Practice Exam III

Problem 1  Graph the inequality.

        2x - 5y  > 10

Solution

First find the intercepts of the line:

First, when x = 0 we get

        -5y  =  10

Divided both sides by -5 to get

        y  =  -2

Next, when y = 0 we get

        2x  =  10

Divide both sides by 2 to get

        x  =  5

In T-table form, this gives

so we get the two points

        (0,-2)  and (5,0)

Now plot the two points and connect with a dashed line as shown below:

Now test the origin (0,0):

        2(0) - 5(0) = 0 > 10

is a false statement, so we shade in the part of he plane that is away from the origin.  The solution is shown below:

Problem 2  Perform the indicated operations.

A.  (3v⁴ - 5v³ + 9v - 1)  +  (2v⁴ + 7v² - 8v - 3)

Solution

We just combine like terms.  Notice that we can drop the parentheses, since we are adding:

        3v⁴ - 5v³ + 9v - 1 + 2v⁴ + 7v² - 8v - 3

        =  3v⁴ + 2v⁴ - 5v³ + 7v² + 9v - 8v - 1 - 3

        =  5v⁴ - 5v³ + 7v² + v - 4

B.  (2x⁵ - x⁴ - 5x³ + 3x² - 6)  -  (4x⁵ - x⁴ - 2x³ + 3x² - 7x - 10)

Solution

We first distribute the "-" through by changing all of the signs of the second expression

        =  2x⁵ - x⁴ - 5x³ + 3x² - 6 - 4x⁵ + x⁴ + 2x³ - 3x² + 7x + 10

Now combine like terms

        =  2x⁵ - 4x⁵ - x⁴ + x⁴  - 5x³ + 2x³ + 3x² - 3x² + 7x - 6  + 10

        =  -2x⁵ - 3x³ + 7x + 4

Problem 3  Find each product

A.  3x⁸yz²(5xy² + 3x³z² - 6y + 1)

Solution

We use the distributive property (multiply through) and use the product rule for exponents:

        =  15x⁸⁺¹ y¹⁺² z² + 9x⁸⁺³ y z²⁺² - 18x⁸y¹⁺¹z² + 3x⁸yz²

        =  15x⁹y³z² + 9x¹¹yz⁴ - 6x⁸y²z² + 3x⁸yz²

B.  (2x - 3)(x³ + 4x - 10)

Solution

Use the distributive law.  First with the 2x and then with the -3:

        =  2x(x³ + 4x - 10) + (-3)(x³ + 4x - 10)

        =  2x⁴ + 8x² - 20x - 3x³ - 12x + 30

Now combine like terms

        =  2x⁴ - 3x³ + 8x² - 20x  - 12x + 30    

        =  2x⁴ - 3x³ + 8x² - 32x + 30    

Problem 4  Find each product

A.  (2x + 9)(3x - 1)

Solution

Use FOIL

        =  (2x)(3x) + (2x)(-1) + (9)(3x) + (9)(-1)

        =  6x² - 2x + 27x - 9

Now combine like terms

        =  6x² + 25x - 9

B.  (3x - 1/2)(4x - 2/3)

Solution

Use FOIL

        =  (3x)(4x) + (3x)(-2/3) + (-1/2)(4x) + (-1/2)(-2/3)

        =  12x² - 2x - 2x + 1/3

Now combine like terms

        =  12x² - 4x + 1/3

C.  (x + 1)(x + 2)(x + 3)

Solution

Use FOIL on the first two factors

        (x + 1)(x + 2)  =  x² + 2x + x + 2  =  x² + 3x + 2

Now multiply this by the third term

        (x + 3)(x2 + 3x + 2)  =  (x)(x2 + 3x + 2) + 3(x2 + 3x + 2)

        =  x3 + 3x2 + 2x + 3x2 + 9x + 6

Now combine like terms

        =  x3 + 6x2 + 11x  + 6

Problem 5  Use the special product formulas to find

A.  (3x - 4)²

Solution

This is the square of a binomial.  We have

        (3x)² + 2(3x)(-4) + (-4)²

        =  9x² = 24x + 16

B.  (7z - 3)(7z + 3)

Solution

This is the difference of squares.  We have

        =  (7z)² - (3)²

        =  49z² - 9

Problem 6  Simplify

A.  (x²y⁵x⁴)¹⁰

Solution

First use the product rule on the x's.

        =  (x²⁺⁴y⁵)¹⁰  =  (x⁶y⁵)¹⁰

Use the power rule. 

        =  x^(6)(10)y^(5)(10)

        =  x⁶⁰y⁵⁰

B.       -20(a²b)⁷ 
                            
          (2ab⁵)²a⁵

Solution

First use the power rule (multiply the 7 through the top and the 2 through the bottom.)

               -20a¹⁴b⁷ 
       =                        
               4a²b¹⁰a⁵

Now use the product rule on the a's.

               -20a¹⁴b⁷ 
       =                       
                4a⁷b¹⁰

Now use the quotient rule.

               -5a⁷
       =               
                b³

Problem 7  Simplify and write without negative exponents.

        (x ^-3 y) ^-2(x ^-4)

Solution

First use the power rule

         =  (x⁶ y^-2)(x ^-4)

Now use the product rule

         =  x² y^-2

Now remember that we can write without negative exponents.      

               x²
       =          
             y²

Problem 8  The Milky Way galaxy is about 120,000 light years across.  One light year is 950,000,000,000,000 meters long.

A.  Write both of these numbers in scientific notation.

Solution

The first number is

        1.2 x 10⁵

and the second number is

        9.5 x 10¹⁴

B.  How many meters wide is the milky way galaxy?  Write your answer in scientific notation.

Solution

We just multiply the two numbers.

        (1.2 x 10⁵)(9.5 x 10¹⁴)  =  (1.2)(9.5)(10⁵)(10¹⁴)

        =  11.4 x 10¹⁹  =  1.14 x 10¹ x 10¹⁹  =  1.14 x 10²⁰

Problem 9  Solve each compound inequality.  Write the solution set using interval notation and graph it.

A.  x - 4 > 1   or    7 - x  > 5

Solution

We add 4 to both sides of the first inequality and -7 to both sides of the second.

        x  >  5  or   -x  >  -2

Now multiply both sides of the second inequality, remembering to switch the inequality sign

        x  >  5  or  x  <  2

In interval notation we get

        (- , 2)  U  (5, )

Now put this on a number line

B.  2/3 x  < 4   and   x - 5  > 4

Solution

Multiply the both sides of the first by 3/2 and add 5 to both sides of the second to get

        x  <  6  and  x  >  9

Notice that these cannot happen simultaneously, so the solution is the empty set. 

Problem 10  Solve the absolute value inequality and graph the solution set.

A.  3|2 - x| + 5  >  14

Solution

First subtract 5 from both sides to get

        3|2 - x|  >  9

Then divide both sides by 3 to get

        |2 - x|  >  3

Now turn it into an or statement to find the endpoints

        2 - x  =  3    or     2 - x  =  -3

Subtract 2 from both sides to get

        -x  =  1     or     -x  =  -5

Now multiply both sides by -1 to get

        x  =  -1     or     x  =  5

Now check test points.  First if x = -2, the equation gives

        3|-2 - 2| + 5  =  3|-4| + 5  =  3(4) + 5  =  12 + 5  =  17

which is greater than 14, so the region to the left of -1 is included.  Now test x = 0.

        3|0 - 2| + 5  =  3|-2| + 5  =  3(2) + 5  =  6 + 5  =  11

which is not greater than 14, so the middle region is not included.  Finally test x = 6 to get

        3|6 - 2| + 5  =  3|4| + 5  =  3(4) + 5  =  12 + 5  =  17

which is greater than 14, so the region to the right of -1 is included.  In interval notation, we get

        (- , -1)  U  (5, )

The graph is shown below

B.  5|6 - x| +7  >  3

Solution

First subtract 7 from both sides to get

        5|6 - x|  >  -4

Then divide both sides by 5 to get

        |6 - x|  >  -4/3

Now notice that the right hand side is negative.  The absolute value if always greater than a negative number, so the solution is all real numbers.

Problem 11  Find the domain and range of each relation.  Then determine whether it is a function.

A.  {(1,2), (2,4), (4,7), (4,8)}

Solution

The domain is the set of first coordinates:

        {1,2,4}

and the range is the set of second coordinates:

        {2,4,7,8}

Since the number 4 gets sent to both 7 and 8, this is not a function.

B.  x + 6y  = 12

Solution

This is the equation of a line.  Notice that the slope is defined:

        6y  =  -x + 12

        y  =  -1/6 x  + 2

Since the slope is defined, the line is not vertical, hence the graph passes the vertical line test, and the equation is a function.

C.   

Solution

This graph passes the Vertical Line Test, since every vertical line passes through the graph at most once, hence it is the graph of a function.

Problem 12  Let

    f(x) = 3x - 5     and     g(x)  =  x² - x

Find

A.  (f + g)(x)

Solution

Just add

        (3x - 5) + (x² - x)

        =  3x - 5 + x² - x

Now combine like terms and put in descending order to get

        =  x² + 2x - 5

B.  (f - g)(3)

Solution

Subtract to get

        (3x - 5) - (x² - x)

Now multiply the "-" through to get

        3x - 5 - x² + x

Now combine like terms and put into descending order to get

        -x² + 4x - 5

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