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2) {4, 5} 2. The product (2a 3)(2a 3) can be written as (3) {4, 5, 6} (4) {5, 6} (1) 2a2 9 (2) 4a2 9 (3) 4a2 9 (4) 4a2 12a 9 3. When 0.00034 is written in the form 3.4 10n, the value of n is (4) 4 (1) 3 (2) 4 (3. When (1) 5, x equals (2) 1 (3) 1 2 5. The mean of the set of even integers from 2 to 100 is (1) 49 (2) 50 (...
III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. The lengths ...
–55 verbal phrases involving, 89 Addition method in solving systems of linear equations, 416–420 Addition property of equality, 118 of inequality, 147–148 of zero, 49–50 Additive identity, 49–50 Additive inverse (opposite), 50 Adjacent angles, 250 Adjacent side of an angle, 307 Algebraic equation, 565 Algebraic express...
(postulate), 246 Axis, graphing line parallel to, 352–353 Axis of symmetry of the parabola, 510 B Bar (fractional line), 41 Base(s), in geometry, angles of an isosceles triangle, 265 of a prism, 282 of a trapezoid, 275 of an isosceles triangle, 265 in number theory, division of powers with the same, 186–187 of exponen...
195 squaring numbers, 18, 470–472 probability, combinations, 642 factorials, 628–629 permutations, 631–634 statistics, 1-Var Stats, 684, 695 box-and-whisker plot, 700 five statistical summary, 700 frequency histogram, 676–678 line of best fit, 715–716 mean, 684 median, 684 scatter plot, 711–712 trigonometry, cosine rat...
11–715 Corresponding angles, 259 parallel lines and, 259–260 Cosine of an angle, 317 Cosine ratio, 318 applications of, 323–324 finding, on calculator, 319–320 Counting numbers, 2, 11 Counting principle, 610–611, 612 Cross-multiplication, 217 Cross-product, 217 Cube root, 472 Cumulative frequency, 702–703 Cumulative fr...
Disjoint sets, 72 Distance formula, 136 Distributive property, 48–49 of multiplication over addition or subtraction, 168, 178, 183 Division, 4, 38 of fractions, 548–549 of a monomial by a monomial, 197–198 of a polynomial by a binomial, 200–201 by a monomial, 198–199 of powers that have the same base, 186–187 of signe...
double-blind, 664 in probability, 579–581 single-blind, 663 Exponent(s), 39, 40, 95–96 negative integral, 189–190 zero, 188–189 Exponential decay, 389 Exponential function, 387–391 exponential decay, 389–391 exponential growth, 388–391 Exponential growth, 388 Expression. See Algebraic expression(s); numerical expressi...
ic, 540 common, 13 decimal, 13 division of, 548–549 division property of, 541 equivalent, 13–14, 541, 216 lowest terms, 541 multiplication of, 545–547 multiplication property of a, 542 reducing to lowest terms, 541–543 subtraction of, 550–554 writing probability of events as, 586 Frequency distribution table, 668 Frequ...
of, 151–155 with fractional coefficients, 562–563 graphing first-degree, in one variable, 151–155 in two variables, 378–381 systems of, 431–434 in problem solving, 157–159 properties of, 146–149 addition, 147–148 multiplication, 148–149 order, 146–147 transitive property, 147 solution set of, 151 symbols of, 7–8 verba...
triangles, 263, 307 Letters, using, to represent numbers, 89–90 Length of a line segment, 247 Like radicals, 487 addition of, 487 subtraction of, 487 Like terms (similar terms), 123, 168 Line, in geometry, 246–247. See also Linear function Line of best fit, 715–720 Line segment, 247 Linear equation, 347. See also Firs...
Monomial square roots, division of, 494–495 Multiplication, 3–4, 38 associative property of, 48 of binomials, 454–455 commutative property of, 47 distributive property of, over addition, 49, 168, 178, 183 of fractions, 545–547 grouping symbols and, 179–180 of monomial by a monomial, 177–178 of polynomials, 183–184 by ...
42–43 O Obtuse angle, 249 Obtuse triangle, 263 One, multiplication property of, 50 Open sentence, 104–105 Index 735 Operation(s) binary, 38 computations with more than one, 41 order of, 38–43 properties of, 45–52 with sets, 71–73 solving equations using more than one, 117–121 Operational symbols, 3 Opposite angles, 27...
equation given slope and one, 402 writing equations given two, 404–405 Polygon(s), 78, 262 angles and, 275 area of irregular, 279–280 graphing, 78–79 sides of, 262 vertices of, 262 permutations and, 627–634 as sums, 605–606 theoretical, 584–587 with two or more activities with replacement, 617–618 without replacement,...
, 50 associative of addition, 47–48 of multiplication, 48 closure, 45–47 commutative of addition, 47 of multiplication, 47 distributive, 48–49 multiplication property of one, 50 multiplication property of zero, 52 multiplicative inverses, 51–52 Proportion, 216–220 cross products in, 217 extremes of, 216 general form of...
properties of, 12–13 set of, 11–15 Ratio(s), 208 constant, 222 continued, 209–210 equivalent, 208–209 expression of, in simplest form, 208–209 using, to express a rate, 212 verbal problems involving, 214–215 Ray(s), 247–248 opposite, 248 Real number line, 25 Real numbers, 2, 25–26 completeness property of, 25 ordering...
joint, 72 empty, 3, 72 finite, 3, 339–340 infinite, 3, 340–344 of integers, 4–5 of irrational numbers, 17–18 null, 3 operations with, 71–73 intersection of, 71–72, 597 graphing, 153–154 complement of, 73 union of, 72–73 of rational numbers, 11–15 of real numbers, 25 types of, 3 graphing, 154–155 universal, 71 of whole ...
, 494–495 multiplication of, 491–492 simplest form of, 485 simplifying, 484–485 Standard deck of cards, 580 Standard form, 503 first-degree equation in, 347 Standard number line, 6 Statistical summary (five statistical summary), 699 Statistics, 660–730 bivariate data, 710–720 correlation, 711–715 causation, 711–715 lin...
293 of rectangular solids, 283 of solids, 282–284 Symbol(s) approximately equal to (), 20 factorial (!), 628 grouping, 41–43 of inequality, 7–8 infinity (∞), 151–152 is an element of (∈), 338 is not an element of (βˆ‰), 338 minus (), 61–62 for numbers, 2 operational, 3 for permutations, 631 translating verbal phrases in...
ratio in, 307–317 Trinomial(s), 170 factoring, 457–460 Turning point, 509 Two-valued statistics, 710. See also Bivariate statistics U Undefined term, 246 Unfavorable event, 584 Uniform probability, 586 Union of sets, 72–73 graphing, 154–155 Unit measure, 6 Unit price, 212 Unit rate, 212 of change, 356 Units of measure...
1.2 Practice - Two-Step Problems Solve each equation. 1) 5 + n 4 = 4 3) 102 = 7r + 4 βˆ’ 8n + 3 = 5) βˆ’ 7) 0 = 6v βˆ’ 8 = x 9) βˆ’ 11 βˆ’ 13) 12 + 3x = 0 βˆ’ 15) 24 = 2n 8 βˆ’ 12 + 2r 17) 2 = βˆ’ 3 + 7 = 10 19) b 77 βˆ’ 6) 8) βˆ’ βˆ’ 4 βˆ’ 2 + x 2) 2 = 2m + 12 βˆ’ βˆ’ 4) 27 = 21 3x βˆ’ b = 8 2 = 4 10) 12) 14) 16) βˆ’ βˆ’ βˆ’ βˆ’ 18) βˆ’ 20) x 5 = a 1 4 βˆ’ 6...
. Example 59. 4(2x 8x βˆ’ βˆ’ 6) = 16 Distribute 4 through parenthesis 24 = 16 Focus on the subtraction first + 24 + 24 Add 24 to both sides 8x = 40 Now focus on the multiply by 8 8 Divide both sides by 8 8 x = 5 Our Solution! Often after we distribute there will be some like terms on one side of the equation. Example 2 sho...
(8) + 10 Multiply 4(8) and 2(8) first βˆ’ 6 = 16 + 10 βˆ’ 26 = 26 Add and Subtract True! The next example illustrates the same process with negative coefficients. Notice first the smaller term with the variable is moved to the other side, this time by adding because the coefficient is negative. 38 Example 63. 3x + 9 = 6x βˆ’ + 3x ...
on each side of the equation. 3. Get the variables on one side by adding or subtracting 4. Solve the remaining 2-step equation (add or subtract then multiply or divide) 5. Check your answer by plugging it back in for x to find a true statement. The order of these steps is very important. World View Note: The Chinese de...
left with a true βˆ’ statement, this indicates that the equation is always true, no matter what x is. Thus, for our solution we say all real numbers or R. βˆ’ Example 68. 2(3x 6x βˆ’ βˆ’ βˆ’ 5) 10 2x 2x βˆ’ βˆ’ βˆ’ 4x = 2x + 7 Distribute 2 through parenthesis 4x = 2x + 7 Combine like terms 6x 10 = 2x + 7 Notice the variable is on bot...
v + 3) + 8v = 5v βˆ’ 4(1 βˆ’ βˆ’ 6v) 2) = 4 6(x 1) βˆ’ βˆ’ 6(8k + 4) = βˆ’ βˆ’ 8(6k + 3) βˆ’ 7p) = 8(p 7) βˆ’ 2(1 βˆ’ 2 βˆ’ 2) 2( 3n + 8) = 20 βˆ’ 4 + 3x) = 34 βˆ’ 4) 2 8( βˆ’ 6) 32 = 2 βˆ’ βˆ’ 8) 10) 12) (3 3n βˆ’ βˆ’ βˆ’ 14) 56p βˆ’ 5( 4n + 6) βˆ’ 55 = 8 + 7(k βˆ’ 5) βˆ’ 5n) = 12 27 = βˆ’ 48 = 6p + 2 βˆ’ 27 3n βˆ’ 16) 4 + 3x = 12x + 4 βˆ’ 16n + 12 = 39 7n 18) βˆ’ 20) 17 2...
equations we will need to work with an equation with fraction coefficients. We can solve these problems as we have in the past. This is demonstrated in our next example. Example 69 Focus on subtraction Add 7 2 to both sides Notice we will need to get a common denominator to add 5 6 + 7 common denominator of 6. So we bui...
2)2x βˆ’ (6)2 = (3)3x + (1)1 Multiply out each term 4x 4x βˆ’ βˆ’ βˆ’ 12 = 9x + 1 Notice variable on both sides Subtract 4x from both sides Focus on addition of 1 Subtract 1 from both sides Focus on multiplication of 5 Divide both sides by 5 4x 12 = 5x + 1 1 1 βˆ’ 13 = 5x 5 5 13 5 = x Our Solution βˆ’ βˆ’ βˆ’ βˆ’ We can use this same pr...
βˆ’ 2 2 Our Solution βˆ’ World View Note: The Egyptians were among the first to study fractions and linear equations. The most famous mathematical document from Ancient Egypt is the Rhind Papyrus where the unknown variable was called β€œheap” 45 1.4 Practice - Fractions Solve each equation. 1) 3 5(1 + p) = 21 20 6 5 ) 5 4(x ...
οΏ½οΏ½c variable. For example, we may have a formula such as A = Ο€r2 + Ο€rs (formula for surface area of a right circular cone) and we may be interested in solving for the varaible s. This means we want to isolate the s so the equation has s on one side, and everything else on the other. So a solution might look like s = A....
that group. However, if we are searching for what is inside the parenthesis, we will have to break up the parenthesis by distributing. The following example is the same formula, but this time we will solve for x. Example 77. βˆ’ βˆ’ y) = b for x Solving for x, we need to distribute to clear parenthesis a(x ax ay = b + ay ...
equation from there. Example 80. h = 2m n (n)2m n (n)h = nh = 2m 2 2 nh 2 = m for m To clear the fraction we use LCD = n Multiply each term by n Reduce n with denominators Divide both sides by 2 Our Solution The same pattern can be seen when we have several fractions in our problem. Example 81. (b)a b + + c b = e (b) ...
) Multiply (or divide) each term by 1 βˆ’ Our Solution Both answers to the last two examples are correct, they are just written in a different form because we solved them in different ways. This is very common with formulas, there may be more than one way to solve for a varaible, yet both are equivalent and correct. World...
ax + b = c for a 35) lwh = V for w a + b = c a for a 37) 1 39) at βˆ’ bw = s for t 41) ax + bx = c for a 43) x + 5y = 3 for y 45) 3x + 2y = 7 for y 47) 5a 49) 4x βˆ’ βˆ’ 7b = 4 for b 5y = 8 for y 34) rt = d for r 36) V = Ο€r2h a + b = c 38) 1 3 a for b for h 40) at βˆ’ bw = s for w 42) x + 5y = 3 for x 44) 3x + 2y = 7 for x 46...
Example 86. = x | 4 βˆ’ 4 | βˆ’ βˆ’ βˆ’ 20 Notice absolute value is not alone 4 Divide both sides by 4 βˆ’ 52 x = 5 | x = 5 or x = | βˆ’ Absolute value can be positive or negative 5 Our Solution Notice we never combine what is inside the absolute value with what is outside the absolute value. This is very important as it will oft...
numbers outside cannot be combined with the numbers inside the absolute value. Thus we get the absolute value alone in the following way: βˆ’ βˆ’ βˆ’ 4 | βˆ’ 2 2 2x + 3 = | βˆ’ | 4 = βˆ’ 2x + 3 4 2x + 3 | βˆ’ = 5 | 2x + 3 = 5 or 2x + 3 = | βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ 18 Notice absolute value is not alone Subtract 2 from both sides 2 20 Absolute valu...
2x + 6 6 6 βˆ’ βˆ’ βˆ’ βˆ’ 13 = 2x βˆ’ 2 2 13 βˆ’ 2 = x or 6 βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ (4x + 6) 2x 7 = 2x 4x 7 = + 4x + 4x 6x 6 7 = βˆ’ + 7 + 7 6x = 1 6 6 1 6 x = βˆ’ This gives us our two solutions, x = βˆ’ 13 2 or x = 1 6. 55 1.6 Practice - Absolute Value Equations Solve each equation + 8a = 53 | 3k + 8 | 9 + 7x | = 2 = 30 1) 3) 5) 7) | | | | 9) | 1...
| | | 32) 34) 36) | | | 56 1.7 Solving Linear Equations - Variation Objective: Solve variation problems by creating variation equations and finding the variation constant. One application of solving linear equations is variation. Often different events are related by what is called the constant of variation. For example...
the relationship 57 The formula for the area of a triangle has three variables in it. If we divide the area by the base times the height we will also get a constant, 1. This relationship 2 is called joint variation or jointly proportional. If we see this phrase in the problem we know to divide the first variable by the...
uate to find our constant = k Substitute known values for price and tax 40 t = 12.5 Using our constant, substitute 40 for price to find the tax = 12.5(t) Multiply by LCD = t to clear fraction 40 = 12.5t Reduce the t with the denominator (t)40 t 12.5 12.5 Divide by 12.5 3.2 = t Our solution: Tax is S3.20 Example 99. The s...
variable is squared in our formula. This is shown in the next example. Example 101. The area of a circle is directly proportional to the square of the radius. A circle with a radius of 10 has an area of 314. What will the area be on a circle of radius 4? β€²β€²Directβ€²β€² tells us to divide, be sure we use r2 for the denomin...
inversely proportional to n and m = 1.8 when n = 2.1 Solve each of the following variation problems by setting up a formula to express the relationship, finding the constant, and then answering 61 the question. 19. The electrical current, in amperes, in a circuit varies directly as the voltage. When 15 volts are applie...
a fixed distance varies inversely as the speed. It takes 5 hr at a speed of 80 km/h to drive a fixed distance. How long will it take to drive the same distance at a speed of 70 km/h? 29. The weight of an object on Mars varies directly as its weight on Earth. A person weighs 95lb on Earth weighs 38 lb on Mars. How much w...
33.5 cm3, what is the volume of a cone with a height of 6 centimeters and a radius of 4 centimeters? 37. The intensity of a television signal varies inversely as the square of the dis- tance from the transmitter. If the intensity is 25 W/m2 at a distance of 2 km, how far from the trasmitter are you when the intensity ...
same as ten less than six times the number. What is the number 3x + 15 6x 3x + 15 = 6x 3x 3x βˆ’ 15 = 3x βˆ’ βˆ’ βˆ’ First, addition is built backwards 10 Then, subtraction is also built backwards 10 Is between the parts tells us they must be equal Subtract 3x so variable is all on one side 10 Now we have a two βˆ’ βˆ’ + 10 Add 1...
ally add two more to get the third, x + 4. The same is 65 true for consecutive odd numbers, if the first is x, the next will be x + 2, and the third would be x + 4. It is important to note that we are still adding 2 and 4 even when the numbers are odd. This is because the phrase β€œodd” is refering to our x, not to what i...
6 Divide both sides by 6 6 x = 23 Our solution for x First 23 Replace x with 23 in the original list Second (23) + 2 = 25 The numbers are 23, 25, and 27 Third (23) + 4 = 27 66 When we started with our first, second, and third numbers for both even and odd we had x, x + 2, and x + 4. The numbers added do not change with...
There are two lengths and two widths in a rectangle (opposite sides) so we add 8 + 8 + 3 + 3 = 22. As there are two lengths and two widths in a rectangle an alternative to find the perimeter of a rectangle is to use the formula P = 2L + 2W. So for the rectangle of length 8 and width 3 the formula would give, P = 2(8) +...
sofa costs S296. Be careful on problems such as these. Many students see the phrase β€œdouble” and believe that means we only have to divide the 444 by 2 and get S222 for one or both of the prices. As you can see this will not work. By clearly labeling the variables in the original list we know exactly how to set up and...
the measure the angles. 18. Two angles of a triangle are the same size. The third angle is 3 times as large as the first. How large are the angles? 19. The third angle of a triangle is the same size as the first. The second angle is 4 times the third. Find the measure of the angles. 20. The second angle of a triangle is...
cost of each? 32. A horse and a saddle cost S5000. If the horse cost 4 times as much as the saddle, what was the cost of each? 33. A bicycle and a bicycle helmet cost S240. How much did each cost, if the bicycle cost 5 times as much as the helmet? 34. Of 240 stamps that Harry and his sister collected, Harry collected ...
�nd their current age. There can be a lot of information in these problems and we can easily get lost in all the information. To help us organize and solve our problem we will fill out a three by three table for each problem. An example of the basic structure of the table is below Age Now Change Person 1 Person 2 Table ...
the xβ€²s with 38 and simplify. Adam is 18 and Brian is 38 36 Solving age problems can be summarized in the following five steps. These five steps are guidelines to help organize the problem we are trying to solve. 1. Fill in the now column. The person we know nothing about is x. 2. Fill in the future/past collumn by addi...
old as Kristin. How old are they now? 32 Age Now + 2 Nicole Kristen x 32 x βˆ’ Nicole Kristen Age Now x + 2 x + 2 32 x βˆ’ x + 2 32 βˆ’ The change is + 2 for two years in the future The total is placed above Age Now The first person is x. The second becomes 32 x βˆ’ Add 2 to each cell fill in the change column Nicole Kristen Ag...
to the t Subtract 8 from both sides In 18 years she will be double her daughterβ€²s age Age problems have several steps to them. However, if we take the time to work through each of the steps carefully, keeping the information organized, the problems can be solved quite nicely. World View Note: The oldest man in the wor...
34 years old, and B is 4 years old. In how many years will A be twice as old as B? 14. A man’s age is 36 and that of his daughter is 3 years. In how many years will the man be 4 times as old as his daughter? 15. An Oriental rug is 52 years old and a Persian rug is 16 years old. How many years ago was the Oriental rug ...
If the sum of their ages 3 years ago was 63 years, find the present age of the father. 28. The sum of Clyde and Wendy’s age is 64. In four years, Wendy will be three times as old as Clyde. How old are they now? 29. The sum of the ages of two ships is 12 years. Two years ago, the age of the older ship was three times th...
problems by creating and solving a linear equation. An application of linear equations can be found in distance problems. When solving distance problems we will use the relationship rt = d or rate (speed) times time equals distance. For example, if a person were to travel 30 mph for 4 hours. To find the total distance ...
hours they are 30 miles apart. How fast did each walk? Rate Time Distance Bob Fred 3 3 The basic table with given times filled in Both traveled 3 hours 80 Rate Time Distance Bob r + 2 Fred r 3 3 Bob walks 2 mph faster than Fred We know nothing about Fred, so use r for his rate Bob is r + 2, showing 2 mph faster Bob r +...
) for βˆ’ 4t With equal sign, distance colum is equation 12t = 4 4t βˆ’ 12t = 4 βˆ’ + 4t + 4t 16t = 4 16 16 1 4 t = Add 4t to both sides so variable is only on one side Variable is multiplied by 16 Divide both sides by 16 Our solution, turn around after 1 4 hr (15 min ) Another type of a distance problem where we do some wor...
will solve. The final example clearly illustrates this. Example 118. On a 130 mile trip a car travled at an average speed of 55 mph and then reduced its speed to 40 mph for the remainder of the trip. The trip took 2.5 hours. For how long did the car travel 40 mph? Rate Time Distance 55 Fast Slow 40 Basic table for fast...
a freight train start toward each other at the same time from two points 300 miles apart. If the rate of the passenger train exceeds the rate of the freight train by 15 miles per hour, and they meet after 4 hours, what must the rate of each be? 6. Two automobiles started at the same time from a point, but traveled in ...
later a cabin cruiser leaves the same harbor and travels at an average speed of 16 mph toward the same island. In how many hours after the cabin cruiser leaves will the cabin cuiser be alongside the motorboat? 16. A long distance runner started on a course running at an average speed of 6 mph. One hour later, a second...
miles apart at the end of 4 hours, what is the rate of each? 25. As part of flight traning, a student pilot was required to fly to an airport and then return. The average speed on the way to the airport was 100 mph, and the average speed returning was 150 mph. Find the distance between the two airports if the total fligh...
motorcycle was being driven at 45 mph, and the rider walks at a speed of 6 mph. The distance from home to work is 25 miles, and the total time for the trip was 2 hours. How far did the motorcycle go before if broke down? 35. A student walks and jogs to college each day. The student averages 5 km/hr walking and 9 km/hr...
in the center is called the origin. This center origin is where x = 0 and y = 0. As we move to the right the numbers count up from zero, representing x = 1, 2, 3. To the left the numbers count down from zero, representing x = 3. Similarly, as we move up the number count up from zero, y = 1, 2, 3., and as we move down ...
at (3, 2) this means x = 3 (right 3) and y = 2 (up 2). Following these instructions, starting from the origin, we get our point. The second point, B ( 2, 1), is left 2 (negative moves backwards), up 1. This is also illustrated on the graph. βˆ’ A Right 3 B Left 2 Down 3 D Down 4 C The third point, C (3, down 4 (negative...
. For example, notice the graph also goes through the point (2, 1). If we use 2x x = 2, we should get y = 1. Sure enough, y = 2(2) 3 = 1, just as the graph suggests. Thus we have the line is a picture of all the solutions for y = 2x 3. We can use this table of values method to draw a graph of any linear equation. 3 = 4...
βˆ’ 15) 4x + y = 5 15 17) 2x y = 2 βˆ’ 19) x + y = 21) y = βˆ’ 8) y = 5 3 x 10) y = βˆ’ 12) y = 1 x 2 2 βˆ’ 14) 8x y = 5 βˆ’ 16) 3x + 4y = 16 18) 7x + 3y = βˆ’ 20) 3x + 4y = 8 12 22) 9x y = 4 βˆ’ βˆ’ 94 2.2 Graphing - Slope Objective: Find the slope of a line given a graph or two points. As we graph lines, we will want to be able to id...
the coordinate plane and the idea of graphing lines (and other functions) the y-axis was not a verticle line! Example 124. 95 Run 3 Rise 6 To find the slope of this line, the rise is up 6, the run is right 3. Our slope is run or 6 3. then written as a fraction, This fraction reduces to 2. This will be our slope. rise 2...
following examples. Example 126. Find the slope between ( 4, 3) and (2, 9) Identify x1, y1, x2, y2 βˆ’ (x1, y1) and (x2, y2) Use slope formula) βˆ’ βˆ’ 12 m = βˆ’ 6 m = βˆ’ Simplify Reduce 2 Our Solution Example 127. Find the slope between (4, 6) and (2, 1) Identify x1, y1, x2, y2 βˆ’ (x1, y1) and (x2, y2) Use slope formula, m = ...
sides βˆ’ y Divide both sides by 1 1 βˆ’ 8 = y Our Solution 98 Example 131. Find the value of x between the points ( 3, 2) and (x, 6) with slope 2 5 βˆ’ m = 2 5 = y2 βˆ’ x2 βˆ’ 2 6 βˆ’ ( y1 x1 We will plug values into slope formula Simplify βˆ’ = 3x + 3) = 4 Multiply by 5 to clear fraction Multiply both sides by (x + 3) 2 5 (5) 2 5...
) (2, 6) and (x, 2); slope: 4 7 2) and (x, 6); slope: 3, βˆ’ βˆ’ 1, 1); slope: 6 7 8 5 34) ( βˆ’ 36) (x, 33) ( 35) ( βˆ’ βˆ’ 37) (x, 8, y) and ( 7) and ( βˆ’ βˆ’ βˆ’ 9, βˆ’ 9); slope: 2 5 38) (2, βˆ’ 32) (8, y) and ( 2, 4); slope: βˆ’ βˆ’ 2, y) and (2, 4); slope: 1 4 1) and ( 4, 6); slope: βˆ’ βˆ’ 5) and (3, y); slope: 6 1 5 7 10 βˆ’ 39) (x, 5) and...
, y βˆ’ intercept = 3 Use the slope intercept equation βˆ’ βˆ’ y = mx + b m is the slope, b is the y intercept βˆ’ y = 3 4 x βˆ’ 3 Our Solution We can also find the equation by looking at a graph and finding the slope and yintercept. Example 134. 102 Identify the point where the graph crosses the y-axis (0,3). This means the y-int...
each term by 4 3x + 12 Put the x term first 4 3 4 y = mx + b 3 4, b = 3 Make the graph x + 3 Recall slope Idenfity m and b βˆ’ βˆ’ βˆ’ intercept equation Starting with a point at the y-intercept of 3, Then use the slope rise run, but its negative so it will go downhill, so we will drop 3 units and run 4 units to find the next ...
6, y-intercept = 4 1, y-intercept = βˆ’ 1 4, y-intercept = 3 2 5, y-intercept = 5 Write the slope-intercept form of the equation of each line. 9) 11) 13) 10) 12) 14) 105 15) x + 10y = 37 βˆ’ 17) 2x + y = 1 βˆ’ 3y = 24 19) 7x βˆ’ 21) x = 8 βˆ’ 23) y βˆ’ 25) y βˆ’ 27) y + 5 = 29) y + 1 = (x + 5) 4 = βˆ’ 4 = 4(x βˆ’ 4(x 1 2(x 1) 2) 4) βˆ’ βˆ’...
second equation. Example 139. m, (x1, y1), (x, y) Recall slope formula y2 y1 βˆ’ x2 βˆ’ x1 y y1 βˆ’ x1 x βˆ’ y1 = m(x βˆ’ y βˆ’ = m Plug in values = m Multiply both sides by (x x1) βˆ’ x1) Our Solution If we know the slope, m of an equation and any point on the line (x1, y1) we can easily plug these values into the equation above w...
1 = m(x y1 x1 4 3 slope formula βˆ’ x 4 3 βˆ’ 5 = βˆ’ 2)) ( βˆ’ Simplify signs (x + 2) Our Solution βˆ’ 4 3 Example 143. 108 Find the equation of the line through the points ( intercept form. βˆ’ 3, 4) and ( 1, βˆ’ βˆ’ 2) in slope- m = y1 x1 First we must find the slope y2 βˆ’ x2 βˆ’ 6 = = βˆ’ 2 βˆ’ x1) With slope and either point, point 3)) 3...
through (2, 1), slope = 5) through ( 1, βˆ’ βˆ’ 5), slope = 9 6) through (2, βˆ’ 4, 1), slope = 3 4 8) through (4, βˆ’ 2), slope = 1 2 2 βˆ’ βˆ’ 2 3), slope = βˆ’ 1, 1), slope = 4 3 10) through ( βˆ’ 7) through ( βˆ’ 9) through (0, βˆ’ 11) through (0, 2), slope = βˆ’ 5), slope = βˆ’ 1 4 βˆ’ 13) through ( 15) through ( 5, βˆ’ 3), slope = 1 5 1, 4...
βˆ’ 41) through: (3, 3) and ( 4, 5) βˆ’ βˆ’ 34) through: (1, 3) and ( 3, 3) βˆ’ 36) through: ( 38) through: ( 40) through: ( 42) through: ( βˆ’ βˆ’ βˆ’ βˆ’ 4, 5) and (4, 4) 4, 1) and (4, 4) 4) and ( 5) and ( 1, 1, βˆ’ βˆ’ βˆ’ βˆ’ 5, 0) 4) 5, βˆ’ Write the slope-intercept form of the equation of the line through the given points. 43) through: (...
geometries! As the above graphs illustrate, parallel lines have the same slope and perpendicular lines have opposite (one positive, one negative) reciprocal (flipped fraction) slopes. We can use these properties to make conclusions about parallel and perpendicular lines. Example 146. Find the slope of a line parallel t...
the slope, coefficient of Perpendicular lines have opposite reciprocal slopes We will use this slope and our point (6, 9) βˆ’ y1 = m(x y βˆ’ βˆ’ x1) Plug this information into point slope formula βˆ’ y ( βˆ’ βˆ’ 9x βˆ’ (x βˆ’ 6) Simplify signs 6) Distribute slope βˆ’ βˆ’ 10 Solve for y 9 Subtract 9 from both sides βˆ’ 19 Our Solution Zero sl...
through: (1, 1), parallel to y = βˆ’ 21) through: (2, 3), parallel to βˆ’ 22) through: ( βˆ’ 1, 3), parallel to y = 3x 1 βˆ’ βˆ’ 23) through: (4, 2), parallel to x = 0 24) through: (1, 4), parallel to y = 7 5 x + 2 25) through: (1, 26) through: (1, βˆ’ 5), perpendicular to βˆ’ 2), perpendicular to x + y = 1 βˆ’ x + 2y = 2 βˆ’ 115 27) t...
( βˆ’ 6 βˆ’ 9 βˆ’ 116 Chapter 3 : Inequalities 3.1 Solve and Graph Inequalities.................................................................118 3.2 Compound Inequalities..........................................................................124 3.3 Absolute Value Inequalities..............................................
finity we will always use a curved bracket for that value. βˆ’ ∞ Example 151. Graph the inequality and give the interval notation x < 2 Start at 2 and shade below Use ) for less than Our Graph Interval Notation ( βˆ’ ∞, 2) Example 152. Graph the inequality and give the interval notation y > 1 βˆ’ Start at 1 and shade above Us...
we solve can get as complex as the linear equations we solved. We will use all the same patterns to solve these inequalities as we did for solving equations. Just remember that any time we multiply or divide by a negative the symbol switches directions (multiplying or dividing by a positive does not change the symbol!...
βˆ’ 8(n βˆ’ 60 > 5) > 0 4( βˆ’ βˆ’ 2n) > 8(2 βˆ’ 36 + 6x > 34) βˆ’ βˆ’ 36) 3(n + 3) + 7(8 6x 3) βˆ’ 16 + n βˆ’ 8(x + 2) + 4x 8n) < 5n + 5 + 2 βˆ’ 5p) + 3 > 2(8 βˆ’ βˆ’ 5p) 33) 5v 5 < 5(4v + 1) βˆ’ βˆ’ βˆ’ 35) 4 + 2(a + 5) < 2( a βˆ’ βˆ’ 4) 37) (k βˆ’ βˆ’ 2) > βˆ’ βˆ’ βˆ’ k 20 38) (4 βˆ’ βˆ’ 123 3.2 Inequalities - Compound Inequalities Objective: Solve, graph and gi...
direction as in the graph below on the left, or pointing opposite directions, but overlapping as in the graph below on the right. Notice how interval notation works for each of these cases. 124 As the graphs overlap, we take the largest graph for our solution. When the graphs are combined they cover the entire number ...
their is no overlap, no values make it to the final number line. Interval Notation: No Solution or βˆ… The third type of compound inequality is a special type of AND inequality. When our variable (or expression containing the variable) is between two numbers, we can write it as a single math sentence with three parts, su...
) 10r > 0 or r βˆ’ 6) 9 + n < 2 or 5n > 40 βˆ’ 12 8) βˆ’ 9x < 63 and x 4 < 1 10) βˆ’ 6n 6 12 and n 3 6 2 6 + v > 0 and 2v > 4 12) βˆ’ 14) 0 > x 9 16) βˆ’ 18) 1 6 p 8 > 1 βˆ’ 11 6 n 6 0 9 6 5 βˆ’ βˆ’ 20) 3 + 7r > 59 or 6r 3 > 33 22) 6 8x > βˆ’ βˆ’ 24) n + 10 > 15 or 4n βˆ’ βˆ’ βˆ’ 6 or 2 + 10x > 82 5 < 1 βˆ’ βˆ’ 6 or 10n 26) 4n + 8 < 3n βˆ’ 2a > 2a + 1 ...
by changing the problem to an OR inequality, the first inequality looking just like the problem with no absolute value, the second flipping the inequality symbol and 2, as the graph changing the value to a negative. So above illustrates. > 2 becomes x > 2 or x < x | βˆ’ | World View Note: The phrase β€œabsolute value” comes...
2 βˆ’ 4x + 1 | 2 | βˆ’ 9 9 | > 4x + 1 | 2 4x + 1 3 < 4x + 1 < 3 1 1 βˆ’ | 1 | βˆ’ βˆ’ βˆ’ βˆ’ Subtract 9 from both sides > 3 9 6 Divide both sides by 2 Dividing by negative switches the symbol βˆ’ βˆ’ βˆ’ < 3 Absolute value is less, use three part βˆ’ 2 Solve Subtract 1 from all three parts βˆ’ 4 < 4x < 2 Divide all three parts by 4 4 4 4 1 ...
| > 5 βˆ’ | 3x | x = 3 > = 3 | 3x 5 | βˆ’ > > 3 > = 10 6 15 βˆ’ > 2x | βˆ’ βˆ’ 7 < | > 1 βˆ’ 5 | 21) 4 + 3 x | βˆ’ 23 | 25) 2 3 | βˆ’ βˆ’ 27) 4 29) 3 5 2 | βˆ’ 4x | βˆ’ βˆ’ 31) 5 2 | βˆ’ βˆ’ 4 βˆ’ 2x βˆ’ 3x βˆ’ 2x + 6 6 < > 8 4 βˆ’ βˆ’ | | 33) 4 4 βˆ’ | βˆ’ 10 + x 35) | βˆ’ > 8 | 8 1 26) 28) 30) x 2) | 6 8 | x + 3 4) | 6) x 8) | 10) < 4 | < 12 6 4 8 βˆ’ | x + 3 |...
solution to the system 3x 2, 1) Identify x and y from the orderd pair x = 2, y = 1 Plug these values into each equation 3(2) (1) = 5 First equation βˆ’ 6 βˆ’ 1 = 5 Evaluate 5 = 5 True (2) + (1) = 3 Second equation, evaluate 3 = 3 True As we found a true statement for both equations we know (2,1) is the solution to the sys...
Now we can graph both lines on the same plane 3 βˆ’ To graph each equation, we start at the y-intercept and use the slope rise run to get the next point and connect the dots. Remember a negative slope is downhill! Find the intersection point, ( 2, βˆ’ βˆ’ 1) 2, ( βˆ’ βˆ’ 1) Our Solution As we are graphing our lines, it is possi...
of equations with two variables. However, their work with systems was quickly passed by the Greeks who would solve systems of equations with three or four variables and around 300 AD, developed methods for solving systems with any number of unknowns! 137 4.1 Practice - Graphing Solve each equation by graphing. 1) y = ...
. For these reasons we will rarely use graphing to solve our systems. Instead, an algebraic approach will be used. The first algebraic approach is called substitution. We will build the concepts of substitution through several example, then end with a five-step process to solve problems using this method. Example 170. We...
15y + 2y = 1 Combine like terms 15y + 2y Subtract 18 from both sides 18 + 17y = 1 18 18 17 Divide both sides by 17 17 17y = 17 βˆ’ βˆ’ βˆ’ x = 6 + 5( y = βˆ’ x = 6 1 We have our y, plug this into the x = equation to find x βˆ’ 1) Evaluate, multiply first 5 Subtract x = 1 We now also have x βˆ’ (1, βˆ’ 1) Our Solution The process in t...
is one or the other, the process takes an interesting turn as shown in the following example. Example 174. Solve for the lone variable, subtract 4 from both sides Find the lone variable, y in the first equation βˆ’ βˆ’ y + 4 = 3x 6x = 2y 8 y + 4 = 3x 4 4 βˆ’ y = 3x 4 Plug into untouched equation 6x = 8 8 Combine like terms 6...
we will solve for x in the first equation Solve for our variable, add 6y to both sides Divide each term by 5 Plug into untouched equation Solve, distribute through parenthesis Clear fractions by multiplying by 5 + 4y(5) = 12(5) Reduce fractions and multiply βˆ’ 12y + 20y = 60 28 + 8y = 60 28 28 βˆ’ 8y = 32 8 8 βˆ’ Combine li...