - 2) ) * ( (x - 2) (3 + x) / (x^2 - 1) ) (cancels out)

= (x - 1) (x + 3) / (x^2 - 1 ) = (x - 1) (x + 3) / ((x - 1 ) (x + 1)

= (x + 3) / (x + 1)

3. For what values of p is #1 undefined?

(2 + p) / ( p - 1)

An expression is undefined when it does not have meaning.

The function is not defined when the denominator is 0, p = 1

4. For what values of x is #2 undefined?

(x + 3) / (x + 1)

The function is not defined when the denominator is 0, x = -1

5. Find the LCD of the following: 1/(x + 2) and 1/(x^2 + 2x).

LCD = lowest common denominator

Factor (x^2 + 2x) = x ( x + 2)

LCD: x ...

#### Solution Summary

Basic algebra problems are solved.