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Graphs/140316: Find the domain of the function 1 solutions Answer 102186 by jim_thompson5910(28550)   on 2008-05-06 11:09:04 (Show Source): You can put this solution on YOUR website! Start with the given function Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain. Take the square root of both sides Subtract 2 from both sides Combine like terms on the right side Divide both sides by 3 to isolate x Reduce Since makes the denominator equal to zero, this means we must exclude from our domain So our domain is: which in plain English reads: x is the set of all real numbers except So our domain looks like this in interval notation note: remember, the parenthesis excludes -2/3 from the domain If we wanted to graph the domain on a number line, we would get: Graph of the domain in blue and the excluded value represented by open circle Notice we have a continuous line until we get to the hole at (which is represented by the open circle). This graphically represents our domain in which x can be any number except x cannot equal -2/3

Graphs/140315: Graph the solution to the system 1 solutions Answer 102185 by jim_thompson5910(28550)   on 2008-05-06 11:06:09 (Show Source): You can put this solution on YOUR website! Start with the given system of inequalities In order to graph this system of inequalities, we need to graph each inequality one at a time. First lets graph the first inequality In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (note: if you need help with graphing, check out this solver) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of Firefox to see these images.) Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green) --------------------------------------------------------------- Now lets graph the second inequality In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (note: if you need help with graphing, check out this solver) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green) (note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line) --------------------------------------------------------------- So we essentially have these 2 regions: Region #1 Graph of Region #2 Graph of When these inequalities are graphed on the same coordinate system, the regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in. Here is a cleaner look at the intersection of regions Here is the intersection of the 2 regions represented by the series of dots

Graphs/140314: Graph the equation 1 solutions Answer 102184 by jim_thompson5910(28550)   on 2008-05-06 11:03:47 (Show Source): You can put this solution on YOUR website! Start with the given equation Rearrange the terms Subtract from both sides Complete the square for the x terms (let me know if you need help completing the square) Complete the square for the y terms Combine like terms Add to both sides Combine like terms Subtract from both sides Take the square root of both sides Subtract 1 from both sides So we have the two equations and When we graph the two, we get Graph of (red) and (green)

Graphs/140313: Graph the equation 1 solutions Answer 102182 by jim_thompson5910(28550)   on 2008-05-06 10:56:41 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract from both sides Take the square root of both sides Subtract 1 from both sides So we have the two equations and So when we graph the two equations we get Graph of (red) and (green)

Graphs/140312: Graph the equation 1 solutions Answer 102181 by jim_thompson5910(28550)   on 2008-05-06 10:54:14 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract from both sides Take the square root of both sides Add 1 to both sides So we have the two equations and So when we graph the two equations we get Graph of (red) and (green)

Quadratic_Equations/140238: x^2+4x+1=0 1 solutions Answer 102153 by jim_thompson5910(28550)   on 2008-05-06 00:19:27 (Show Source): You can put this solution on YOUR website! Let's use the quadratic formula to solve for x: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=1, b=4, and c=1 Square 4 to get 16 Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 1 to get 2 So now the expression breaks down into two parts or Now break up the fraction or Simplify or So these expressions approximate to or So our solutions are: or Notice when we graph , we get: when we use the root finder feature on a calculator, we find that and .So this verifies our answer

Linear-equations/140258: (2,-3) (-5,4) find the slope , show the linear equation, draw a graph 1 solutions Answer 102152 by jim_thompson5910(28550)   on 2008-05-06 00:17:46 (Show Source): You can put this solution on YOUR website! First lets find the slope through the points (,) and (,) Start with the slope formula (note: is the first point (,) and is the second point (,)) Plug in ,,, (these are the coordinates of given points) Subtract the terms in the numerator to get . Subtract the terms in the denominator to get Reduce So the slope is ------------------------------------------------ Now let's use the point-slope formula to find the equation of the line: ------Point-Slope Formula------ where is the slope, and is one of the given points So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Distribute Multiply and to get Subtract from both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line which goes through the points (,) and (,) is: The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver) Graph of through the points (,) and (,) Notice how the two points lie on the line. This graphically verifies our answer.

Polynomials-and-rational-expressions/140099: I am having trouble solving this type of problems. x^3+5x^2-6x+10 ______________ x+3 1 solutions Answer 102085 by jim_thompson5910(28550)   on 2008-05-05 10:24:00 (Show Source): You can put this solution on YOUR website! Let's simplify this expression using synthetic division Start with the given expression First lets find our test zero: Set the denominator equal to zero Solve for x. so our test zero is -3 Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero. -3 | 1 5 -6 10 | Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1) -3 | 1 5 -6 10 | 1 Multiply -3 by 1 and place the product (which is -3) right underneath the second coefficient (which is 5) -3 | 1 5 -6 10 | -3 1 Add -3 and 5 to get 2. Place the sum right underneath -3. -3 | 1 5 -6 10 | -3 1 2 Multiply -3 by 2 and place the product (which is -6) right underneath the third coefficient (which is -6) -3 | 1 5 -6 10 | -3 -6 1 2 Add -6 and -6 to get -12. Place the sum right underneath -6. -3 | 1 5 -6 10 | -3 -6 1 2 -12 Multiply -3 by -12 and place the product (which is 36) right underneath the fourth coefficient (which is 10) -3 | 1 5 -6 10 | -3 -6 36 1 2 -12 Add 36 and 10 to get 46. Place the sum right underneath 36. -3 | 1 5 -6 10 | -3 -6 36 1 2 -12 46 Since the last column adds to 46, we have a remainder of 46. This means is not a factor of Now lets look at the bottom row of coefficients: The first 3 coefficients (1,2,-12) form the quotient and the last coefficient 46, is the remainder, which is placed over like this Putting this altogether, we get: So which looks like this in remainder form: remainder 46 You can use this online polynomial division calculator to check your work

Equations/140072: I would be so grateful if someone could help me with this. 4=3(x-3)+4-2x Thank you in advance. 1 solutions Answer 102084 by jim_thompson5910(28550)   on 2008-05-05 10:22:01 (Show Source): You can put this solution on YOUR website! Start with the given equation Distribute Combine like terms on the right side Add 5 to both sides Combine like terms on the left side -------------------------------------------------------------- Answer: So our answer is

Functions/140040: determine the domain of the function f(x)=sqrt of 3-x 1 solutions Answer 102070 by jim_thompson5910(28550)   on 2008-05-04 22:02:17 (Show Source): You can put this solution on YOUR website! Start with the given expression Remember you cannot take the square root of a negative value. So that means the argument must be greater than or equal to zero (i.e. the argument must be positive) Set the inner expression greater than or equal to zero Subtract 3 from both sides Combine like terms on the right side Divide both sides by -1 to isolate x (note: Remember, dividing both sides by a negative number flips the inequality sign) Divide So that means x must be less than or equal to in order for x to be in the domain So the domain in set-builder notation is So here is the domain in interval notation: (-,3] Notice if we graph , we get notice how the graph never crosses the line and we can see that x must be less than or equal to in order to lie on the graph. So this graphically verifies our answer.

Linear-equations/140012: f(x)=-4x-5 Find the Slope and the y-intercept 1 solutions Answer 102050 by jim_thompson5910(28550)   on 2008-05-04 20:23:00 (Show Source): You can put this solution on YOUR website! Notice how is in slope-intercept form where m is the slope and b is the y-intercept. So this shows us that the slope is and the y-intercept is which is the point (0,-5).

test/140019: Reduce the given expression to lowest terms. 3x2+7x+2 -------- 4x2-16 1 solutions Answer 102049 by jim_thompson5910(28550)   on 2008-05-04 20:21:42 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor to get Factor to get Combine the fractions Cancel like terms Simplify ------------------------------------------- Answer: So simplifies to

Graphs/140018: 2. One number exceeds another by 5. The sum of their reciprocals equal to 19 divided by the product of the two numbers. Find the two numbers. (a) How will you set up the problem? (b) What is the equation that the one number exceeds another by 5? (c) What is the product of two numbers, in terms of x? (d) What is x? Also check your answer. 1 solutions Answer 102047 by jim_thompson5910(28550)   on 2008-05-04 20:17:20 (Show Source): You can put this solution on YOUR website! a) Let x=first number and y=second number b) Since "One number exceeds another by 5", this means that the first equation is c) The product of the two numbers in terms of x is d) The value of x is 7 ---------------------------- So let's solve the problem Let x=first number and y=second number Since "One number exceeds another by 5", this means that the first equation is Also, because "The sum of their reciprocals equal to 19 divided by the product of the two numbers" we have the second equation Plug in Multiply both sides by the LCD to clear out the fractions Distribute and multiply Combine like terms on the left side Subtract 5 from both sides Combine like terms on the right side Divide both sides by 2 to isolate x Divide -------------------------------------------------------------- Answer: So our answer is

Graphs/140017: 1. The perimeter of a rectangular box is 42 inches. The length of the box is 15 inches more than the width. Determine the dimensions of the rectangle in terms of feet Also find the area of the box in terms of feet. Draw the diagram by showing the dimensions. (a) How will you set up the problem. (b) What are the two linear equations. (c) What is the product of two dimensions of the rectangle. (d) How is this product different from perimeter. Interpret on that. 1 solutions Answer 102045 by jim_thompson5910(28550)   on 2008-05-04 20:16:25 (Show Source): You can put this solution on YOUR website! a) Let L=length, W=width b) Since "The length of the box is 15 inches more than the width", this means the first equation is Remember the perimeter formula is If we plug in , we get the second equation: So the two equations are: c) the product of the two dimensions of the rectangle is d) The perimeter is a linear equation while the product is a nonlinear equation

Exponential-and-logarithmic-functions/140001: simplify 4m^4n^3p^3/3m^2n^2p^4 1 solutions Answer 102040 by jim_thompson5910(28550)   on 2008-05-04 19:34:43 (Show Source): You can put this solution on YOUR website! Start with the given expression. Remember when you divide monomials, you subtract their corresponding exponents. For instance Simplify. Remember to reduce the coefficients (which are numbers in front of the variables) to get . Flip the expression with a negative exponent. Simplify -------------------------------------------- Answer: So simplifies to . In other words, where and

Polynomials-and-rational-expressions/140005: Multiply and match your result to the correct answer below: (z – 8)(z + 8) A)z2 + 16 B)z2 – 64 C)z2 – 16 D)z2 – 16y – 64 1 solutions Answer 102039 by jim_thompson5910(28550)   on 2008-05-04 19:31:33 (Show Source): You can put this solution on YOUR website! Start with the given expression Now let's FOIL the expression Remember, when you FOIL an expression, you follow this procedure: Multiply the First terms: Multiply the Outer terms: Multiply the Inner terms: Multiply the Last terms: Now collect every term to make a single expression Now combine like terms --------------------- Answer: So foils and simplifies to In other words,

test/140011: Factor completely and match your result to the correct answer below. r2 + 2r – 80 A)(r + 10)(r + 8) B)(r – 10)(r + 8) C)(r – 10)(r – 8) D)(r + 10)(r – 8) 1 solutions Answer 102038 by jim_thompson5910(28550)   on 2008-05-04 19:30:54 (Show Source): You can put this solution on YOUR website! Looking at we can see that the first term is and the last term is where the coefficients are 1 and -80 respectively. Now multiply the first coefficient 1 and the last coefficient -80 to get -80. Now what two numbers multiply to -80 and add to the middle coefficient 2? Let's list all of the factors of -80: Factors of -80: 1,2,4,5,8,10,16,20,40,80 -1,-2,-4,-5,-8,-10,-16,-20,-40,-80 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to -80 (1)*(-80) (2)*(-40) (4)*(-20) (5)*(-16) (8)*(-10) (-1)*(80) (-2)*(40) (-4)*(20) (-5)*(16) (-8)*(10) note: remember, the product of a negative and a positive number is a negative number Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2 First Number Second Number Sum 1 -80 1+(-80)=-79 2 -40 2+(-40)=-38 4 -20 4+(-20)=-16 5 -16 5+(-16)=-11 8 -10 8+(-10)=-2 -1 80 -1+80=79 -2 40 -2+40=38 -4 20 -4+20=16 -5 16 -5+16=11 -8 10 -8+10=2 From this list we can see that -8 and 10 add up to 2 and multiply to -80 Now looking at the expression , replace with (notice adds up to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) ------------------------------------------------------------ Answer: So factors to

test/140010: Factor: st + 5b – bs – 5t A) (s – 5)(t + b) B) (s – 5)(t – b) C) (s + 5)(t – b) D) The expression is prime. 1 solutions Answer 102037 by jim_thompson5910(28550)   on 2008-05-04 19:29:15 (Show Source): You can put this solution on YOUR website! Start with the given expression Rearrange the terms Group like terms Factor out the GCF "s" from the first group. Factor out the GCF -5 from the second group Rearrange in the second group to get Combine like terms ----------------------------- Answer: So factors to

test/140008: Factor completely: 12p2 + 20p A)2(6p2 + 10) B)4p(3p + 20) C)4p(3p + 5) D)6p(2p + 5) 1 solutions Answer 102036 by jim_thompson5910(28550)   on 2008-05-04 19:23:36 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor out the GCF ----------------------------------------------------------- Answer: So factors to

test/140007: Factor: z2 – 14z + 49 A) (z – 7)2 B) (z – 7)(z + 7) C) (z – 14)(z + 1) D) The expression is prime. 1 solutions Answer 102035 by jim_thompson5910(28550)   on 2008-05-04 19:21:53 (Show Source): You can put this solution on YOUR website! Looking at we can see that the first term is and the last term is where the coefficients are 1 and 49 respectively. Now multiply the first coefficient 1 and the last coefficient 49 to get 49. Now what two numbers multiply to 49 and add to the middle coefficient -14? Let's list all of the factors of 49: Factors of 49: 1,7 -1,-7 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to 49 1*49 7*7 (-1)*(-49) (-7)*(-7) note: remember two negative numbers multiplied together make a positive number Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14 First Number Second Number Sum 1 49 1+49=50 7 7 7+7=14 -1 -49 -1+(-49)=-50 -7 -7 -7+(-7)=-14 From this list we can see that -7 and -7 add up to -14 and multiply to 49 Now looking at the expression , replace with (notice adds up to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) note: is equivalent to since the term occurs twice. So also factors to ------------------------------- Answer: So factors to

test/140006: Factor: y2 – 64 A) (y – 8)(y + 8) B) (y – 8)2 C) (8 – y)(8 + y) D) The expression is prime. 1 solutions Answer 102034 by jim_thompson5910(28550)   on 2008-05-04 19:21:05 (Show Source): You can put this solution on YOUR website! Start with the given expression Rewrite as Rewrite as Now use the difference of squares. Remember, the difference of squares formula is where in this case and Plug in and So the expression factors to Notice that if you foil the factored expression, you get the original expression. This verifies our answer.

Geometric_formulas/139954: 1 solutions Answer 102009 by jim_thompson5910(28550)   on 2008-05-04 13:25:13 (Show Source): You can put this solution on YOUR website! Let x=supplement to angle A Remember, supplement angles add to 180. So if we want to find the supplement to angle A, then Subtract 72 from both sides Combine like terms on the right side -------------------------------------------------------------- Answer: So our answer is So the supplement to angle A is 108 degrees

Geometric_formulas/139952: 7. Find the measure of angle x and measure of angle y. 1 solutions Answer 102008 by jim_thompson5910(28550)   on 2008-05-04 13:20:11 (Show Source): You can put this solution on YOUR website! Notice how angle x and 135 degrees form a 180 degree angle. So this means that Subtract 135 from both sides Combine like terms on the right side Also, notice how angle x and angle y also form a 180 degree angle. So But we know what x is, so plug in the value of x Plug in Subtract 45 from both sides Combine like terms on the right side -------------------------------------------------------------- Answer: So the values are and

Geometric_formulas/139951: 6. Find the missing angle. 1 solutions Answer 102006 by jim_thompson5910(28550)   on 2008-05-04 13:14:15 (Show Source): You can put this solution on YOUR website! To find the missing angle, first add up the two given angles So the sum of the two angles is 148 degrees. Now subtract this answer from 180 (remember the sum of the three angles of a triangle is 180 degrees) So the missing angle is 32 degrees

Geometric_formulas/139950: 5. Which two triangles are similar? 1 solutions Answer 102005 by jim_thompson5910(28550)   on 2008-05-04 13:08:48 (Show Source): You can put this solution on YOUR website! First, we need to find the missing angles of each triangle. For the first triangle, first add up the two given angles 69 and 48 to get Now subtract 117 from 180 to get So the first triangle has these angles --------------------------------------------------------- For the second triangle, first add up the two given angles 63 and 48 to get Now subtract 111 from 180 to get So the second triangle has these angles ------------------------------------------------------------ For the third triangle, first add up the two given angles 63 and 48 to get Now subtract 111 from 180 to get So the third triangle has these angles ---------------------------------------------------------- Summary: So the three triangles have these angles ---------------------------------------------------------- Answer: Remember, similar triangles have equal angles. From the figure, we can see that triangles a) and b) have equal angles. So triangles a) and b) are similar triangles.

Geometric_formulas/139948: 4. The two triangles are similar. Find the indicated side. Find y. 1 solutions Answer 102003 by jim_thompson5910(28550)   on 2008-05-04 12:54:18 (Show Source): You can put this solution on YOUR website! Since the triangles are similar, this means that the length of the sides are dependent on one another. In fact, these sides form the ratio: Multiply both sides by y Multiply both sides by 8 Multiply Divide both sides by 10 Simplify So our answer is

Geometric_formulas/139946: 3. Identify the hypotenuse of the triangle by giving its letter 1 solutions Answer 102002 by jim_thompson5910(28550)   on 2008-05-04 12:48:23 (Show Source): You can put this solution on YOUR website! The hypotenuse is the longest side. So in this case, the hypotenuse is z.

Geometric_formulas/139945: Find the missing length of the right triangle 1 solutions Answer 102001 by jim_thompson5910(28550)   on 2008-05-04 12:45:30 (Show Source): You can put this solution on YOUR website! Let's use Pythagoreans theorem to solve this problem Pythagoreans theorem: where a and b are the legs of the triangle and c is the hypotenuse Plug in a=7, b=24. Now lets solve for c. Square each individual term Combine like terms Take the square root of both sides Simplify the square root So our answer is So the length of the unknown side is 25 units.

Geometric_formulas/139941: 1. Approximate by giving the two whole numbers that it lies between 1 solutions Answer 102000 by jim_thompson5910(28550)   on 2008-05-04 12:10:17 (Show Source): You can put this solution on YOUR website! First, note that the numbers 25 and 36 are perfect squares. In other words, and . So since 31 is in between 25 and 36, this means that the square root of 31 is in between 5 and 6. In other words, since , this means . Notice how 31 is 6 units away from 25 and 5 units away from 36. So 31 is about the halfway point from 25 to 36. So a good approximation for is 5.5 since 5.5 is halfway between 5 and 6. So If we take the square root of 31 with a calculator, we get . So this shows us that our approximation is very close.

Equations/139872: Find the slope of a line whose equation is y= -6x+3 1 solutions Answer 101938 by jim_thompson5910(28550)   on 2008-05-03 15:09:29 (Show Source): You can put this solution on YOUR website! Notice how is in slope-intercept form where m is the slope and b is the y-intercept. So this shows us that the slope is and the y-intercept is

Quadratic_Equations/139853: Please help me solve this equation: 5x+4y=12 I have tried the equation by doing this: 5x+4y=12 5x-5x+4y=12-5x 4y=12-5x 4y/4=12-5x/4 My answer came totaled to: y=3-5x 1 solutions Answer 101934 by jim_thompson5910(28550)   on 2008-05-03 14:14:02 (Show Source): You can put this solution on YOUR website! Do you want to solve for y? Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce