Diana Pell
Section 4.1: Solve Linear Inequalities Using Properties of Inequality
Example 1. Solve each inequality. Graph the solution set and write it
using interval notation.
a) 2x − 9 − 10x ≤ 3 + 4x + 12
b) −7 >
16
15 t
+1
1
c) 7 <
10
9s
+2
d) 32 (x + 2) > 45 (x − 3)
2
Section 4.2: Solving Compound Inequalities
Solve Compound Inequalities Containing the Word And.
Example 2. Solve x + 3 ≤ 2x − 1 and 3x − 2 < 5x − 4. Graph the solution
set and write it using interval notation.
Example 3. Solve x − 1 > −3 and 2x < −8, if possible.
3
Solve Double Linear Inequalities.
Example 4. Solve −3 ≤ 2x + 5 < 7. Graph the solution set and write it
using interval notation.
Note: Solve −15 < −5x ≤ 25.
Solve Compound Inequalities Containing the Word Or.
x 2
Example 5. Solve > or −(x − 2) > 3. Graph the solution set and
3
3
write it using interval notation.
4
x
Example 6. Solve > 2 or −3(x − 2) > 0. Graph the solution set and
2
write it using interval notation.
Example 7. Solve x + 3 ≥ −3 or −x > 0. Graph the solution set and
write it using interval notation.
5
Example 8. Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation.
a)
x x 1
− >
3 4
6
b) 3(x + 23 ) ≤ −7
x 2 3
+ ≤
2 3 4
or
and
2(x + 2) ≥ −2
6
Section 4.3: Solving Absolute Value Equations and Inequalities
Example 9. Solve |x| = 3
Example 10. Solve.
a) |x| =
1
2
b) |3x − 2| = 5
c) |10 − x| = −40
7
d) |2x − 3| = 7
2
e) + 3 + 4 = 10
3
1
f) 3 x − 5 − 4 = −4
2
8
2
g) −5 x + 4 + 1 = 1
3
Example 11. Let f (x) = |x + 4|. For what value(s) of x is f (x) = 20?
9
Example 12. Solve: |5x + 3| = |3x + 25|
Solve Inequalities of the Form |x| < k
For any positive number k and any algebraic expression X:
To solve |X| < k, solve the equivalent double inequality −k < X < k.
To solve |X| ≤ k, solve the equivalent double inequality −k ≤ X ≤ k.
Example 13. Solve |x| < 5 and graph the solution set.
10
Example 14. Solve |2x − 3| < 9 and graph the solution set.
Example 15. Solve |4x − 5| < −2 and graph the solution set.
Solve Inequalities of the Form |x| > k
For any positive number k and any algebraic expression X:
To solve |X| > k, solve the equivalent compound inequality X > k or
X < −k.
To solve |X| ≥ k, solve the equivalent compound inequality X ≥ k or
X ≤ −k.
11
Example 16. Solve |x| > 5 and graph the solution set.
3 − x
≥ 6 and graph the solution set.
Example 17. Solve 5 12
2 − x
≥ 1 and graph the solution set.
Example 18. Solve 4 2
Example 19. Solve 6 < x − 2 − 3 and graph the solution set.
3
13
3
Example 20. Solve 3 < x + 2 − 1 and graph the solution set.
4
x
Example 21. Solve x − 1 ≥ −4 and graph the solution set.
8
14
Section 4.4: Linear Inequalities in Two Variables
Exercise 1. Graph each inequality.
a) y > 3x + 2
15
b) (You Try!) y > 2x − 4.
c) 2x − 3y ≤ 6
16
d) (You Try!) 3x − 2y ≥ 12
e) y < 2x
17
Section 4.5: Systems of Linear Inequalities
Exercise 2. Graph the solution set of each system of inequalities on a
rectangular coordinate system.
a)
y ≤ −x + 1
2x − y > 2
18
b)

x≥1
y≥x

4x + 5y < 20
19
Exercise 3. A homeowner has a budget of $300 to $600 for trees and
bushes to landscape his yard. After shopping, he finds that trees cost
$150 each and bushes cost $75 each. What combination of trees and
bushes can he afford to buy?
Let x = the number of trees purchased and
y = the number of bushes purchased.
20
Section 5.1: Exponents
Properties of Exponents
Let a, b ∈ R and r, s ∈ Z
1) ar as = ar+s
x11 x5
2) (ar )s = ar·s
(x11 )5
3) (ab)r = ar · br
(xy)3
4) a−r =
1
provided that a 6= 0 and r ∈ Z+
r
a
2−3
5)
a r
b
ar
= r,
b
b 6= 0
x 2
3
6)
ar
= ar−s
s
a
x5
x3
7) a1 = a and a0 = 1 (a 6= 0)
21
−n x
y n
8)
=
.
y
x
−4
2
a)
3
y2
x3
a−2 b3
a2 a3 b4
b)
c)
−3
−3
22
d)
2x2
3y −3
−4
Exercise 4. Evaluate each of the following.
a) (−4)2
b) −42
c) −(−4)2
3
1
d)
2
Exercise 5. Use the properties of exponents to simplify each of the following as much as possible.
a) x5 · x4
b) (23 )2
c)
2
− x2
3
3
d) −3a2 (2a4 )
23
Exercise 6. Write each of the following with positive exponents. Then
simplify as much as possible.
a) 3−2
b) (−2)−5
−2
3
c)
4
−2 −3
1
1
d)
+
3
2
Exercise 7. Simplify each expression. Write all answers with positive
exponents only.
a) x−4 x7
b) (a4 b−3 )3
c) (3y 5 )−2 (2y −4 )3
d)
1 −3
x
7
7 −5
x
8
8 8
x
9
24
e) (4x−4 y 9 )−2 (5x4 y −3 )2
Exercise 8. Simplify each expression. Write all answers with positive
exponents only.
a)
a5
a−2
b)
t−8
t−5
c)
x7
x4
5
(x−4 )3 (x3 )−4
d)
x10
(6x−3 y −5 )2
e)
(3x−4 y −3 )4
f)
x−8 y −3
x−5 y 6
−1
25
Section 5.3: Polynomials and Polynomial Functions
Definition 22. A term, or monomial, is a constant or the product of
a constant and one or more variables raised to whole-number exponent.
Exercise 9. The following are monomials (or terms):
2
− ab2 c
2x
3
Definition 23. A polynomial is any finite sum of terms.
−14
3x2 y
Exercise 10. The following are polynomials:
2x2 + 6x − 3
− 5x2 y + 2xy
4a − 5b + 6c
Definition 24. The degree of a polynomial with one variable is the
highest power to which the variable is raised in any one term.
Addition and Subtraction of Polynomials
Exercise 11. Add:
3 4 7 3
1 4 1 3
m + m +
m − m .
4
2
4
3
Exercise 12. Subtract 4x2 − 9x + 1 from −3x2 + 5x − 2.
26
Section 5.4: Multiplying Polynomials
Exercise 13. Multiply:
1. (3x2 )(6x3 )
2. −2ab(3a3 b − 2a2 b + 4b2 )
3. (3x + 2)(4x + 9)
4. (2a + b)(3a2 − 4ab − b2 )
Exercise 14. Multiply: 5cd(c + 6d)(3c − 8d)
27
Special Products
(x + y)2 = x2 + 2xy + y 2
(x − y)2 = x2 − 2xy + y 2
Exercise 15. Multiply.
a) (5c + 3d)2
b)
1 4
a − b2
2
2
c) [(5x + y) + 4]2
28
Exercise 16. If f (x) = x2 + 9x − 5, find f (a + 4).
Exercise 17. If f (x) = x2 − 6x + 1, find f (a − 8).
Exercise 18. Simplify: (5x − 4)2 − (x − 7)(x + 1)
29
Section 5.5: The Greatest Common Factor and Factoring by
Grouping
The greatest common factor is the largest factor that is common to
all terms of the expression.
Example 25. Find the GCF of 6a2 b3 c, 9a3 b2 c, and 18a4 c3 .
Example 26. Factor.
a) 16y 2 + 24y
b) 3xy 2 z 3 + 6xyz 3 + 3xz 2
Example 27. Factor out −1 from −n3 + 2n2 − 8
30
Example 28. Factor.
a) x(x + 1) + y(x + 1)
b) a(x − y + z) − b(x − y + z) + 3(x − y + z)
c) 2m − 2n + mn − n2
d) 7r − 7s + rs − s2
31
e) y 3 + 3y 2 + y + 3
f) x2 − bx − x + b
g) 5x3 − 8 + 10x2 − 4x
32
h) 3x3 y − 4x2 y 2 − 6x2 y + 8xy 2
Section 5.6:Factoring Trinomials
Multiply: (x + 8)(x − 6)
Exercise 19. Factor each trinomial, if possible.
a) n2 + 20n + 100
b) x2 + 10x + 24
33
c) x2 + 11x + 24
d) 5x2 + 7x + 2
e) −8t2 + t4 + 12
f) d4 + 12d2 + 27
g) 3p2 − 4p − 4
34
h) 2q 2 − 17q − 9
i) 2x2 y 2 + 4xy 3 − 30y 4
j) 3a2 b2 + 6ab3 − 105b4
k) 7t2 − 15t + 11
35
l) −15x2 + 25xy + 60y 2
m) −6x2 − 57xy − 72y 2
n) 6y 3 + 13x2 y 3 + 6x4 y 3
Section 5.7: The Difference of Two Squares; the Sum and Difference of Two Cubes
Difference of Squares
x2 − y 2 = (x − y)(x + y)
Exercise 20. Factor each expression
a) x2 − 16
b) 25x2 − 36
36
c) 100w4 − 9z 4
d) 75x2 − 3
e) x4 − 1
f) a4 − 81
g) (x + y)4 − z 4
h) 2x4 y − 32y
Factor the Sum and Difference of Two Cubes
x3 + y 3 = (x + y)(x2 − xy + y 2 )
x3 − y 3 = (x − y)(x2 + xy + y 2 )
Exercise 21. Factor each expression
a) a3 + 8
37
b) p3 + 27
c) 27a3 − 64b6
d) a3 − (c + d)3
e) (p + q)3 − r3
f) x6 − 64
Exercise 22. Factor each expression completely.
a) 60q 2 r2 s4 + 78qr2 s4 − 18r2 s4
38
b) ax2 − 2axy + ay 2 − x2 + 2xy − y 2
c)
81 4
x − y 40
16
d) 8(4 − a2 ) − x3 (4 − a2 )
e) (3z + 2)2 − 12(3z + 2) + 36
39
Section 5.9: Solving Equations by Factoring
Exercise 23. Solve each equation.
a) 2y(4y + 3) = 9
b)
x2
9
= 98 x −
7
9
c) b3 − 5b2 − 9b + 45 = 0
40
d)
x2 (6x+37)
35
=x
41