Calculus Homework
Algebra Practice Problems for Precalculus and Calculus
Solve the following equations for the unknown x :
1. 5 = 7x 16
2. 2x 3 = 5 x
3. 12 (x 3) + x = 17 + 3(4 x )
4. 5x =
2
x 3
Multiply the indicated polynomials and simplify.
5. (4x 1)(3x + 2)
6. (x 1)(x 2 + x + 1)
7. (x + 1)(x 2 x + 1)
8. (x 2)(x + 2)
9. (x 2)(x 2)
10. (x 3 + 2x 1)(x 3 5x 2 + 4)
Find the domain of each of the following functions in 11-15.
11. f (x ) =
1 + x
12. f (x ) = 11+x
13. f (x ) = 1
x
14. f (x ) = 1
1+x
15. f (x ) = 1
1+x 2
16. Given that f (x ) = x 2 3x + 4, find and simplify f (3), f (a), f (t ), and f (x 2 + 1).
Factor the following quadratics
17. x 2 x 20
18. x 2 10x + 21
19. x 2 + 10x + 16
20. x 2 + 8x 105
21. 4x 2 + 11x 3
22. 2x 2 + 7x + 15
23. x 2 2
1
Solve the following quadratic equations in three ways: 1) factor, 2) quadratic formula, 3) complete the square
24. x 2 + 6x 16 = 0
25. x 2 3x 2 = 0
26. 2x 2 + 2x 4 = 0
Solve the following smorgasbord of equations and inequalities
27.
x =
2x 1
28.
x 2 3 =
2x
29. |x 5| = 4
30. 2x + 4 3
31. 2x + 4 3
32. x +4x 3 = 2
33. x 2 x 2 > 0
Add/Subtract the following rational expressions:
34. xx +2 +
3
x 4
35. x
2+1
(x 1)(x 2)
x 3
x 3
Simplify the following rational expressions (if possible):
36. x
2+x 2
x 21
37. x
2+5x +6
x 23x +2
38.
x
x +2 + 3
x +1
x 1
Solutions
1. Given that 5 = 7x 16, add 16 to both sides to get 7x = 21. Now divide both sides by 7 to get x = 3. Checking, we see that
7(3) 16 = 21 16 = 5.
2. Given that 2x 3 = 5 x , add x to both sides and then add 3 to both sides to get 3x = 8. Now divide both sides by 3 to get
x = 8/3 = 2.6. Checking, we see that 2(8/3) 3 = 163 3 =
16
3
9
3 =
7
3 and 5 (8/3) =
15
3
8
3 =
7
3 .
3. Given that 12 (x 3) + x = 17 + 3(4 x ), we first simplify the left and right hand sides using the distributive property to get
1
2 x
3
2 + x = 17 + 12 3x . Combining like terms on both sides gives
3
2 x
3
2 = 29 3x . Now we add 3x and
3
2 to both
sides, obtaining 92 x =
61
2 . Dividing both sides by
9
2 (or multiplying both sides by
2
9 ) gives x =
61
2
2
9 =
61
9 . Checking we see
that L H S = 12 (
61
9
27
9 ) +
61
9 =
1
2
34
9 +
61
9 =
17
9 +
61
9 =
78
9 and R H S = 17 + 3(
36
9
61
9 ) = 17 + 3
(
25
9
)
= 17 759 =
153
9
75
9 =
78
9 .
2
