text stringlengths 235 3.08k |
|---|
) b. ( 5 _ −1 w 3 ) _ Simplifying Exponential Expressions Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions. Example 9 Simplifying Exponential Exp... |
ative and associative laws of multiplication The product rule Simplify. The product rule Simplify. The zero exponent rule The power of a product rule The power rule Simplify. The quotient rule and reduce fraction Simplify. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.2 EXPONE... |
× 106 Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right. 0.00000000000047 00000000000004.7 13 places right Be careful not to include the leading 0 in your count. We move the deci... |
): 7,158,000,000 c. World gross national income (April 2014): $85,500,000,000,000 d. Time for light to travel 1 m: 0.00000000334 s e. Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715 Converting from Scientific to Standard Notation To convert a number in scientific notation to standard notati... |
(1.22 × 104) ≈ 4.83 × 1025 atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation! When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consi... |
and associative properties of multiplication Product rule of exponents Scientific notation Commutative and associative properties of multiplication Quotient rule of exponents Scientific notation e. (3.33 × 104)(−1.05 × 107) (5.62 × 105) = [3.33 × (−1.05) × 5.62] (104 × 107 × 105) ≈ (−19.65) (1016) = −1.965 × 1017 Down... |
-to-end. Write the answer in both scientific and standard notations. Access these online resources for additional instruction and practice with exponents and scientific notation. • Exponential Notation (http://openstaxcollege.org/l/exponnot) • Properties of Exponents (http://openstaxcollege.org/l/exponprops) • Zero Exp... |
simplify the given expression. Write answers with positive exponents. x−3 _ 2 25. a3a2 _ a 26. mn2 _ m−2 −5 28. ( y 2 ) 27. ( b3c4) 29. ab 2 ÷ d−3 33. p−4q2 _ p 2q−3 37. 52m ÷ 50m 41. (b−3c) 3 −1 30. (w0x5) 34. (l × w)2 — 2 x ) 38. ( 16 √ _ y−1 42. (x 2y13 ÷ y0) 2 31. m4 ___ n0 3 35. (y7) ÷ x 14 39. 23 _____ (3a)−2 −2... |
152x3 EXTENSIONS For the following exercises, simplify the given expression. Write answers with positive exponents. 32 ) 53. ( __ a3 −2 2 a4 ) ( __ 22 54. (62 − 24) −5 2 x __ ÷ ( ) y 55. m2n3 _____ a2c−3 · a−7n−2 ______ m2c4 56. ( x 6y 3 _____ · x 3y−3 y −7 ___ x −3 ) 10 2 57. ( (ab2c)−3 ) _______ b−3 58. Avogadro’s c... |
the square root of a number is squared, the result is the original number. Since 42 = 16, the square root of 16 is 4. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root. In general terms, if a is a positive real num... |
144 = 13. The order of operations requires us to add the terms in the radicand before finding the square root. Try It #1 a. √ — 225 b. √ — — 81 √ c. √ — 25 − 9 d. √ — 36 + √ — 121 Using the Product Rule to Simplify Square Roots To simplify a square root, we rewrite it such that there are no perfect squares in the radi... |
of multiple radical expressions as a single radical expression. 2. Simplify. Example 3 Using the Product Rule to Simplify the Product of Multiple Square Roots Simplify the radical expression. — 12 ∙ √ — 3 √ Solution — 12 ∙ 3 — 36 √ √ 6 Express the product as a single radical expression. Simplify. Try It #3 Simplify √ ... |
________. — 3a4b5 √ Adding and Subtracting Square Roots We can add or subtract radical expressions only when they have the same radicand and when they have the same — 2. However, it is often possible to radical type such as square roots. For example, the sum of √ — simplify radical expressions, and that may change the ... |
4 √ = 14(2) ∣ a ∣ b2 √ = 28 ∣ a ∣ b2 √ — 2ac — a √ 2ac — Now the terms have the same radicand so we can subtract. — — 120 ∣ a ∣ b2 √ 2ac − 28 ∣ a ∣ b2 √ 2ac = 92 ∣ a ∣ b2 √ — 2ac Try It #7 Subtract 3 √ — 80x − 4 √ — 45x. Rationalizing Denominators When an expression involving square root radicals is written in simples... |
It #8 Write — 3 12 √ ______ in simplest form. — 2 √ CHAPTER 1 PREREQUISITES How To… Given an expression with a radical term and a constant in the denominator, rationalize the denominator. 1. Find the conjugate of the denominator. 2. Multiply the numerator and denominator by the conjugate. 3. Use the distributive prope... |
one nth root, then the principal nth root of a, written as the same sign as a that, when raised to the nth power, equals a. The index of the radical is n. n — a, is the number with Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.3 RADICALS AND RATIONAL EXPRESSIONS 37 Example 10... |
as the radicand, raise the radicand to the power and use the root as the index. Example 11 Writing Rational Exponents as Radicals 2 __ Write 343 as a radical. Simplify. 3 Solution The 2 tells us the power and the 3 tells us the root. 2 = ( _ 343 3 √ 3 — 343 ) 2 = √ 3 — 3432 Download the OpenStax text for free at http:... |
(http://openstaxcollege.org/l/simpradical) • Rationalize Denominator (http://openstaxcollege.org/l/rationdenom) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.3 SECTION EXERCISES 39 1.3 SECTION EXERCISES VERBAL 1. What does it mean when a radical does not have 2. Where would r... |
p2q6) 2 289 — 5 _ 2 39. m √ ______ 225x3 _ 49x ____ 32 _ 14d 43. √ 47. √ 40. 9 √ — 3m2 + √ — 27 41. 3 √ — ab2 − b √ — a 44. 3 √ — 44z + √ — 99z 45. √ — 50y8 3 _ 48. q √ 2 — 63p 49. — √ 8 ________ 1 − √ 3x — 3 _ 2 51. w √ — 3 _ 32 − w 2 √ — 50 52. √ — 108x4 + √ — 27x4 53. — 12x √ ________ 3 2 + 2 √ — 55. √ — 125n10 56. ... |
___________ − 16 68. 4 _________ 1 _ 3 8 66. A car accelerates at a rate of 6 − — 4 √ ____ m/s2 where t t √ — is the time in seconds after the car moves from rest. Simplify the expression. — 69. mn3 √ ________ · c−3 a2 √ — a−7n−2 _______ m2c4 √ — 70. a _______ a − √ c — 71. — — 64y + 4 √ x √ __ 128y √ — y — — 72. ( 25... |
x − x ft 2, or 4x 2 + subtracting the area of the rectangle. When we do this, we get 4x 2 + x ft 2. 2 2 In this section, we will examine expressions such as this one, which combine several variable terms. Identifying the Degree and Leading Coefficient of Polynomials The formula just found is an example of a polynomial,... |
coefficient. How To… Given a polynomial expression, identify the degree and leading coefficient. 1. Find the highest power of x to determine the degree. 2. Identify the term containing the highest power of x to find the leading term. 3. Identify the coefficient of the leading term. Example 1 Identifying the Degree and... |
. 4x 3 + 20x 2 + 4x − 1 Simplify. Analysis We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make t... |
/content/col11759/latest. 44 CHAPTER 1 PREREQUISITES How To… Given the multiplication of two polynomials, use the distributive property to simplify the expression. 1. Multiply each term of the first polynomial by each term of the second. 2. Combine like terms. 3. Simplify. Example 4 Multiplying Polynomials Using the Di... |
the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.4 POLYNOMIALS 45 Example 5 Using FOIL to Multiply Binomials Use FOIL to find the product. (2x − 10)(3x + 3) Solution Find the product of the first terms. 2x − 18 3x + 3 2x · 3x = 6x2 Find the product of the outer terms. 2x − 18 3x + 3 2x ·... |
Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 46 CHAPTER 1 PREREQUISITES How To… Given a binomial, square it using the formula for perfect square trinomials. 1. Square the first term of the binomial. 2. Square the last term of the binomial. 3. For the middle term of the trinomial, doub... |
binomial with the same terms but the opposite sign, find the difference of squares. 1. Square the first term of the binomials. 2. Square the last term of the binomials. 3. Subtract the square of the last term from the square of the first term. Download the OpenStax text for free at http://cnx.org/content/col11759/late... |
these online resources for additional instruction and practice with polynomials. • Adding and Subtracting Polynomials (http://openstaxcollege.org/l/addsubpoly) • Multiplying Polynomials (http://openstaxcollege.org/l/multiplpoly) • Special Products of Polynomials (http://openstaxcollege.org/l/specialpolyprod) Download ... |
binomial. 24. (4x + 5)2 28. (2m − 3)2 25. (3y − 7)2 29. (3y − 6)2 For the following exercises, multiply the binomials. 32. (9a − 4)(9a + 4) 31. (4c + 1)(4c − 1) 36. (14p + 7)(14p − 7) 35. (4 + 4m)(4 − 4m) 7. −625a8 + 16b4 8. 200p − 30p2m + 40m3 12. (4z3 + 8z2 − z) + (−2z2 + z + 6) 14. (7a3 + 6a2 − 4a − 13) + (−3a3 − 4... |
) 46. (4x − 6y) (6x − 4y) 49. (4t − x)(t − x + 1) 52. (x + y) (x2 − xy + y2) REAL-WORLD APPLICATIONS 53. A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: (4x + 1)(8x − 3) where x is measured in meters. Multiply the binomials to find the ar... |
area of A = s2 = 42 = 16 units 2. The other rectangular region has one side of length 10x − 8 and one side of length 4, giving an area of A = lw = 4(10x − 8) = 40x − 32 units 2. So the region that must be subtracted has an area of 2(16) + 40x − 32 = 40x units 2. The area of the region that requires grass seed is found... |
3 + 45x 2y 2 + 21xy. Solution First, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of x3, x 2, and x is x. (Note that the GCF of a set of expressions in the form xn will always be the exponent of lowest degree.) And the GCF of y3, y 2, and y is y. Combine these to find the GCF of the polynomia... |
where pq = c and p + q = b. Q & A… Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.5 FACTORING POLYNOMIALS 51 How To… Given a trinomial in... |
3(x + 1). We then pull out the GCF of (x + 1) to find the factored expression. factor by grouping To factor a trinomial in the form ax 2 + bx + c by grouping, we find two numbers with a product of ac and a sum of b. We use these numbers to divide the x term into the sum of two terms and factor each portion of the expr... |
Perfect Square Trinomial A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term. a2 + 2ab + b2 = (a + b)2 and a2 − 2ab + b2 ... |
that the first and last term are perfect squares. 2. Write the factored form as (a + b)(a − b). Example 5 Factoring a Difference of Squares Factor 9x 2 − 25. Solution Notice that 9x 2 and 25 are perfect squares because 9x 2 = (3x)2 and 25 = 52. The polynomial represents a difference of squares and can be rewritten as ... |
factored form as (a + b)(a2 − ab + b2). For a difference of cubes, write the factored form as (a − b)(a2 + ab + b2). Example 6 Factoring a Sum of Cubes Factor x3 + 512. Solution Notice that x 3 and 512 are cubes because 83 = 512. Rewrite the sum of cubes as (x + 8)(x2 − 8x + 64). Analysis After writing the sum of cube... |
+ 2) − (7x + 8) 3 Factor out the GCF. Simplify. Try It #8 3 1 __ _ + 7a (5a − 1) −. Factor 2(5a − 1) 4 4 Access these online resources for additional instruction and practice with factoring polynomials. • Identify GCF (http://openstaxcollege.org/l/findgcftofact) • Factor Trinomials when a Equals 1 (http://openstaxcolle... |
25b − 247 19. 9d 2 −73d + 8 20. 90v2 −181v + 90 21. 12t2 + t − 13 22. 2n2 − n − 15 24. 25y2 − 196 25. 121p2 − 169 26. 4m2 − 9 23. 16x2 − 100 27. 361d2 − 81 28. 324x2 − 121 29. 144b2 − 25c2 30. 16a2 − 8a + 1 31. 49n2 + 168n + 144 32. 121x2 − 88x + 16 33. 225y2 + 120y + 16 34. m2 − 20m + 100 35. m2 − 20m + 100 36. 36q2 ... |
and width of the park are perfect factors of the area. l × w = 98x 2 + 105x − 27 51. Factor by grouping to find the length and width of the park. 53. At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is 9x2 − 25 m2. Factor the area to find the lengths of... |
omial expressions. Simplifying Rational Expressions The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first n... |
ly the rational expressions and show the product in simplest form: x 2 + 4x − 5 __ ∙ 3x + 18 2x − 1 _______ x + 5 Solution (2x − 1) (x + 5)(x − 1) _______ ____________ ∙ (x + 5) 3(x + 6) (x + 5)(x − 1)(2x − 1) ___________________ 3(x + 6)(x + 5), (x + 5) (x − 1)(2x − 1) ___________________ 3(x + 6), (x + 5) (x − 1)(2x ... |
) (2x − 3)(x + 2) __ ∙ (x + 1)(x − 1) (2x − 3)(x + 2)(x + 1)(x + 1) ___ (x + 1)(x − 1)(x + 2)(x − 2) (2x − 3)(x + 2)(x + 1)(x + 1) ___ (x + 1)(x − 1)(x + 2)(x − 2) (2x − 3)(x + 1) __ (x − 1)(x − 2) Rewrite as multiplication. Factor the numerator and denominator. Multiply numerators and denominators. Cancel common facto... |
EXPRESSIONS 61 How To… Given two rational expressions, add or subtract them. 1. Factor the numerator and denominator. 2. Find the LCD of the expressions. 3. Multiply the expressions by a form of 1 that changes the denominators to the LCD. 4. Add or subtract the numerators. 5. Simplify. Example 4 Adding Rational Expres... |
PREREQUISITES Try It #4 Subtract the rational expressions −#3 Simplifying Complex Rational Expressions A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator... |
://openstaxcollege.org/l/multdivratex) • Add and Subtract Rational Expressions (http://openstaxcollege.org/l/addsubratex) • Simplify a Complex Fraction (http://openstaxcollege.org/l/complexfract) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 64 CHAPTER 1 PREREQUISITES 1.6 SECTION EXERCI... |
19h + 24 h2 − 16h + 64 _____________ 5h2 − 37h − 24 18. 6b2 + 13b + 6 ____________ · 4b2 − 9 6b2 + 31b − 30 _____________ 18b2 − 3b − 10 19. 2d 2 + 15d + 25 _____________ · 4d 2 − 25 2d 2 − 15d + 25 _____________ 25d 2 − 1 20. 6x 2 − 5x − 50 ______________ · 15x 2 − 44x − 20 20x 2 − 7x − 6 ____________ 2x 2 + 9x + 10 ... |
d 2 − 15d + 2 3d 2 + 29d − 44 _____________ 9d 2 − 15d + 4 29. 144b2 − 25 _____________ ÷ 72b2 − 6b − 10 18b2 − 21b + 5 ______________ 36b2 − 18b − 10 31. 22y2 + 59y + 10 ______________ ÷ 12y2 + 28y − 5 11y2 + 46y + 8 _____________ 24y2 − 10y + 1 Download the OpenStax text for free at http://cnx.org/content/col11759/la... |
10x + 25 ft 2. Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard. 53. Aaron wants to mulch his garden. His garden is x2 + 18x + 81 ft 2. One bag of mulch covers x2 − 81 ft 2. Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.... |
two numbers may be multiplied in any order without affecting the result; in symbols, a · b = b · a constant a quantity that does not change value degree the highest power of the variable that occurs in a polynomial difference of squares the binomial that results when a binomial is multiplied by a binomial with the sam... |
be expressed as a fraction of two integers leading coefficient the coefficient of the leading term leading term the term containing the highest degree least common denominator the smallest multiple that two denominators have in common Download the OpenStax text for free at http://cnx.org/content/col11759/latest. CHAPT... |
a ∙ b)n = an ∙ bn a __ ) ( b n = an __ bn perfect square trinomial (x + a)2 = (x + a)(x + a) = x2 + 2ax + a2 difference of squares difference of squares (a + b)(a − b) = a2 − b2 a2 − b2 = (a + b)(a − b) perfect square trinomial a2 + 2ab + b2 = (a + b)2 sum of cubes difference of cubes a3 + b3 = (a + b)(a2 − ab + b2) a3... |
simplified by multiplying exponents. See Example 3. • An expression with exponent zero is defined as 1. See Example 4. • An expression with a negative exponent is defined as a reciprocal. See Example 5 and Example 6. • The power of a product of factors is the same as the product of the powers of the same factors. See ... |
the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See Example 1. • We can add and subtract polynomials by combining like terms. See Example 2 and Example 3. • To multiply polynomials, use the distributive... |
ES REAL NUMBERS: ALGEBRA ESSENTIALS For the following exercises, perform the given operations. 1. (5 − 3 · 2)2 − 6 2. 64 ÷ (2 · 8) + 14 ÷ 7 3. 2 · 52 + 6 ÷ 2 For the following exercises, solve the equation. 4. 5x + 9 = −11 5. 2y + 42 = 64 For the following exercises, simplify the expression. 6. 9(y + 2) ÷ 3 · 2 + 1 7. ... |
− 2y − 5) 37. (2x 2 + 3x − 6) + (3x 2 − 4x + 9) 38. (6a2 + 3a + 10) − (6a2 −3a + 5) 39. (k + 3)(k − 6) 41. (x + 1)(x 2 + 1) 43. (a + 2b)(3a − b) 40. (2h + 1)(3h − 2) 42. (m − 2)(m2 + 2m − 3) 44. (x + y)(x − y) FACTORING POLYNOMIALS For the following exercises, find the greatest common factor. 45. 81p + 9pq − 27p2q 2 4... |
6c + 12d _ dc 64. d − 4 ______ · d 2 − 9 d − 3 _______ d 2 − 16 66. 4d 2 − 7d − 2 _____________ 6d 2 − 17d + 10 ÷ 8d 2 + 6d + 1 ____________ 6d 2 + 7d − 10 68. 12 __________ − a2 + 2a + 1 3 _____ a2 −1 70. 3 7 _ y _ x − _______ 2 _ x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 7 2 CH... |
− + 2(x − 6) 4 4 For the following exercises, simplify the expression. 27. 2z2 + 7z + 3 ___________ · z2 − 9 4z2 − 15z + 9 ____________ 4z2 − 1 x 2 _ x _ y + 28. 29. a 2b ___ ___ − 9a 2b ________ 3a − 2b _______ 6a Download the OpenStax text for free at http://cnx.org/content/col11759/latest. Equations and Inequalitie... |
In this section, we will learn how to use grid lines to describe locations and changes in locations. Plotting Ordered Pairs in the Cartesian Coordinate System An old story describes how seventeenth-century philosopher/mathematician René Descartes invented the system that has become the foundation of algebra while sick... |
-coordinate, or horizontal displacement from the origin, and its y-coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form (x, y). An ordered pair is also known as a coordinate pair because it consists of x- and y-coor... |
the point (0, −3), begin again at the origin. The x-coordinate is 0. This tells us not to move in either direction along the x-axis. The y-coordinate is –3, so move three units down in the negative y direction. See the graph in Figure 5. (–2, 4) –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 (3, 3) 321 4 5 x (0, –3) Figure... |
can connect them. See Figure 6. This is not true for all equations. (4, 7) (3, 5) (2, 3) (1, 1) 321 (0, –11 –2 –3 –4 –5 –6 –7 –5 –4 –3 –2 –1 (–1, –3) (–2, –5) (–3, –7) Figure 6 Note that the x-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require partic... |
/latest. 7 8 CHAPTER 2 EQUATIONS AND INEQUALITIES Now, plot the points. Connect them if they form a line. See Figure 7. (–5, 7) (–3, 5) (–1, 30, 2) (1, 1) –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 321 (3, –1) 4 5 6 7 (5, –3) Figure 7 x Try It #1 1 _ x + 2. Construct a table and graph the equation by plotting points: y = 2 Graphing... |
We can clearly view the intercepts in the new window. Example 3 Using a Graphing Utility to Graph an Equation 2 4 _ _ Use a graphing utility to graph the equation: y = − x −. 3 3 Solution Enter the equation in the y = function of the calculator. Set the window settings so that both the x- and y-intercepts are showing ... |
intercepts. Solution Set y = 0 to find the x-intercept. Set x = 0 to find the y-intercept. y = −3x − 4 0 = −3x − 4 4 = −3x 3x − 4 y = −3(0) − 4 y = −4 (0, −4) x−intercept y−intercept Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 80 CHAPTER 2 EQUATIONS AND INEQUALITIES Plot both points,... |
)2 + (y2 − y1)2 → d = √ —— (x2 − x1)2 + (y2 − y1)2 the distance formula Given endpoints (x1, y1) and (x2, y2), the distance between two points is given by d = √ —— (x2 − x1)2 + (y2 − y1)2 Example 5 Finding the Distance between Two Points Find the distance between the points (−3, −1) and (2, 3). Solution Let us first lo... |
, 1). After that, she traveled 3 blocks east and 2 blocks north to (8, 3). Lastly, she traveled 4 blocks north to (8, 7). We can label these points on the grid as in Figure 15. y Schiller Avenue (8, 7) Mannhelm Road (8, 3) Wolf Road x Bertau Avenue McLean Street North Avenue (1, 1) (5, 1) (0, 0) Figure 15 Download the ... |
)2 = √ — 64 + 49 = √ — 113 ≈ 10.63 units At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point (8... |
of its diameter. Thus, the midpoint formula will yield the center point. x1 + x2 ( _______, 2 y1 + y2 ) _______ 2 −4 −4 −1 + 5 ( ______ _______, 2 2 8 4 ) = (2, −4) ) = ( __ __, − 2 2 Access these online resources for additional instruction and practice with the Cartesian coordinate system. • Plotting Points on the Co... |
. (−4, 1) and (3, −4) 20. (−4, 3) and (10, 3) 18. (2, −5) and (7, 4) 19. (5, 0) and (5, 6) 21. Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth. (19, 12) and (41, 71) For each of the following exercises, find the coordinates of the midpoint of the line... |
Name the quadrant in which the following points would be located. If the point is on an axis, name the axis. a. (−3, −4) b. (−5, 0) c. (1, −4) d. (−2, 7) e. (0, −3) For each of the following exercises, construct a table and graph the equation by plotting at least three points. 1 _ 37. 2y = x + 3 35. y = x + 2 3 36. y ... |
the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in betwe... |
coordinates of the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length. midpoint for each diagonal. (–6, 5) y 6 5 4 3 2 1 (10, 5) –5 –7 –4 (–6, –1) –3 –2 –1 –1 –2 –3 321 4 5 6 7 8 9 11 (10, –1) x REAL-WORLD APPLICATIONS 60. The coordinates on a map for San Francisco are (53, 17) ... |
or neither. • Find the equation of a line parallel or perpendicular to a gien line. 2.2 LINEAR EQUATIONS IN ONE VARIABLE Figure 1 y Figure 1 x Solving Linear Equations in One Variable linear equationTh ax+b = identity equation x =x +x solution set x conditional equation x +=x − x +=x − x =− x =− Th− inconsistent equat... |
2x + 1 = −9. Example 2 Solving an Equation Algebraically When the Variable Appears on Both Sides Solve the following equation: 4(x − 3) + 12 = 15 − 5(x + 6). Solution Apply standard algebraic properties. 4(x − 3) + 12 = 15 − 5(x + 6) 4x − 12 + 12 = 15 − 5x − 30 Apply the distributive property. 4x = −15 − 5x Combine li... |
rs; 2x, 3x, and 3. The LCD must contain 2x, 3x, and 3. An LCD of 6x contains all three denominators. In other words, each denominator can be divided evenly into the LCD. Next, multiply both sides of the equation by the LCD 6x. − 7 5 22 ) = ( (6x) ( ) (6x) _ _ _ 2x 3x 3 5 7 22 ) (6x) ) = ( ) − (6x) ( (6x) ( _ _ _ 3 3x 2... |
content/col11759/latest. 90 CHAPTER 2 EQUATIONS AND INEQUALITIES Another example is a problem with two denominators, such as x and x 2 + 2x. Once the second denominator is factored as x 2 + 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 i... |
1} written in set notation. Try It #3 Solve the rational equation: 2 _ 3x 1 _ − = 4 1 _. 6x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 91 Example 5 Solving a Rational Equation by Factoring the Denominator 1 1 _ _ x = Solve the following ra... |
text for free at http://cnx.org/content/col11759/latest. 92 CHAPTER 2 EQUATIONS AND INEQUALITIES c. The least common denominator is 2(x − 2). Multiply both sides of the equation by x(x − 2). 2(x − 2(x − 2) _ − 2 2x = 10 − (x − 2) 2x = 12 − x 3x = 12 x = 4 The solution is 4. The excluded value is 2. Try It #5 Solve −3 ... |
the slope increases, the line becomes steeper. Some examples are shown in Figure 2. The lines indicate the following slopes: m = −3, 1 _ m = 2, and m =. 3 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 93 y y = 2x + 1 1 y = x + 2 3 42 6 8 10 ... |
. Or, just substitute x = 0 and solve for y. The Point-Slope Formula Given the slope and one point on a line, we can find the equation of the line using the point-slope formula. y − y1 = m(x − x1) This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equati... |
_ (x − 3. Distribute the 3 7 _ x − 3. In slope-intercept form, the equation is written as y = 3 Analysis To prove that either point can be used, let us use the second point (0, −3) and see if we get the same equation. 7 _ y − (−3) = (x − 0 We see that the same line will be obtained using either point. This makes sense... |
3), and (−3, 5). First, we will find the slope. m = 5 − 3 2 _________ __ = −3 − (−3) 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 ... |
-intercepts. Lines that are parallel to each other will never intersect. For example, Figure 4 shows the graphs of various lines with the same slope, m = 2. y = 2x − 3 y = 2x + 1 y = 2x + 5 21 1 –1 –2 –3 –4 –5 –5 –4 –3 –2 Figure 4 Parallel lines All of the lines shown in the graph are parallel because they have the sam... |
lines are perpendicular. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 98 CHAPTER 2 EQUATIONS AND INEQUALITIES Try It #11 Graph the two lines and determine whether they are parallel, perpendicular, or neither: 2y − x = 10 and 2y = x + 4. Writing the Equations of Lines Parallel or Perpe... |
the point (−1, −2). Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 99 Example 16 Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point Find the equation of the line perpendicular to 5x − 3y + 4 = 0 and pas... |
= − x−x− x = + 21. x 22. 23. −− 24. − 25. −− 26. − 27. 30. 33. 36. 39. 28. 1 3 31. 34. 37. 40. 29. 32. 35. 38. 41. 42. 44. 46. − −− 43. − 45. − 47. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.2 SECTION EXERCISES 101 48. −− 49. 50. 52. (,) 51. − 53. −− GRAPHICAL 54. y =x + x... |
an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems. Setting up a Linear Equation to Solve a Real-World Application To set up or model a linear equation to fit a real... |
Solve the equation. Be sure the solution can be explained in words, including the units of measure. Example 1 Modeling a Linear Equation to Solve an Unknown Number Problem Find a linear equation to solve for the following unknown quantities: One number exceeds another number by 17 and their sum is 31. Find the two num... |
the $92 monthly cost offered by Company A when the average number of minutes used each month is 1,160. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 104 CHAPTER 2 EQUATIONS AND INEQUALITIES c. If the average number of minutes used each month is 420, we have the following: Company A = 0... |
; and the volume of a rectangular solid, V = LWH. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time. Example 3 Solving an Application Using a Formula It takes Andrew 30 min to drive to work in the morning. He drives home using the same rou... |
, it took Jennifer 3.6 h to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 h to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday? Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 106 ... |
W + 12 + 2W 48 = 4W + 12 36 = 4W 9 = W (9 + 6) = L 15 = L Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.3 MODELS AND APPLICATIONS 107 Now, we find the area given the dimensions of L = 15 in. and W = 9 in. A = LW A = 15(9) = 135 in.2 The area is 135 in.2. Try It #5 A game room ... |
ISES VERBAL 1. To set up a model linear equation to fit real-world applications, what should always be the first step? 2. Use your own words to describe this equation where n is a number: 5(n + 3) = 2n 3. If the total amount of money you had to invest was 4. If a man sawed a 10-ft board into two sections and $2,000 and... |
$40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use P for the number of devices that need data plans as part of their cost. 13. ... |
His intent is to earn 11% interest on his investment. He can invest part of his money at 8% interest and part at 12% interest. How much does Paul need to invest in each option to make get a total 11% return on his $20,000? For the following exercises, use this scenario: A truck rental agency offers two kinds of plans.... |
previous question to find m when the coordinates of the point are (4, 7) and b = 12. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 110 CHAPTER 2 EQUATIONS AND INEQUALITIES 1 _ h(b1 + b2). 38. The area of a trapezoid is given by A = 2 Use the formula to find the area of a trapezoid with... |
a height of 36 and a volume of 324π. 54. Solve the formula from the previous question for π. Notice why π is sometimes defined as the ratio of the circumference to its diameter. 51. Solve for r: V = πr2h 53. The formula for the circumference of a circle is C = 2πr. Find the circumference of a circle with a diameter of... |
number as a multiple of i. Consider the square root of −49. We use 7i and not −7i because the principal root of 49 is the positive root. √ — −49 = √ = √ = 7i — 49 ċ (−1) 49 √ −1 — — A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + bi ... |
is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs (a, b), where a represents the coordinate for the horizontal axis and b represents the coordinate for the v... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.