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     Find all solutions x to the equation 5(2x + 1) + 5*2 = 5*x + 5(x + 3).     

                                  5(2x + 1) + 5*2  =  5*x + 5(x + 3)     

     multiply it out         10x + 5   + 10   =   5x  +  5x + 15     

     combine like terms          10x + 15  =  10x + 15     

     There isn't enough time left in the universe to write down all the solutions,     

     because there are infinite solutions to an equation that amounts to x = x.     

     Every x satisfies the given equation.     

_.     

Let S be the set {1, 2, 3, \dots, 10, 11, 12}.     

How many subsets of the set S have no two consecutive primes as members?     

I don't know how many subsets exist in set S, but the only consecutive primes in the set are 2 and 3.     

So figure up all the subsets, then eliminate the ones that contain 2, 3.  BTW, 1 is not a prime number.   

_.      

     In the diagram below, we know \tan \theta = \frac{3}{4}. Find the area of the triangle.     

     The triangle is right, and the hypotenuse is 80.     

     Area of a triangle equals half the product of the base times the height.     

     A  =  (1/2)(48)(64)     

     A  =  1536     

     Recognize that 5 is a factor of 80.  And that 3, 4, 5 is a Pythagorean triplet.     

     So, just multiply all the way across by 16 and you get a 48, 64, 80 right triangle.     

     You know that the legs are 48 and 64 because the longest one is the hypotenuse.     

_.     

A G.P has its 3rd term and 7th term as 45 and 3645 respectively. what is the product of its 2nd and 4th     

Call the common ratio "r"     

1st term     

2nd term    45 / r      

3rd term     45                              45     

4th term     45 • r     

5th term     45 • r • r     

6th term     45 • r • r • r     

7th term     45 • r • r • r • r        3645     

8th term     45 • r • r • r • r • r     

Taking what we know, then         45 • r⁴  =  3645     

                                                           r⁴  =  3645 / 45  =  81     

                                                           r¹  =  3     

4th term is  45 • 3  =  135     

2nd term is 45 / 3   =   15             Ha ha, I bit.  It wasn't even necessary to solve for r.     

                                                     No matter what r is, it cancels out in the 2nd and 4th.     

product of 2nd times 4th  =  2025     

_.     

a)  vertex at (1,6)     

b)  opens downward     

c)  symmetry about x=1     

d)  two x-intercepts:  a little short of x= –1     

                                  a little short of x= +3     

e)  y-intercept at y= +4     

_.     

if a parabola has its vertex at (3,-5) and one of its x-intercepts at x=8, what is the other x-intercept?     

Assuming the parabola is symmetrical     

The x-coord of the vertex would be halfway between the x-intercepts     

From vertex x-coord 3 to x-coord 8 is 5 units in the positive direction    

For the other x-intercept, go the same distance

from the vertex x-coord in the other direction     

From 3 go 5 units in the negative direction     

The other x-intercept is at x = – 2     

_.     

Karen paid 1.20 for 6 pencils and 4 erasers at the school store. Trey paid 1.10 for 8 pencils and 2 erasers at the same store.  How much does a pencil cost?  How much does an eraser cost?     

1.20 for 6 pencils and 4 erasers     —>>     6P + 4E  =  1.20     (equation 1)     

1.10 for 8 pencils and 2 erasers     —>>     8P + 2E  =  1.10     (equation 2)     

multiply both sides (eq 2) by 2                  16P  + 4E  =  2.20     (eq 3)     

and subtract (eq 1) from it                          6P  +  4E  =  1.20     (eq 1)     

                                                                 10P             =  1.00     

                                                                                P  =  0.10     Pencils are 10¢ apiece     

substitute 10¢ back into (eq 1)             (6)(0.10) + 4E  =  1.20     

                                                                   0.60  + 4E  =  1.20     

                                                                               4E  =  0.60     

                                                                                 E  =  0.15     Erasers are 15¢ apiece     

_.     

  In five years, Angela will be the age that Khalil is now. In three years, Khalil will be twice as old as Angela. How old is each now?     

  In five years, Angela will be the age that Khalil is now     

  (A + 5)  =  K     

  In three years, Khalil will be twice as old as Angela     

  (K + 3)  =  2 • (A + 3)     

  Substitute (A + 5) in place of K in the second equation.     

  A + 5 + 3  =  2A + 6     

        A + 8  =  2A + 6     

               2  =  A     

         A + 5  =  K     

         2 + 5  =  K     

               7  =  K     

  Angela is 2 and Khalil is 7 at present.     

_.     

The measures of the angles of a triangle are shown in the figure below.  Solve for x.          

The sum of all three angles of a triangle is 180^o.     

The other two angles total is (59^o + 41^o) = 100^o.     

Therefore, x = 180^o – 100^o     

                  x  =  80^o     

_.     

What is KJL?     

Sides KJ and KL are equal (isoceles triangle).     

Since it's isoceles, angles KJL and KLJ are equal.     

The sum of all three angles of a triangle is 180^o.     

So the sum of KJL and KLJ is (180^o – 72^o) = 108^o     

Since KJL and KLJ are equal, each is half of 108^o.     

Therefore angle KJL is 54^o (i.e., half of 108^o).     

_.     

What is the measure of x?     

I'm saying that the triangles are similar, so     

3 : z  =  z : 30      i.e., ratio of short side to hypotenuse of small triangle     

                           equals ratio of short side to hypotenuse of large triangle.     

                           If they are not similar, don't bother reading any further.    

                                    3 / z  =  z / 30     

crossmultiply to get      z² =  90     

by Pythagoras              x²  =  30² – z²     

                                     x²  =  900 – 90     

                                     x   =  sqrt(810)     

This answer looks counterintuitive because the figure is not drawn to scale.     

_.     

     What is the surface area?     

     four sides plus the base

     [4 times (1/2) • 9 • 10)]  plus  (9 • 9)     

     A  =  180 + 81     

     A  =  262 cm²    

_.     

At the city museum child museum child admission is 5.70 and adult admission is 9.30 on Saturday three times as many adult tickets as child tickets were sold for a total of 1075.20 how many child tickets were sold that day     

9.30  =  cost of one adult ticket     

5.70  =  cost of one child ticket     

A  =  number of adult tickets sold  =  3C     

C  =  number of child tickets sold     

(A • 9.30)   + (C • 5.70)  =  1075.20     

(3C • 9.30) + (C • 5.70)  =  1075.20     

27.90A + 5.70A  =  1075.20     If the decimal points throw you off, remove them all,

                                                 just remember to put one back in the final answer.     

33.60C  =  1075.20     

         C  =  32                           Child tickets sold that Saturday     

_.     

When Lara was 13 years old, she weighed 38 kg.     
Two years later, her weight increased by 10%.     
Find Lara's weight when she was 15 years old.     

At age 13 Lara weighed 38 kg     

Her weight increased by10%                   10% of 38 kg     

                                                                  0.10 x 38 kg = 3.8 kg     

To find Lara's weight after the increase,     

add the increase to her original weight      38 kg + 3.8 kg  =  41.8 kg     

_.     

     The figure below consists of a rectangle and a right triangle.     

     What is the area of the shaded region?     

     A_rectangle  =  7 • 8     

     A_rectangle  =  56     

     A_triangle  =  (1/2) • 4 • 4     

     A_triangle  =  8     

     A_shadedarea  =  A_rectangle – A_triangle     

     A_shadedarea  =  56 – 8     

     A_shadedarea  =  48     

_.     

Josie's friend is skinnier than a fashion model.     

A.  No.  Hyperbole is exaggerated statements or claims not meant to be taken literally.     

              Josie's friend could easily be skinnier than a fashion model.     

B.  No.  Personification is the attribution of a personal nature or human characteristics     

             to something nonhuman.  Neither Josie nor Josie's friend is nonhuman.     

             All that leaves is C. Neither for the answer.     

_.     

               y  =  (180 – 75)     

               y  =  105     

     (4x–17)  =  180 – 105     

              4x  =  92     

                x  =  23     This is not the size of the angle, it's only what x equals.     

                                  The angle equals 180 – 105 = 75.     

_.     

3 yards of cotton and 12 yards of denim cost 45 dollars. On the other hand, 5 yards of cotton and 8 yards of denim cost 51 dollars. What is the price of a yard of each?     

3C + 12D  =  45     (eq 1)     

5C +   8D  =  51     (eq 2)     

multiply both sides of     

(equation 2) by 3                        15C +  24D  =  153     

multiply both sides of     

(equation 1) by 2                          6C + 24D  =     90     

subtract                                         9C            =     63     

                                                                  C  =  7     

substitute 7 back into     

(equation 1)                               (3)(7) + 12D  =  45     

                                                      21  +  12D  =  45     

                                                                 12D  =  24     

                                                                     D  =  2     

.    

The 3rd angle in the big triangle  =  180 – (70 + 74)     

                                                     =  36     

                                                 y  =  180 – (36 + 78)     

                                                 y  =  66^o     

                                                 x  =  180 – (66 + 90)     

                                                 x  =  24^o     

_.     

Find the vertex of the parabola y = 2x^2 + 10x + 8     

                                                  y  =  2x² + 10x + 8     

take the 1st derivative               y'  =  4x + 10     

set it equal to zero                     4x + 10  =  0     

                                                           4x  =  –10     

                                                             x  =  –2.5     this is the x-coordinate     

find y when x = 2.5                     y  =  (2)(–2.5²) + (10)(–2.5) + 8     

                                                   y  =  12.5 – 25 + 8     

                                                   y  =  –4.5     this is the y-coordinate     

the vertex is at                           (–2.5 , –4.5)     

_.