section 4.1: solve linear inequalities using properties of...

Diana Pell

Section 4.1: Solve Linear Inequalities Using Properties of In-equality

Example 1. Solve each inequality. Graph the solution set and write itusing interval notation.

a) 2x− 9− 10x ≤ 3 + 4x + 12

b) −7 > 1615t + 1

1

c) 7 < 109 s + 2

d) 23(x + 2) > 4

5(x− 3)

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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 solutionset and write it using interval notation.

Example 3. Solve x− 1 > −3 and 2x < −8, if possible.

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Solve Double Linear Inequalities.

Example 4. Solve −3 ≤ 2x + 5 < 7. Graph the solution set and write itusing interval notation.

Note: Solve −15 < −5x ≤ 25.

Solve Compound Inequalities Containing the Word Or.

Example 5. Solvex

3>

2

3or −(x − 2) > 3. Graph the solution set and

write it using interval notation.

4

Example 6. Solvex

2> 2 or −3(x − 2) > 0. Graph the solution set and

write it using interval notation.

Example 7. Solve x + 3 ≥ −3 or −x > 0. Graph the solution set andwrite it using interval notation.

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Example 8. Solve each compound inequality, if possible. Graph the solu-tion set (if one exists) and write it using interval notation.

a)x

3− x

4>

1

6or

x

2+

2

3≤ 3

4

b) 3(x + 23) ≤ −7 and 2(x + 2) ≥ −2

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Section 4.3: Solving Absolute Value Equations and Inequalities

Example 9. Solve |x| = 3

Example 10. Solve.

a) |x| = 12

b) |3x− 2| = 5

c) |10− x| = −40

7

d) |2x− 3| = 7

e)

∣∣∣∣23 + 3

∣∣∣∣+ 4 = 10

f) 3

∣∣∣∣12x− 5

∣∣∣∣− 4 = −4

8

g) −5

∣∣∣∣23x + 4

∣∣∣∣+ 1 = 1

Example 11. Let f(x) = |x + 4|. For what value(s) of x is f(x) = 20?

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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.

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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 orX < −k.

To solve |X| ≥ k, solve the equivalent compound inequality X ≥ k orX ≤ −k.

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Example 16. Solve |x| > 5 and graph the solution set.

Example 17. Solve

∣∣∣∣3− x

5

∣∣∣∣ ≥ 6 and graph the solution set.

12

Example 18. Solve

∣∣∣∣2− x

4

∣∣∣∣ ≥ 1 and graph the solution set.

Example 19. Solve 6 <

∣∣∣∣23x− 2

∣∣∣∣− 3 and graph the solution set.

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Example 20. Solve 3 <

∣∣∣∣34x + 2

∣∣∣∣− 1 and graph the solution set.

Example 21. Solve∣∣∣x8x− 1

∣∣∣ ≥ −4 and graph the solution set.

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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

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Section 4.5: Systems of Linear Inequalities

Exercise 2. Graph the solution set of each system of inequalities on arectangular coordinate system.

a) {y ≤ −x + 12x− y > 2

18

b) x ≥ 1y ≥ x4x + 5y < 20

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Exercise 3. A homeowner has a budget of $300 to $600 for trees andbushes to landscape his yard. After shopping, he finds that trees cost$150 each and bushes cost $75 each. What combination of trees andbushes can he afford to buy?

Let x = the number of trees purchased and

y = the number of bushes purchased.

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Section 5.1: Exponents

Properties of Exponents

Let a, b ∈ R and r, s ∈ Z

1) aras = ar+s

x11x5

2) (ar)s = ar·s

(x11)5

3) (ab)r = ar · br

(xy)3

4) a−r =1

arprovided that a 6= 0 and r ∈ Z+

2−3

5)(ab

)r=

ar

br, b 6= 0

(x3

)26)

ar

as= ar−s

x5

x3

7) a1 = a and a0 = 1 (a 6= 0)

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8)

(x

y

)−n=(yx

)n.

a)

(2

3

)−4

b)

(y2

x3

)−3

c)

(a−2b3

a2a3b4

)−3

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d)

(2x2

3y−3

)−4

Exercise 4. Evaluate each of the following.

a) (−4)2

b) −42

c) −(−4)2

d)

(1

2

)3

Exercise 5. Use the properties of exponents to simplify each of the fol-lowing as much as possible.

a) x5 · x4

b) (23)2

c)

(−2

3x2)3

d) −3a2(2a4)

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Exercise 6. Write each of the following with positive exponents. Thensimplify as much as possible.

a) 3−2

b) (−2)−5

c)

(3

4

)−2

d)

(1

3

)−2+

(1

2

)−3

Exercise 7. Simplify each expression. Write all answers with positiveexponents only.

a) x−4x7

b) (a4b−3)3

c) (3y5)−2(2y−4)3

d)

(1

7x−3)(

7

8x−5)(

8

9x8)

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e) (4x−4y9)−2(5x4y−3)2

Exercise 8. Simplify each expression. Write all answers with positiveexponents only.

a)a5

a−2

b)t−8

t−5

c)

(x7

x4

)5

d)(x−4)3(x3)−4

x10

e)(6x−3y−5)2

(3x−4y−3)4

f)

(x−8y−3

x−5y6

)−1

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Section 5.3: Polynomials and Polynomial Functions

Definition 22. A term, or monomial, is a constant or the product ofa constant and one or more variables raised to whole-number exponent.

Exercise 9. The following are monomials (or terms):

−14 3x2y − 2

3ab2c 2x

Definition 23. A polynomial is any finite sum of terms.

Exercise 10. The following are polynomials:

2x2 + 6x− 3 − 5x2y + 2xy 4a− 5b + 6c

Definition 24. The degree of a polynomial with one variable is thehighest power to which the variable is raised in any one term.

Addition and Subtraction of Polynomials

Exercise 11. Add:

(1

4m4 +

1

2m3

)+

(3

4m4 − 7

3m3

).

Exercise 12. Subtract 4x2 − 9x + 1 from −3x2 + 5x− 2.

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Section 5.4: Multiplying Polynomials

Exercise 13. Multiply:

1. (3x2)(6x3)

2. −2ab(3a3b− 2a2b + 4b2)

3. (3x + 2)(4x + 9)

4. (2a + b)(3a2 − 4ab− b2)

Exercise 14. Multiply: 5cd(c + 6d)(3c− 8d)

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Special Products

(x + y)2 = x2 + 2xy + y2

(x− y)2 = x2 − 2xy + y2

Exercise 15. Multiply.

a) (5c + 3d)2

b)

(1

2a4 − b2

)2

c) [(5x + y) + 4]2

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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)

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Section 5.5: The Greatest Common Factor and Factoring byGrouping

The greatest common factor is the largest factor that is common toall terms of the expression.

Example 25. Find the GCF of 6a2b3c, 9a3b2c, and 18a4c3.

Example 26. Factor.

a) 16y2 + 24y

b) 3xy2z3 + 6xyz3 + 3xz2

Example 27. Factor out −1 from −n3 + 2n2 − 8

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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

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e) y3 + 3y2 + y + 3

f) x2 − bx− x + b

g) 5x3 − 8 + 10x2 − 4x

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h) 3x3y − 4x2y2 − 6x2y + 8xy2

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

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c) x2 + 11x + 24

d) 5x2 + 7x + 2

e) −8t2 + t4 + 12

f) d4 + 12d2 + 27

g) 3p2 − 4p− 4

34

h) 2q2 − 17q − 9

i) 2x2y2 + 4xy3 − 30y4

j) 3a2b2 + 6ab3 − 105b4

k) 7t2 − 15t + 11

35

l) −15x2 + 25xy + 60y2

m) −6x2 − 57xy − 72y2

n) 6y3 + 13x2y3 + 6x4y3

Section 5.7: The Difference of Two Squares; the Sum and Dif-ference of Two Cubes

Difference of Squares

x2 − y2 = (x− y)(x + y)

Exercise 20. Factor each expression

a) x2 − 16

b) 25x2 − 36

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c) 100w4 − 9z4

d) 75x2 − 3

e) x4 − 1

f) a4 − 81

g) (x + y)4 − z4

h) 2x4y − 32y

Factor the Sum and Difference of Two Cubes

x3 + y3 = (x + y)(x2 − xy + y2)

x3 − y3 = (x− y)(x2 + xy + y2)

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) 60q2r2s4 + 78qr2s4 − 18r2s4

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b) ax2 − 2axy + ay2 − x2 + 2xy − y2

c)81

16x4 − y40

d) 8(4− a2)− x3(4− a2)

e) (3z + 2)2 − 12(3z + 2) + 36

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Section 5.9: Solving Equations by Factoring

Exercise 23. Solve each equation.

a) 2y(4y + 3) = 9

b) x2

9 = 89x−

79

c) b3 − 5b2 − 9b + 45 = 0

40

d) x2(6x+37)35 = x

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