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Find all values of x such that x/(x - 5) = -4/(x - 4)     

                                                      x                – 4     

                                                 ––––––   =   ––––––     

                                                  (x – 5)          (x – 4)     

crossmultiply                              x² – 4x  =  – 4x + 20     

rearrange for clarity                    x² – 4x  =  20 – 4x     

                                                           x²  =  20     

                                                           x  =  + sqrt(20)     

                                                           x  =  – sqrt(20)     

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Let a, b, and c, be nonzero real numbers such that a+b+c=0. Compute the value of . . .      

     a(b – c)  multiplied out is  ab – ac     

     b(c – a)  multiplied out is  bc – ba     

     c(a – b)  multiplied out is  ca – cb     

add up the column on the right     ab – ac + bc – ba + ca – cb     

change ba to ab     

change ca to ac     

change cb to bc                             ab – ac + bc – ab + ac – bc     

rearrange                                       ab – ab + ac – ac + bc – bc     

add parentheses for clarity            (ab – ab) + (ac – ac) + (bc – bc)     

Everything cancels out and the      final total is zero.     

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Let a and b be real numbers such that a^3 + 3ab^2 = 679 and 3a^2 b + b^3 = -679. Find a + b.     

                               a³ + 3ab²  =    679     

                               3a²b + b³  =  –679     

add them to get       a³ + 3ab² + 3ba² + b³  =  0     

this factors to           (a + b)³  =  0     

                                  a + b    =  0      

Verified this by plotting (a+b)³ = 0 on Desmos.     

The curve is identical to the curve for y = –x.      

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Suppose that p and q are prime numbers such that p divides q+1 and q divides p+3. Determine p+q.     

I think p = 2 and q = 5 will do it.     

(q+1) / p       (p+3) / q     

(5+1) / 2       (2+3) / 5     

_.     

If 5 = –4b + 5 and 3a = –2b + 3 + 6b, what is b?     

consider only the first equation                     5 = –4b + 5    

subtract 5 from both sides and you have  –4b = 0  therefore b = 0     

what about that other equation . . . plug zero in for instances of b     

                                                         3a = –2b + 3 + 6b     

                                                         3a =     0 + 3 + 0  and a = 1    

_.     

A positive integer divisor of 11 is chosen at random.     

What is the probability that this divisor is prime?     

Express your answer as a common fraction.     

11 is prime, therefore its only divisors are 1 and 11.     

By the definition of prime, 1 is not a prime number.     

That leaves 11 as the only positive divisor of 11.     

The probability of choosing 11 at random out of all numbers is                             1     

                                                                                                            ––––––––––––––––––––     

                                                                                                             every integer that exists     

_.     

The average of x + 6, 6x + 2, and 4x is 4x-7. What is x?     

      (x + 6)  +  (6x + 2)  +  (4x)           (4x – 7)     

     ––––––––––––––––––––––   =   –––––––     

                        3                                    1     

      11x  +  8           4x – 7     

     ––––––––   =   ––––––      cross multiply to obtain   11x + 8  =  12x – 21      

             3                   1                                                          – x  =  – 29     

                                                                                                 x  =  29     

check answer     

(29+6) + (174+2) + (116) / 3  =?  (116–7)     

             (35 + 176 + 116) / 3  =?  109     

                                 327 / 3  =?  109

                                      109  =  109    True     

_.     

A nonzero number of goats, each with four legs, and a nonzero number of chickens, each with two legs, are living on a farm. Between all the animals, there are 1/3 as many heads as there are legs. Assuming that all animals have all of their limbs, what is the fewest possible number of chickens on the farm?      

      No. Chickens Legs     No. Goats Legs     Total Heads     Total Legs     

                  1           2                 1         4                 2                     6     

_.     

What is the median of the data set {4, 16, 0, 8, 3, 11, 7, 25, 20}?      

Arrange them in order:     0  3  4  7  8  11  16  20  25     

The median is the middle value in a dataset that has been arranged in order from smallest to largest.     

If the dataset has an odd number of values, the median is the single middle number.     

If the dataset has an even number of values, the median is the average (mean) of the two middle numbers.     

Since our problem has nine values, the median is the middle one:     8          

_.     

the sale proce of the home was 375,000 at 5.5% commission     

So what are we looking for?     

If it's the agent's commission, then proceed as follows:     

                                                  C  =  ($375,000)(5.5%)     

                                                  C  =  ($375,000)(0.055)     

                                                  C  =   $20,625     

.     

Brian has earned 65%, 80% and 60% on his three pre-final exams. These exams are not weighed equally: the lowest counts for only 25% of his overall grade, while the other two count for 30% each. If the final exam is the remainder of the overall grade and there are no opportunities for extra credit, what is the highest grade Brian can earn in the class? Express your answer to the nearest whole percent.      

                                        65% x 30%  =  19.5     

                                        80% x 30%  =  24.0     

                                        60% x 25%  =  15.0     

If Brian makes 100%     

on the fourth test  —>   100% x 15%  =  15.0     

                                                               –––––     

                                                                 73.5 is the highest grade he can make in the class.     

              rounded to the whole number is 74     

.     

A bag contains red, blue and green marbles. 2/3 of the marbles are not red and 3/4 of the marbles are not blue. What fraction of the marbles are not green? Express your fraction in lowest terms.     

Given that 2/3 of the marbles are not red, then 1/3 of the marbles are red.     

Given that 3/4 of the marbles are not blue, then 1/4 of the marbles are blue.     

1/3  =  4/12 are red         

1/4  =  3/12 are blue        

total    7/12 are either red or blue, so if they're red or blue they're not green     

_.     

The number a is 5 times as large as b.     

The sum of a and b is 18.     

What is the value of the product ab? Express your answer as a common fraction.      

                    a + b  =  18     

                          a  =  5b     

                    5b + b  =  18     

                          6b  =  18     

                            b  =  3     

                    a + 3  =  18     

                          a  =  15     

                    ab  =  (15)(3)     

                    ab  =  45     

_.     

A furniture store had a sale with 30%  off all items. If the sale price for a chair was $42 , what was the original price, in dollars?     

     Sale Price  =  Original Price minus Discount     

                                     SP  =  OP –  30%     

                                     SP  =  0.70 OP     

                                     SP / 0.70  =  OP     

                                   $42 / 0.70  =  OP     ——>     OP  = $60     

_.     

Which of the following sets of numbers could not represent the three sides of a triangle?     

\{8,14,20\}     

\{4,6,7\}     

\{6,8,10\}     

\{4,14,19\}      

     4, 14, 19  does not comply with the Triangle Inequality Theorem.     

_.     

     Determine the missing side length.     

     ?²  =  15² – 9²     

     ?²  =  225 – 81     

     ?²  =  144     

     ?    =  12     

_.     

     Determine the missing side length.     

     x²  =  14² – 9²      per Pythagoras Theorem     

     x²  =  196 – 81     

     x²  =  115     

     x    =  sqrt(115)     

_.     

What non-zero integer must be placed in the square so that the simplified product of these two binomials is a binomial?     

(3x+2)(15x-n)     

     consider (3x + 2)(15x – n)     

      2 • 15x gives you + 30x     

      n needs to be a number such that 3x • n gives you – 30x     

      So, n = 10     

      demonstrate answer:  (3x + 2)(15x – 10)  =  45x² – 30x + 30x – 20     

                                                                             45x² – 20     

_.     

The range of the function.

\(h(x) = 5x^2 + 20x + 33/x^2\\ h'(x)=10x+20-\dfrac{66x}{x^4}=0\\ 10x^4+20x^3-66=0\\ x\in\{-2.4492, 1.2645\}\)

The function has minima at \(M_1(-2.4492, -13.4897)\) and \(M_2(1.2645, 53.9232)\).

At x = 0, the function value grows to infinity.

\(\color{blue}So\ R\in \{-13.4897\le \mathbb R< infinity\}\)

  !

  C is the center of the circle.  Find arc DGE.     

      going CCW            Arc DGE  =  360^o – 115^o     

                                                     =  245^o     

      therefore        Distance DGE  =  (²⁴⁵/₃₆₀) π (2 • 14)     

                                                      =  59.86     

_.     

The diameter of a sphere is 12cm. What is the volume of the sphere, in cubic centimeters?     

Use 3.14 for π\piπ . Round your answer to the nearest tenth.      

                 V_sphere  =  (⁴/₃) π r³     

                           V  =  (⁴/₃) (3.14) (6²)     

                           V  =  (⁴/₃) (3.14) (36)     

cancel 3's            V  =  (4) (3.14) (12)     

                            V  =  150.72   —>  rounded is 150.7 cm³     

_.     

  I put in 4x = 16 (which I know x = 4), and it gives me (without quotation marks), "Error:" with no other explanation.     

     Don't use any spaces, that's your other explanation.     

     I just tried 4x=16 with no spaces, and it worked fine.     

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