Concept explainers
In Exercises 1–12, solve each equation.
1. −5 + 3(x + 5) = 2(3x − 4)
To solve: The equation
Answer to Problem 1MC
The solution set of the equation
Explanation of Solution
The given equation is
Simplify the algebraic expression on each side of the equation.
Obtain an equivalent equation by isolating x on one side and the constant terms on the other side as follows.
On further simplifications,
Thus, the solution of the equation is 6.
Replace x with 6 in the original solution and check the correctness of the solution.
The left hand side of the equation is,
The right hand side of the equation is,
That is,
Thus, the solution set of the equation
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Chapter 1 Solutions
COLLEGE ALGEBRA ESSENTIALS
- For Exercises 5–10, a. Simplify the expression. b. Substitute 0 for h in the simplified expression. 2(x + h)? + 3(x + h) · 5. (2x + 3x) 3(x + h - 4(x + h) – (3x - 4x) 6. h 1 1 1 1 (x + h) – 2 7. x - 2 2(x + h) + 5 8. 2x + 5 h (x + h) – x 9. (x + h) 10. - X h harrow_forwardExercises 105-120: Complete the following. (a) Write the equation as ax² + bx + e = 0 with a > 0. (b) Calculate the discriminant b² – 4ac and determine the number of real solutions. (c) Solve the equation. 105. 3x² = 12 106. 8x - 2 = 14 107. x² – 2x = -1 108. 6x² = 4x 109. 4x = x? 110. 16x + 9 = 24x 111. x² + 1 = x 112. 2x² + x = 2 113. 2x² + 3x = 12 – 2x 114. 3x² + 3 = 5x 115. x(x – 4) = -4 116. + 3x = x – 4 117. x(x + 2) = -13 118. 4x = 6 + x? 119. 3x = 1- x 120. x(5x – 3) = 1arrow_forwardFor Exercises 81–100, make an appropriate substitution and solve the equation. (See Examples 10–11) 81. (2x + 5)? – 7(2x + 5) - 30 = 0 82. (Зх — 7)? - 6(3х — 7)-16 3D 0 83. (x + 2x)? – 18(r + 2x) = -45 84. (x + 3x)? - 86. (у? — 3)? — 9(y? — 3) — 52 %3D 0 14(x + 3x) = -40 85. (x + 2)2 + (x + 2) – 42 = 0 10 2 10 - 61 m - - 27 = 0 x + + 35 = 0 87. 88. - 121 x + т - m m 89. 2 + 2 + = 12 90. + 3 + 6 + 3 = -8 91. 5c2/5 11c/5 + 2 = 0 92. З3 d'/3 – 4 = 0 93. y'/2 – y/4 6 = 0 94. n'/2 + 6n/4 – 16 = 0 95. 9y 10y + 1 = 0 96. 100х-4 29x-2 + 1 = 0 | 97. 4t – 25 Vi = 0 98. 9m – 16Vm = 0 100. 392 + 16q -1 99. 30k-2 – 23k- + 2 = 0 + 5 = 0arrow_forward
- Exercises 149–151 will help you prepare for the material covered in the next section. 149. Multiply: (Vx + 4 + 1)2. 150. Solve: 4x2 16x + 16 = 4(x + 4). 151. Solve: 26 – 11x 16 – 8x + x?.arrow_forwardFor Exercises 73–80, (a) evaluate the discriminant and (b) determine the number and type of solutions to each equation. (See Example 9) 73. Зх? 4х + 6 3D 0 74. 5x - 2x + 4 = 0 75. - 2w? + 8w = 3 76. -6d + 9d = 2 77. Зx(х — 4) 3D х — 4 78. 2x(x – 2) = x + 3 79. –1.4m + 0.1 = -4.9m² 80. 3.6n + 0.4 = -8.1n?arrow_forwardFor Exercises 39–42, multiply the radicals and simplify. Assume that all variable expressions represent positive real numbers. 39. (6V5 – 2V3)(2V3 + 5V3) 40. (7V2 – 2VIT)(7V2 + 2V1T) 41. (2c²Va – 5ď Vc) 42. (Vx + 2 + 4)²arrow_forward
- In Exercises 1-2, solve each equation. 1. 5(x + 1) + 2 = x – 3(2x + 1) 2(x + 6) 2. 4х — 7 = 1 + 3 3arrow_forwardSolve the following quadratic equation for all values of x in simplest form. 2(x + 10)2 – 37 = 13 - Answer: Submit Answer 士arrow_forwardFor Exercises 107–108, solve the equation in two ways. a. Solve as a radical equation by first isolating the radical. b. Solve by writing the equation in quadratic form and using an appropriate substitution. 107. y + 4Vy = 21 108. w – 3Vw = 10arrow_forward
- In Exercises 125-134, solve each equation. 125. 25 - [2 + 5x – 3(x + 2)] = -3(2x – 5) – [5(x – 1) – 3x + 3] 126. 45 - [4 - 2x – 4(x + 7)] = -4(1 + 3x) – [4 – 3(x + 2) – 2(2x – 5)] 127. 7 - 7x = (3x + 2)(x - 1) 128. 10x - 1 = (2r + 1) 129. |x + 2x – 36| = 12 130. |x + 6x + 1| = 8 1 + 1 5 131. 3x + 2 x + 2 x? - 4 1 132. X - 2 X - 3 x - 5x + 6 133. Vx + 8 - Vx 4 = 2 134. Vx + 5 – Vx - 3 = 2 + 1.arrow_forwardIn Exercises 65–74, factor by grouping to obtain the difference of two squares. 6x + 9 – y? 12x + 36 – y? 65. x? 66. x2 67. x + 20xr + 100 68. x? + 16x + 64 – x4 69. 9x2 70. 25x? – 20x + 4 – 81y? 30x + 25 – 36y? 71. x* - x? – 2x – 1 72. x4 -х2 — бх — 9 x? + 4xy – 4y2 x²+ 10xy - 25y2 73. z? 74. z? - rarrow_forwardExercises 105–107 will help you prepare for the material covered in the next section. 105. Let x represent a number. a. Write an equation in x that describes the following conditions: Four less than three times the number is 32. b. Solve the equation and determine the number. 106. Let x represent the number of countries in the world that are not free. The number of free countries exceeds the number of not-free countries by 44. Write an algebraic expression that represents the number of free countries. 107. You purchase a new car for $20,000. Each year the value of the car decreases by $2500. Write an algebraic expression that represents the car's value, in dollars, after x years.arrow_forward
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