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3 β β β β 2 3 454. 8x3 (2x)2 455. β16y0β β β 2yβ2 456. 441 457. 490 458. 9x 16 459. 121b2 1 + b 460. 6 24 + 7 54 β 12 6 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 107 2 | EQUATIONS AND INEQUALITIES Figure 2.1 Chapter Outline 2.1 The Rectangular Coordi... |
ting Ordered Pairs in the Cartesian Coordinate System An old story describes how seventeenth-century philosopher/mathematician RenΓ© Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized t... |
the point (3, β1) in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See Figure 2.5. 110 Chapter 2 Equations and Inequalities Figure 2.5 When dividing the axes into equally spaced increments, note that the x-axis may be considered sep... |
axis. If the y-coordinate is zero, the point is on the x-axis. Graphing Equations by Plotting Points We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any grap... |
graph by plotting points. 1. Make a table with one column labeled x, a second column labeled with the equation, and a third column listing the resulting ordered pairs. 2. Enter x-values down the first column using positive and negative values. Selecting the x-values in numerical order will make the graphing simpler. 3... |
that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows β10 β€ x β€ 10, and β10 β€ y β€ 10. See Figure 2.9c. Figure 2.9 a. Enter the equation. b. This is the graph in the original window. c. These are the original settings. 116 Chapter 2 Equations and Inequalit... |
that the graph crosses the axes where we predicted it would. Figure 2.12 Given an equation, find the intercepts. 1. Find the x-intercept by setting y = 0 and solving for x. 2. Find the y-intercept by setting x = 0 and solving for y. Example 2.4 Finding the Intercepts of the Given Equation Find the intercepts of the eq... |
a2 + b2 β c = a2 + b2 d 2 = (x2 β x1)2 + (y2 β y1)2 β d = (x2 β x1)2 + (y2 β y1)2 We do not have to use the absolute value symbols in this definition because any number squared is positive. The Distance Formula Given endpoints (x1, y1) and (x2, y2), the distance between two points is given by d = (x2 β x1)2 + (y2 β y1... |
for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 121 Figure 2.16 Next, we can calculate the distance. Note that each grid unit represents 1,000 feet. β’ From her starting location to her first stop at (1, 1), Tracie might have driven north 1,000 feet and then east 1,000 feet, or vic... |
heard the saying βas the crow flies,β which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways. Using the Midpoint Formula When the endpoints of a line segment are known, we can find the point ... |
college.org/l/coordplotpnts) β’ Find x and y intercepts based on the graph of a line (http://Openstaxcollege.org/l/ xyintsgraph) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 125 2.1 EXERCISES Verbal Is it possible for a point plotted in the Cartesian 1. ... |
inear (on the same line). 30. (4, 1)(β2, β3)(5, 0) 5. 6. 7. 8. 9. y = β3x + 6 4y = 2x β 1 3x β 2y = 6 4x β 3 = 2y 3x + 8y = 9 10. 2x β 2 3 = 3 4 y + 3 For each of the following exercises, solve the equation for y in terms of x. 11. 4x + 2y = 8 12. 3x β 2y = 6 13. 2x = 5 β 3y 14. x β 2y = 7 15. 5y + 4 = 10x 16. 5x + 2y ... |
plug in x = 0, thus finding the y-intercept, for each of the following graphs. 48. Y1 = β2x + 5 49. 50. Y1 = 3x β 8 4 Y1 = x + 5 2 For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu. After graphing it, use the 2nd CALC button and 2:zero button, hit enter. At the l... |
any number for A man drove 10 mi directly east from his home, made a 54. left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile? If the road was made in the previous exerc... |
://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 129 2.2 | Linear Equations in One Variable Learning Objectives In this section you will: 2.2.1 Solve equations in one variable algebraically. 2.2.2 Solve a rational equation. 2.2.3 Find a linear equation. 2.2.4 Given the equations of perpendicular. 2.... |
ations and Inequalities An inconsistent equation results in a false statement. For example, if we are to solve 5x β 15 = 5(x β 4), we have the following: 5x β 15 = 5x β 20 5x β 15 β 5x = 5x β 20 β 5x β15 β β20 Subtract 5x from both sides. False statement Indeed, β15 β β20. There is no solution because this is an incons... |
Solve the following equation: 4(xβ3) + 12 = 15β5(x + 6). Solution Apply standard algebraic properties. 4(x β 3) + 12 = 15 β 5(x + 6) 4x β 12 + 12 = 15 β 5x β 30 4x = β15 β 5x 9x = β15 x = β 15 9 x = β 5 3 Apply the distributive property. Combine like terms. Place x β terms on one side and simplify. Multiply both sides... |
6x) β β (6x ) β 7 2x β‘ (6x) β£ β β 7 β β (6x) β 2x β β β β (6x ) β β€ β¦(6x) β β (6x) β β (6 x) β‘ β£ β β β β 22 3 22 3 22 3 β€ 2x β 5 7 β¦ = 3x β 5 β = 3x β 5 β = 3x 3(7) β 2(5) = 22(2x) 21 β 10 = 44x 11 = 44x 11 = x 44 1 4 = x Use the distributive property. Cancel out the common factors. Multiply remaining factors by each num... |
the second denominator is factored as x2 + 2x = x(x + 2), there is a common factor of x in both denominators and the LCD is x(x + 2). Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation. a b = c d We can use ... |
which is not an excluded value, so the solution set contains one number, x = β1, or {β1} written in set notation. 2.7 Solve the rational equation: 2 3x = 1 4 β 1 6x. Example 2.13 Solving a Rational Equation by Factoring the Denominator Solve the following rational equation: 1 x = 1 10 β 3 4x. 134 Chapter 2 Equations a... |
x = 13 β x 3x = 13 x = 13 3. The excluded value is x = 3. The solution is x = 13 3 c. The least common denominator is 2(x β 2). Multiply both sides of the equation by x(x β 2). β‘ 2(x β 2) β£ x x β 2 β€ β¦2(x β 2 2x = 10 β (x β 2) 2x = 12 β x 3x = 12 x = 4 The solution is x = 4. The excluded value is x = 2. 2.9 Solve β3 2x... |
2 β x1 If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in Figure 2.19. The lines indicate the following slopes: m = β3, m = 2, and m = 1 3. Figure 2.19 The Slope of a Line The slope o... |
can find the equation of the line using the point-slope formula. y β y1 = m(x β x1) This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting th... |
7 3 x β 3. Analysis To prove that either point can be used, let us use the second point (0, β3) and see if we get the same equationx β 0) x x β 3 We see that the same line will be obtained using either point. This makes sense because we used both points to calculate the slope. Standard Form of a Line Another way that ... |
find the slope. m = 5 β 3 β3 β (β3) = 2 0 Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through x = β3. See Figure 2.20. The equation of a ho... |
because they have the same slope and different y-intercepts. Lines that are perpendicular intersect to form a 90Β° -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is β1 : m1 β
m2 = β1. For example, Figure 2.22 shows the... |
the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line. Given an equation for a line, write the equation of a line parallel or perpendicular to it. 1. Find the slope of the given line. The easiest way to do this is to write the equation in slope-intercept f... |
-slope formula with this new slope and the given pointx β (β4)β β x β 12 5 x β 12 5 x β 7 5 + 5 5 Access these online resources for additional instruction and practice with linear equations. β’ Solving rational equations (http://openstaxcollege.org/l/rationaleqs) β’ Equation of a line given two points (http://openstaxcol... |
line using the point-slope formula. Write all the final equations using the slope-intercept form. 86. 87. (0, 3) with a slope of 2 3 (1, 2) with a slope of β4 5 88. x-intercept is 1, and (β2, 6) 89. y-intercept is 2, and (4, β1) 90. (β3, 10) and (5, β6) 91. (1, 3) and (5, 5) parallel to y = 2x + 5 and passes through t... |
and (0, 9) (2, 5) and (5, 9) (β1, β1) and (2, 3) Technology For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occu... |
would be the maximum number of miles you could travel? This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 149 2.3 | Models and Applications Learning Objectives In this section you will: 2.3.1 Set up a linear equation to solve a real-world application. 2.3.2 ... |
verbal expressions and their equivalent mathematical expressions. C = 0.10x + 50 150 Verbal One number exceeds another by a Twice a number One number is a more than another number One number is a less than twice another number The product of a number and a, decreased by b Chapter 2 Equations and Inequalities Translati... |
min talk-time. a. Write a linear equation that models the packages offered by both companies. b. c. If the average number of minutes used each month is 1,160, which company offers the better plan? If the average number of minutes used each month is 420, which company offers the better plan? d. How many minutes of talk-... |
renders the plans equal. See Figure 2.26 Figure 2.26 2.18 Find a linear equation to model this real-world application: It costs ABC electronics company $2.50 per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of $350 for utilities and $3,300 for salaries... |
οΏ½οΏ½ 2 3 To work To home As both equations equal the same distance, we set them equal to each other and solve for r. β β 2 3 β rβ β 1 β = (r β 10) β β 2 r β 20 2r = 2 1 3 3 r = β 20 3 r = β 20 3 r = β 20 3 r = 40 (β6 We have solved for the rate of speed to work, 40 mph. Substituting 40 into the rate on the return trip yi... |
The dimensions are L = 15 ft and W = 12 ft. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 155 Find the dimensions of a rectangle given that the perimeter is 110 cm and the length is 1 cm more than 2.20 twice the width. Example 2.29 Solving an Area Probl... |
10 in., and H = 8 in. Analysis Note that the square root of W 2 would result in a positive and a negative value. However, because we are describing width, we can use only the positive result. Access these online resources for additional instruction and practice with models and applications of linear equations. β’ Probl... |
out 8 more applications than 131. Henry. Then each boy filled out 3 additional applications, bringing the total to 28. How many applications did each boy originally fill out? For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company... |
the other 550 mi/h. How long will it take before they are 4,000 mi apart? Ben starts walking along a path at 4 mi/h. One and a 143. half hours after Ben leaves, his sister Amanda begins jogging along the same path at 6 mi/h. How long will it be before Amanda catches up to Ben? 144. Fiora starts riding her bike at 20 m... |
b2 β β If Tim knows he has to travel 300 mi, which plan 150. should he choose? For the following exercises, use the given formulas to answer the questions. 151. A = P(1 + rt) is used to find the principal amount Pdeposited, earning r% interest, for t years. Use this to find what principal amount P David invested at a ... |
of a rectangle whose length is 15 and whose perimeter is 58. Solve for 157. 158. Use the formula from the previous question to find the 170. height to the nearest tenth of a triangle with a base of 15 and an area of 215. 171. The volume formula for a cylinder is V = Οr 2 h. Using the symbol Ο in your answer, find the ... |
. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it. Expressing Square Roots of Negative Numbers as Multip... |
. 3. Write a β
i in simplest form. Example 2.31 Expressing an Imaginary Number in Standard Form Express β9 in standard form. Solution In standard form, this is 0 + 3i. β9 = 9 β1 = 3i 2.22 Express β24 in standard form. Plotting a Complex Number on the Complex Plane We cannot plot complex numbers on a number line as we m... |
combine the imaginary parts. Complex Numbers: Addition and Subtraction Adding complex numbers: Subtracting complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) β (c + di) = (a β c) + (b β d)i 164 Chapter 2 Equations and Inequalities Given two complex numbers, find the sum or difference. 1. Identify the r... |
+ (4 β
5i) = 8 + 20i Multiplying Complex Numbers Together Now, letβs multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The dif... |
β b2 i2 = a2 + b2 The result is a real number. Note that complex conjugates have an opposite relationship: The complex conjugate of a + bi is a β bi, and the complex conjugate of a β bi is a + bi. Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugat... |
bi, or 0 + 1 2 i. This can be written simply as 1 2 i. Analysis Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number... |
cycle of four. Letβs examine the next four powers of i. it is equal to the first power. As we continue to multiply i by i6 = i5 β
i = i β
i = i2 = β1 i7 = i6 β
i = i2 β
i = i3 = βi i8 = i7 β
i = i3 β
i = i4 = 1 i9 = i8 β
i = i4 β
i = i5 = i The cycle is repeated continuously: i, β1, β i, 1, every four powers. Example ... |
Subtracting Complex Numbers (http://openstaxcollege.org/l/addsubcomplex) β’ Multiply Complex Numbers (http://openstaxcollege.org/l/multiplycomplex) β’ Multiplying Complex Conjugates (http://openstaxcollege.org/l/multcompconj) β’ Raising i to Powers (http://openstaxcollege.org/l/raisingi) 170 Chapter 2 Equations and Inequ... |
i)(4 β i) 202. (β1 + 2i)(β2 + 3i) 203. (4 β 2i)(4 + 2i) 204. (3 + 4i)(3 β 4i) 205. 206. 3 + 4i 2 6 β 2i 3 207. β5 + 3i 2i 208. 209. 210. 211. 212. 213. 214. 215. 6 + 4i i 2 β 3i 4 + 3i 3 + 4i 2 β i 2 + 3i 2 β 3i β9 + 3 β16 β β4 β 4 β25 2 + β12 2 4 + β20 2 216. i8 217. i15 Chapter 2 Equations and Inequalities 171 218. i... |
2.5.2 Solve quadratic equations by the square root property. 2.5.3 Solve quadratic equations by completing the square. 2.5.4 Solve quadratic equations by using the quadratic formula. Figure 2.32 The computer monitor on the left in Figure 2.32 is a 23.6-inch model and the one on the right is a 27-inch model. Proportion... |
.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 173 The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form, ax2 + bx + c = 0, where a... |
by looking at the possible factors of β6. 1 β
(β6) (β6) β
1 2 β
(β3) 3 β
(β2) The last pair, 3 β
(β2) sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors. 174 Chapter 2 Equations and Inequalities To solve this equation, we use the zero-product property. Set each f... |
a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property. x2 β 9 = 0 (x β 3)(x + 3) = 0 (x β 3) = 0 x = 3 (x + 3) = 0 x = β3 The solutions are x = 3 and x = β3. 2.31 Solve by factoring: x2 β 25 = 0. 176 Chapter 2 Equations and Inequalities Factoring ... |
, x = β3. See Figure 2.34. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 177 Figure 2.34 2.32 Solve using factoring by grouping: 12x2 + 11x + 2 = 0. Example 2.43 Solving a Higher Degree Quadratic Equation by Factoring Solve the equation by factoring: β3x... |
olate the x2 term on one side of the equal sign. 2. Take the square root of both sides of the equation, putting a Β± sign before the expression on the side opposite the squared term. 3. Simplify the numbers on the side with the Β± sign. Example 2.44 Solving a Simple Quadratic Equation Using the Square Root Property Solve... |
. Add β β 2 bβ β 1 2 to both sides of the equal sign and simplify the right side. We have 4. The left side of the equation can now be factored as a perfect square. x2 + 4x + 4 = β1 + 4 x2 + 4x + 4 = 3 5. Use the square root property and solve. x2 + 4x + 4 = 3 (x + 2)2 = 3 (x + 2)2 Β± 3 6. The solutions are x = β2 + 3, x... |
and use parentheses when inserting a negative number. We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by β1 and obtain a positive a. Given ax2 + bx + c = 0, a β 0, we will complete the square as follo... |
the Quadratic Formula Solve the quadratic equation: x2 + 5x + 1 = 0. Solution Identify the coefficients: a = 1, b = 5, c = 1. Then use the quadratic formula. x = β(5) Β± (5)2 β 4(1)(1) 2(1) = β5 Β± 25 β 4 2 = β5 Β± 21 2 Example 2.48 Solving a Quadratic Equation with the Quadratic Formula Use the quadratic formula to solv... |
under the radical in the quadratic formula: b2 β 4ac. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect. 184 Chapter 2 Equations and Inequalities Example 2.49 Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation Use the dis... |
learned in this section to solve for the missing side. The Pythagorean Theorem is given as a2 + b2 = c2 where a and b refer to the legs of a right triangle adjacent to the 90β angle, and c refers to the hypotenuse, as shown in Figure 2.35. This content is available for free at https://cnx.org/content/col11758/1.5 Chap... |
side, having zero on the other side? 237. In the quadratic formula, what is the name of the expression under the radical sign b2 β 4ac, and how does it determine the number of and nature of our solutions? Describe two scenarios where using the square root 238. property to solve a quadratic equation would be the most e... |
using the quadratic formula. If the solutions are not real, state No Real Solution. Chapter 2 Equations and Inequalities 187 271. 2x2 + 5x + 3 = 0 272. x2 + x = 4 273. 274. 275. 276. 2x2 β 8x β 5 = 0 3x2 β 5x + 1 = 0 x2 + 4x + 2 = 0 4 + 1 x β 1 x2 = 0 Technology For the following exercises, enter the expressions into ... |
given as 285. P = 0.2t 2 β 5.6t + 50.2, where t is the time in months from 1999 to 2001. ( t = 1 is January 1999). Find the two months in which the price of the stock was $30. an that Suppose equation given the is 286. p = β2x2 + 280x β 1000, where x represents number of items sold at an auction and p is the profit ma... |
.6 | Other Types of Equations Learning Objectives In this section you will: 2.6.1 Solve equations involving rational exponents. 2.6.2 Solve equations using factoring. 2.6.3 Solve radical equations. 2.6.4 Solve absolute value equations. 2.6.5 Solve other types of equations. We have solved linear equations, rational equa... |
β β = (2)2 = 4 Solve the Equation Including a Variable Raised to a Rational Exponent Solve the equation in which a variable is raised to a rational exponent: x 5 4 = 32. Solution The way to remove the exponent on x is by raising both sides of the equation to a power that is the reciprocal of 5 4, which is 4 5. 5 4 = 3... |
ations and Inequalities Solving Equations Using Factoring We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equat... |
oring: 12x4 = 3x2. Example 2.55 Solve a Polynomial by Grouping Solve a polynomial by grouping: x3 + x2 β 9x β 9 = 0. Solution This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parent... |
containing terms with a variable in the radicand is called a radical equation. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 195 Given a radical equation, solve it. 1. 2. Isolate the radical expression on one side of the equal sign. Put all remaining te... |
οΏ½4 β x β 2β 2 = β β 2x + 3β β β β 2 Subtract x β 2 from both sides. Square both sides. Use the perfect square formula to expand the right side: (a β b)2 = a2 β2ab + b2. β x β 2β β 2x + 3 = (4)2 β 2(4) x β 2 + β 2x + 3 = 16 β 8 x β 2 + (x β 2) 2x + 3 = 14 + x β 8 x β 2 x β 11 = β8 x β 2 (x β 11)2 = β ββ8 x β 2β x2 β 22x... |
solve independently. 2x β 6 = 8 2x = 14 x = 7 or 2x β 6 = β8 2x = β2 x = β1 Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point. Absolute Value Equations The absolute v... |
5 = β10 3x = β5 x = β 5 3 There are two solutions: x = 5, x = β 5 3. (d) |β5x + 10| = 0 The equation is set equal to zero, so we have to write only one equation. β5x + 10 = 0 β5x = β10 x = 2 There is one solution: x = 2. 2.44 Solve the absolute value equation: |1 β 4x| + 8 = 13. Solving Other Types of Equations There ... |
original term. 6. Solve the remaining equation. Example 2.59 Solving a Fourth-degree Equation in Quadratic Form Solve this fourth-degree equation: 3x4 β 2x2 β 1 = 0. Solution This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substituti... |
.61 Solving a Rational Equation Leading to a Quadratic Solve the following rational equation: β4x 8 x2 β 1. Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 2 Equations and Inequalities 201 We want all denominators in factored form to find the LCD. Two of the denominators cann... |
x + 5| = β7 has no solutions. Explain how to change a rational exponent into the 296. correct radical expression. Algebraic For the following exercises, solve the rational exponent equation. Use factoring where necessary. 297. 298. 299. 300. 301. 302. 303. 2 3 = 16 x 3 4 = 27 x 1 2 β x 1 4 = 0 2x (x β 1) 3 4 = 8 (x + 1... |
alities 203 330. 2 β β βx2 β 1 β β β βx2 β 1 β β 12 = 0 + 331. (x + 1)2 β 8(x + 1) β 9 = 0 332. (x β 3)2 β 4 = 0 Extensions the following exercises, solve for For variable. the unknown 333. 334. 335. 336. xβ2 β xβ1 β 12 = 0 |x|2 = x t 25 β t 5 + 1 = 0 |x2 + 2x β 36| = 12 Real-World Applications For the following exerci... |
as x β₯ 4 can be achieved in several ways. We can use a number line as shown in Figure 2.40. The blue ray begins at x = 4 and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4. Figure 2.40 We can use set-builder notation: ... |
< b} {x|a < x β€ b} {x|a β€ x β€ b} (a, b) (a, β) (ββ, b) [a, β) (ββ, bβ€ β¦ β‘ β£a, b) (a, bβ€ β¦ β‘ β£a, bβ€ β¦ All real numbers less than a or greater than b {x|x < a and x > b} (ββ, a) βͺ (b, β) All real numbers Table 2.8 Example 2.62 {x|x is all real numbers} (ββ, β) Using Interval Notation to Express All Real Numbers Greater ... |
symbol. Properties of Inequalities Addition Property If a < b, then a + c < b + c. Multiplication Property If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. These properties also apply to a β€ b, a > b, and a β₯ b. Example 2.64 Demonstrating the Addition Property Illustrate the addition property for in... |
inequality: 13 β 7x β₯ 10x β 4. Solution Solving this inequality is similar to solving an equation up until the last step. 13 β 7x β₯ 10x β 4 13 β 17x β₯ β4 β17x β₯ β17 x β€ 1 Move variable terms to one side of the inequality. Isolate the variable term. Dividing both sides by β17 reverses the inequality. The solution set i... |
the same time. 3 β€ 2x + 2 < 6 1 β€ 2x < 4 1 2 β€ x < 2 We get the same solution: β‘ β£ β β ., 2 1 2 Isolate the variable term, and subtract 2 from all three parts. Divide through all three parts by 2. 2.54 Solve the compound inequality: 4 < 2x β 8 β€ 10. 210 Chapter 2 Equations and Inequalities Example 2.69 Solving a Compou... |
the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph. Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We c... |
an Absolute Value Inequality Solve |x β 1| β€ 3. Solution |x β 1| β€ 3 β3 β€ x β 1 β€ 3 β2 β€ x β€ 4 [β2, 4] Example 2.72 Using a Graphical Approach to Solve Absolute Value Inequalities Given the equation y = β 1 2|4x β 5| + 3, determine the x-values for which the y-values are negative. Solution We are trying to determine w... |
what happened 341. from Step 1 to Step 2: Step 1 Step 2 β2x > 6 x < β 3 342. When solving an inequality, we arrive at Explain what our solution set is. When writing our solution in interval notation, how 343. do we represent all the real numbers? 344. When solving an inequality, we arrive at 357. |3x β 1| > 11 358. |2... |
the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. 373. |x β 1| > 2 374. |x + 3| β₯ 5 375. |x + 7| β€ 4 376. |x β 2| < 7 377. |x β 2| < 0 Chapter 2 Equations and Inequalities 215 For the following exercises, g... |
the volume for a certain gas is given by 403. V = 20T, where V is measured in cc and T is temperature in ΒΊC. If the temperature varies between 80ΒΊC and 120ΒΊC, find the set of volume values. A basic cellular package costs $20/mo. for 60 min of 404. calling, with an additional charge of $.30/min beyond that time.. The c... |
, typically written in x and y, in which two expressions are equal equations in quadratic form equations with a power other than 2 but with a middle term with an exponent that is one- half the exponent of the leading term extraneous solutions any solutions obtained that are not valid in the original equation graph in t... |
in y-values over the change in x-values solution set the set of all solutions to an equation square root property one of the methods used to solve a quadratic equation, in which the x2 term is isolated so that the square root of both sides of the equation can be taken to solve for x volume in cubic units, the volume m... |
sum of the y-coordinates of the endpoints by 2. See Example 2.7 and Example 2.8. 2.2 Linear Equations in One Variable β’ We can solve linear equations in one variable in the form ax + b = 0 using standard algebraic properties. See Example 2.9 and Example 2.10. β’ A rational expression is a quotient of two polynomials. W... |
and Example 2.30. 2.4 Complex Numbers β’ The square root of any negative number can be written as a multiple of i. See Example 2.31. β’ To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See Example 2.32... |
among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See Example 2.49. 2.6 Other Types of Equations β’ Rational exponents can be rewritten several ways de... |
produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See Example 2.70 and Example 2.71. 220 Chapter 2 Equations and Inequalities β’ Absolute value inequalities can also be solved by graphing. At least we can ch... |
) Models and Applications For the following exercises, construct a table and graph the equation by plotting at least three points. 414. y = 1 2 x + 4 415. 4x β 3y = 6 Linear Equations in One Variable For the following exercises, solve for x. 416. 5x + 2 = 7x β 8 417. 3(x + 2) β 10 = x + 4 For the following exercises, w... |
β 5x + 1 = 0 453. 15x2 β x β 2 = 0 For the following exercises, solve the quadratic equation by the method of your choice. 454. (x β 2)2 = 16 455. x2 = 10x + 3 Other Types of Equations For the following exercises, solve the equations. 456. 3 2 = 27 x 457. 1 2 β 4x 1 4 = 0 x 458. 4x3 + 8x2 β 9x β 18 = 0 Quadratic Equat... |
< 3x β 4 CHAPTER 2 PRACTICE TEST 474. Graph the following: 2y = 3x + 4. 475. Find the x- and y-intercepts for the following: 476. Find the x- and y-intercepts of this equation, and sketch the graph of the line using just the intercepts plotted. Find the exact distance between (5, β3) and (β2, 8). Find the coordinates ... |
through the point (2, 1) and is perpendicular to y = β2 5 x + 3. 490. Add these complex numbers: (3 β 2i) + (4 β i). 491. Simplify: β4 + 3 β16. 492. Multiply: 5i(5 β 3i). 493. Divide: 4 β i 2 + 3i. 494. Solve this quadratic equation and write the two complex roots in a + bi form: x2 β 4x + 7 = 0. 495. Solve: (3x β 1)2... |
ation Learning Objectives In this section, you will: 3.1.1 Determine whether a relation represents a function. 3.1.2 Find the value of a function. 3.1.3 Determine whether a function is one-to-one. 3.1.4 Use the vertical line test to identify functions. 3.1.5 Graph the functions listed in the library of functions. A jet... |
, odd} is not paired with exactly one element in the range, {1, 2, 3, 4, 5}. For example, the term βoddβ corresponds to three values from the domain, {1, 3, 5} and the term βevenβ corresponds to two values from the range, {2, 4}. This violates the definition of a function, so this relation is not a function. Figure 3.2... |
In a particular math class, the overall percent grade corresponds to a grade-point average. Is grade-point average a function of the percent grade? Is the percent grade a function of the grade-point average? Table 3.1 shows a possible rule for assigning grade points. 0β56 57β61 62β66 67β71 72β77 78β86 87β91 92β100 0.0... |
as β f of a.β Remember, we can use any letter to name the function; the notation h(a) shows us that h depends on a. The value a must be put into the function h to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication. We can also give an algebraic expression as... |
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