Algebra Semester 2 Exam Review Notes

Chapter 5: Inequalities

• Solve each inequality and graph its solution.
  1. $n + 17 \le -20$
     • $n \le -37$
  2. \frac{n}{14} < -5
     • n < -70
  3. 5 + 10n > 115
     • 10n > 110
     • n > 11
  4. $11 \ge \frac{x}{2} + 9$
     • $2 \ge \frac{x}{2}$
     • $4 \ge x$
     • $x \le 4$
  5. a - 8 + 6 > 6 + 3a - a
     • a - 2 > 6 + 2a
     • -8 > a
     • a < -8
  6. 24 + 8x < -5(-2 - x) + 5x
     • 24 + 8x < 10 + 5x + 5x
     • 24 + 8x < 10 + 10x
     • 14 < 2x
     • x > 7

Compound Inequalities

• Solve each compound inequality and graph its solution.
  7. $n \ge 4$ or $8n \le -56$
     • $n \ge 4$ or $n \le -7$
  8. $4 + n < 4$ or $n - 1 > 1$
     • $n < 0$ or $n > 2$
  9. -8 \le 6n + 10 < 4
     • -18 \le 6n < -6
     • -3 \le n < -1
  10. 2x - 8 > -2 or 10x - 9 < -49
      • 2x > 6 or 10x < -40
      • x > 3 or x < -4

Graphing Inequalities

• The following problems instruct to graph the inequalities.
  11. $y \le -x - 2$ and y > -5x + 3
  12. y > \frac{1}{2}x - 3 and $y \le -\frac{5}{2}x + 3$
  13. $y \le -x - 2$ and y > -5x + 2
  14. y > x - 3 and $y \le -x + 1$

Chapter 6: Systems of Equations

• Multiple Choice: Which ordered pair is a solution of the system?

  • System:
    • $2x + 3y = -17$
    • $2x + 2y = -8$
  • The correct answer is (c) (-2, -1) (This may be incorrect - needs verification)

• Solve each system by graphing. Tell whether the system has one solution, infinitely many solutions, or no solution.

  2. $y = 3x - 7$ and $y = -x + 1$
     • Solution: $(2, -1)$
  3. $x + 3y = 12$ and $x = y - 8$
     • Solution: $(3, 3)$
  4. $2y - x = 2$ and $x + y = 5$
     • Solution: One Solution
  5. $x + y = -2$ and $y = 2x - 1$
     • Solution: No Solution

• Solve each system using substitution. Tell whether the system has one solution, infinitely many solutions, or no solution.

  6. $y = 4x - 7$ and $y = 2x + 9$
     • $4x - 7 = 2x + 9$
     • $2x = 16$
     • $x = 8$
     • $y = 4(8) - 7 = 25$
     • Solution: One Solution $(8, 25)$
  7. $8x + 2y = -2$ and $x = -2y + 4$
     • $8(-2y + 4) + 2y = -2$
     • $-16y + 32 + 2y = -2$
     • $-14y = -34$
     • $y = \frac{17}{7}$
  8. $y = 3x - 11$ and $3.5x + 7y = 14$
     • $3.5x + 7(3x - 11) = 14$
     • $3.5x + 21x - 77 = 14$
     • $24.5x = 91$
     • $x = \frac{91}{24.5} = \frac{910}{245} = \frac{130}{35} = \frac{26}{7}$
  9. $y - 3x = -13$ and $2y = -8x - 2$
     • $y = 3x - 13$ and $y = -4x - 1$
     • $3x - 13 = -4x - 1$
     • $7x = 12$
     • $x = \frac{12}{7}$
     • No Solution

• Solve each system using elimination. Tell whether the system has one solution, infinitely many solutions, or no solution.

  10. $2x + 6y = 18$ and $x + 3y = 9$
      • Multiply the second equation by -2: $-2x - 6y = -18$
      • Adding the equations results in $0 = 0$
      • Solution: Infinite Solutions
  11. $2x + 5y = 20$ and $3x - 10y = 37$
      • Multiply the first equation by 2: $4x + 10y = 40$
      • Add this to the second equation: $7x = 77$
      • $x = 11$
      • $2(11) + 5y = 20$
      • $5y = -2$
      • Solution: One Solution $(11, -\frac{2}{5})$
  12. $3x + 7y = 11$ and $-5x + 2y = -1$
  13. $-2x + 5y = 7$ and $-2x + 5y = 12$
      • Subtracting the first equation from the second gives $0 = 5$
      • Solution: No Solution

• Application Problems

  14. Jay has written 24 songs to date. He writes an average of 6 songs per year. Jenna started writing songs this year and expects to write about 12 songs per year.
      • a. Write a system of equations to represent the number of songs, y, each has written in x years.
        • Jay: $y = 6x + 24$
        • Jenna: $y = 12x$
      • b. Solve the system you wrote in part a to find how many years from now Jenna will have written as many songs as Jay.
        • $12x = 6x + 24$
        • $6x = 24$
        • $x = 4$
        • In 4 years, both will have written 48 songs.
  15. Three hundred and fifty-eight tickets to the East-West basketball game were sold. Student tickets were $1.50 and non-student tickets were $3.25. The school made $752.25. Write and solve a system to find how many student and non-student tickets were sold.
      • Let x be the number of student tickets and y be the number of non-student tickets.
      • $x + y = 358$
      • $1.50x + 3.25y = 752.25$
      • Multiply the first equation by -1.5: $-1.5x - 1.5y = -537$
      • Add this to the second equation: $1.75y = 215.25$
      • $y = 123$
      • $x = 358 - 123 = 235$
      • 235 student tickets and 123 non-student tickets were sold.
  16. In a talent show of singing and comedy acts, singing acts are 5 min long and comedy acts are 3 min long. The show has 12 acts and lasts 50 min. Write and solve a system to find how many singing acts and how many comedy acts are in the show.
      • Let s be the number of singing acts and c be the number of comedy acts.
      • $5s + 3c = 50$
      • $s + c = 12$
      • Multiply the second equation by -5: $-5s - 5c = -60$
      • Add this to the first equation: $-2c = -10$
      • $c = 5$
      • $s = 12 - 5 = 7$
      • 7 singing acts and 5 comedy acts are in the show.
  17. A cell phone provider offers a plan that costs $40 per month plus $.20 per text message sent or received. A comparable plan costs $60 per month but offers unlimited text messaging.
      • a. Write a system to represent the cost, y, for x text messages.
        • Plan A: $y = 0.20x + 40$
        • Plan B: $y = 60$
      • b. Solve the system you wrote in part a to find how many text messages you would have to send or receive in order for the plans to cost the same each month.
        • $60 = 0.20x + 40$
        • $20 = 0.20x$
        • $x = 100$
        • For 100 texts, both companies charge $60.
      • c. If you send or receive an average of 50 text messages each month, which plan would you choose? Why?
        • Plan A: $y = 0.20(50) + 40 = 10 + 40 = 50$
        • Plan B: $y = 60$
        • Plan A would be cheaper.

Chapter 7: Exponents and Exponential Functions

• Sketch the graph of each function. Give the domain and range.

  1. $y = \frac{1}{4}^x$
     • Domain: All real numbers
     • Range: y > 0
  2. $y = 2^x - 1$
     • Domain: All real numbers
     • Range: y > -1

• Simplify. Your answer should contain only positive exponents.

  3. $3x^3y^3 \cdot 4x^3$
     • $12x^6y^3$
  4. $2x^2y^2 \cdot 2x^4$
     • $4x^6y^2$
  5. $(4v^3)^{-1}$
     • $\frac{1}{4v^3}$
  6. $(2vu^{-2})^3$
     • $8v^3u^{-6} = \frac{8v^3}{u^6}$
  7. $\frac{4x^7y^4}{4x^4y^4}$
     • $x^3$
  8. $\frac{3y}{4x^7y^4}$
     • $\frac{3}{4x^7y^3}$
  9. $x \cdot (x^2)^8$
     • $x \cdot x^{16} = x^{17}$
  10. $a \cdot (a^2b^0)^2$
      • $a \cdot (a^4 \cdot 1) = a^5$
  11. $\frac{(x^{-1}y^4)^4}{2x^3y}$
  12. $\frac{2x^3y}{2x^3y}$
      • $1$
  13. $\frac{(x^{-7}y^3)^{-3}}{x^4y}$
  14. $\frac{(x^2y^{-3})^{-2}}{x^3y^4}$

• Write each number in scientific notation.

  15. 0.179
      • $1.79 \times 10^{-1}$
  16. 230000
      • $2.3 \times 10^5$

• Write each number in standard notation.

  17. $7 \times 10^2$
      • 700
  18. $8.6 \times 10^5$
      • 860000

• Simplify. Write each answer in scientific notation.

  19. $(2.6 \times 10^1)(6.08 \times 10^{-5})$
      • $15.808 \times 10^{-4}$
      • $1.5808 \times 10^{-3}$
  20. $(7.93 \times 10^1)(9.73 \times 10^2)$
      • $77.1589 \times 10^3$
      • $7.71589 \times 10^4$

Chapter 8: Polynomials

• Multiply the following polynomials.

  75. $(x^3)(9x^4)(-3x)$
      • $-27x^8$
  76. $(5r)(-4r^2s^2)(3r^5s^4)$
      • $-60r^8s^6$
  77. $5a^2(2a^3 + a^2)$
      • $10a^5 + 5a^4$
  78. $2x(3x^4 - 2x^2 + x - 1)$
      • $6x^5 - 4x^3 + 2x^2 - 2x$
  79. $-x^2(2x + 3)$
      • $-2x^3 - 3x^2$
  80. $-5v(v + 8) + 4v$
      • $-5v^2 - 40v + 4v = -5v^2 - 36v$
  81. $(6x + 5)(x - 2)$
      • $6x^2 - 12x + 5x - 10 = 6x^2 - 7x - 10$
  82. $(-2x - 3)(-x + 1)$
      • $2x^2 - 2x + 3x - 3 = 2x^2 + x - 3$
  83. $(7x - 1)^2$
      • $(7x - 1)(7x - 1) = 49x^2 - 14x + 1$
  84. $(2m - 7)(m^2 - 3m - 4)$
      • $2m^3 - 6m^2 - 8m - 7m^2 + 21m + 28 = 2m^3 - 13m^2 + 13m + 28$
  85. $(6x - 4)(6x + 4)$
      • $36x^2 - 16$
  86. $(4 + x)(4 - x)$
      • $16 - x^2$
  87. A square tabletop has side lengths $(2x + 7)$ units.
      • a) Write a polynomial to represent the area of the tabletop.
        • $A = (2x + 7)(2x + 7) = 4x^2 + 28x + 49$
      • b) Write a polynomial to represent the perimeter of the tabletop.
        • $P = 4(2x + 7) = 8x + 28$
      • c) Find the area and perimeter of the tabletop if $x = 4$.
        • $A = 4(4)^2 + 28(4) + 49 = 64 + 112 + 49 = 225$
        • $P = 8(4) + 28 = 32 + 28 = 60$
  88. A triangle has a base that is 4 cm longer than its height.
      • a. Write a polynomial that represents the area of the triangle.
        • $A = \frac{1}{2}(h + 4)(h) = \frac{1}{2}h^2 + 2h$
      • b. Find the area of the triangle if the height is 8 cm.
        • $A = \frac{1}{2}(8)^2 + 2(8) = \frac{1}{2}(64) + 16 = 32 + 16 = 48$

• Exponential Growth and Decay

  70. SAVINGS The Fresh and Green Company has a savings plan for its employees. If an employee makes an initial contribution of $1000, the company pays 8% interest compounded quarterly.
      • a. If an employee participating in the plan withdraws the balance of the account after 5 years, how much will be in the account?
        • $y = 1000(1 + \frac{0.08}{4})^{4(5)} = 1000(1.02)^{20} \approx 1485.95$
      • b. If an employee participating in the plan withdraws the balance of the account after 35 years, how much will be in the account?
        • $y = 1000(1 + \frac{0.08}{4})^{4(35)} = 1000(1.02)^{140} \approx 15996.47$
  71. HOUSING Mr. and Mrs. Boyce bought a house for $96,000 in 1995. The real estate broker indicated that houses in their area were appreciating at an average annual rate of 7%. If the appreciation remained steady at this rate, what was the value of the Boyce's home in 2009?
      • $y = 96000(1 + 0.07)^{14} \approx 247539.28$
  72. MANUFACTURING Zeller Industries bought a piece of weaving equipment for $60,000. It is expected to depreciate at an average rate of 10% per year.
      • a. Write an equation for the value of the piece of equipment after t years.
        • $y = 60000(1 - 0.10)^t = 60000(0.9)^t$
      • b. Find the value of the piece of equipment after 6 years.
        • $y = 60000(0.9)^6 \approx 31886.46$
  73. FINANCES Kyle saved $500 from a summer job. He plans to spend 10% of his savings each week on various forms of entertainment. At this rate, how much will Kyle have left after 15 weeks?
      • $y = 500(1 - 0.10)^{15} = 500(0.9)^{15} \approx 102.95$
  74. TRANSPORTATION Tiffany's mother bought a car for $9000 five years ago. She wants to sell it to Tiffany based on a 15% annual rate of depreciation. At this rate, how much will Tiffany pay for the car?
      • $y = 9000(1 - 0.15)^5 = 9000(0.85)^5 \approx 3993.35$

Factoring

*Factor the following polynomials.
89. $16x^4 + 20x^3$
* $4x^3(4x + 5)$
90. $4m^4 - 12m^2 + 8m$
* $4m(m^3 - 3m + 2)$
91. $2x^3 + x^2 - 8x - 4$
* $x^2(2x + 1) - 4(2x + 1)$
* $(2x + 1)(x^2 - 4)$
* $(2x + 1)(x - 2)(x + 2)$

92. $x^2 - 11x + 30$
     *   $(x - 6)(x - 5)$
93. $7p^4 - 2p^3 + 63p - 18$
     *   $(7p - 2)(p^3 + 9)$
94. $x^2 + 10x + 9$
     *   $(x + 9)(x + 1)$
95. $x^2 - 6x - 27$
     *   $(x - 9)(x + 3)$
96. $x^2 + 14x - 32$
     *   $(x + 16)(x - 2)$
97. $5x^2 + 17x + 6$
     *   $(5x + 2)(x + 3)$
98. $2x^2 + 5x - 12$
     *   $(2x - 3)(x + 4)$
99. $6x^2 - 23x + 7$
     *   $(3x - 1)(2x - 7)$
100. $-2x^2 + 7x - 3$
      *   $(-2x + 1)(x - 3)$

101. $8x^2 + 27x + 9$
      *   $(8x + 3)(x + 3)$
102. $4x^2 + 16x - 48$
      *   $4(x^2 + 4x - 12) = 4(x + 6)(x - 2)$

103. $18x^2 - 3x - 3$
      *   $3(6x^2 - x - 1) = 3(3x + 1)(2x - 1)$
104. $x^2 - 81$
      *   $(x + 9)(x - 9)$
105. $18x^2 - 50$
      *   $2(9x^2 - 25) = 2(3x - 5)(3x + 5)$
106. $5x^3 - 20x^3 + 7 - 28x^2$
     *   $-15x^3 - 28x^2 + 7$
107. $x^5 - 81x$
     *   $x(x^4 - 81) = x(x^2 - 9)(x^2 + 9) = x(x - 3)(x + 3)(x^2 + 9)$
108. $5x^3 - 45x$
     *   $5x(x^2 - 9) = 5x(x - 3)(x + 3)$

• Application Problems
  109. A model rocket is fired vertically into the air at 320 ft/sec. The expression $-16t^2 + 320t$ gives the rocket's height after t seconds.
       • a. Factor the expression.
         • $-16t(t - 20)$
       • b. How high is the rocket after 3 seconds?
         • $-16(3)^2 + 320(3) = -144 + 960 = 816$
  110. The area of a rectangular fountain is $(6x^2 + 25x + 14) \text{ ft}^2$. The width is $(3x + 2) \text{ ft}$.
       • a. Find the length of the fountain.
         • Given Area = Length * Width, then Length = Area/Width
         • Area = $6x^2 + 25x + 14 = (3x + 2)(2x + 7)$
         • Given Width = $3x+2$
         • Length = $2x+7$
       • b. Find the perimeter of the fountain.
         • Perimeter = $2(Length + Width) = 2(2x+7 + 3x+2) = 2(5x+9) = 10x+18$
  111. Write a polynomial to represent the area of the shaded region.
       • a. Write a polynomial to represent the area of the shaded region.
         • $A_{total} = (9x)(8x) = 72x^2$
         • $A_{unshaded} = (4y)(8y) = 32y^2$
         • $A<em>{shaded} = A</em>{total} - A_{unshaded} = 72x^2 - 32y^2$
       • b. Completely factor the expression you wrote in part a.
         • $A_{shaded} = 8(9x^2 - 4y^2) = 8(3x - 2y)(3x + 2y)$

Chapter 9: Quadratic Functions

• Tell whether each function is quadratic. Explain.

  112. $y = x^2 - 6$
       • Yes, because it has an $x^2$ term.
  113. $y + x^3 = 4$
       • No, because it has an $x^3$ term.

• Graph each function and identify its parts.

  114. $y = 2x^2 - 6x - 8$
  115. $y = -2x^2 + 8x + 10$
  116. $y = 3x^2 + 6x + 3$
  117. $y = -2x^2 + 3$

• Application Problems

  118. The height in meters of a fireworks rocket launched from a platform 35 feet above the ground can be approximated by $h(t) = -5t^2 + 30t + 35$, where t is the time in seconds.
       • a. Graph the function.
       • b. Find the rocket's maximum height.
         • Rocket's maximum height is 80ft
       • c. How long does it take the rocket to reach its maximum height?
         • It takes 3 seconds for the rocket to reach its Max height
       • d. How long is the rocket in the air?
         • Rocket is in the air for 7 seconds
       • e. What is a reasonable domain and range?
         • D: $0 \le x \le 7$
         • R: $0 \le y \le 80$
  119. Water is shot straight up out of a water soaker toy. The quadratic function $h(t) = -16t^2 + 32t$ models the height in feet of a water droplet after t seconds.
       • a. Graph the function.
       • b. Find the water droplet's maximum height.
         • Maximum height: 16 feet
       • c. How long does it take the droplet to reach its maximum height?
         • Time to max height: 1 second
       • d. How long is the water droplet in the air?
         • Total time in air: 2 seconds
       • e. Factor h(t).
         • $h(t) = -16t(t - 2)$

• Describe the transformation from the parent function.

  11. $f(x) = 2x^2 + 1$
      • Vertical Stretch
      • Up 2
  12. $y = -(x - 1)^2$
      • Reflection
      • Right 1
  13. $h(x) = \frac{1}{2}(x + 5)^2 - 3$
      • Vertical Compression
      • Left 5
        • Down 3

• Solve the following quadratic equations by factoring.

  120. $(2x + 1)(3x - 1) = 0$
       • $2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$
       • $3x - 1 = 0 \Rightarrow x = \frac{1}{3}$
       • $x = -\frac{1}{2}, \frac{1}{3}$
  121. $x^2 - x - 12 = 0$
       • $(x - 4)(x + 3) = 0$
       • $x = 4, -3$
  122. $-4x^2 = 16x + 16$
       • $-4x^2 - 16x - 16 = 0 \Rightarrow x^2 + 4x + 4 = 0$
       • $(x + 2)(x + 2) = 0$
       • $x = -2$
  123. $x^2 - 4x + 4 = 0$
       • $(x-2)(x-2) = 0$
       • $x=2$
  124. -4x+4 = 0 #There's something wrong
  125. $4x^2 - 12x = 0$
       • $4x(x - 3) = 0$
       • $x = 0, 3$

• height Application Problems

  126. The height of a rocket launched upward from a 160-foot cliff is modeled by the quadratic function $h(t) = -16t^2 + 48t + 160$, where h is height in feet and t is time in seconds. Factor h(t) to find the time it takes the rocket to reach the ground at the bottom of the cliff.
       • $0 = -16t^2 + 48t + 160 \Rightarrow 0 = t^2 - 3t - 10 = (t - 5)(t + 2)$
       • $t = 5, -2$
       • Since time cannot be negative, it takes 5 seconds for the rocket to reach the ground.
  127. The height of a skydiver jumping out of an airplane is given by $h = -16t^2 + 3200$. How long will it take the skydiver to reach the ground? Round to the nearest tenth of a second.
       • $0 = -16t^2 + 3200 \Rightarrow t^2 = 200 \Rightarrow t = \pm \sqrt{200} \approx \pm 14.1$
       • Since time cannot be negative, it takes approximately 14.1 seconds for the skydiver to reach the ground.

• Solve each of the following using the quadratic formula.

  128. $x^2 - 7x + 10 = 0$
       • $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(10)}}{2(1)} = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm \sqrt{9}}{2} = \frac{7 \pm 3}{2} = 5, 2$
  129. $$3x^2 + x = 10 \Rightarrow 3x^2 + x - 10 =