Practice Exam 1

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

 

Problem 1  Simplify the following expressions.

A.  |-7|2 - |33|

Solution

|-7|2 - |33|  =  72 - 33  =  49 - 27 = 22

 

B. 37 - 18 - (-11)

Solution

        37 - 18 -(-11)  =  19 - (-11)  =  19 + 11  =  30

C.  (7 + 11) ÷ 6 - 2

Solution

        (7 + 11) ÷ 6 - 2  =  18÷ 6 - 2   =  3 - 2  =  1

D.  -42  + (-4)2

Solution

Notice that for the first term the order of operations tell us that the "-" is performed after the square and for the second term the parentheses force the negative to be performed before the square.  We have

    -(4)(4) + (-4)(-4)  =  -16 + 16  =  0

 

E.  2(2x - 3) + 3(4 - x)

Solution

        2(2x - 3) + 3(4 - x)  =  4x - 6 + 12 - 3x

        =  4x - 3x - 6 + 12  =  x + 6

 

F.  2x(x - 3) + 5(3x + 1)

Solution

Use the distributive property

        (2x)(x) + (2x)(-3) + 5(3x) + (5)(1)

        =  2x2 - 6x + 15x + 5

        =  2x2 + 9x + 5

 

         6x + 3
G.                    
             3

Solution

Notice that the 3's do not cancel, since there are two terms in the numerator.  Instead, we split the numerator:

      6x              3
               +              =  2x + 1
       3              3

H.  0.3(x + 4) - 0.1(x + 2)

Solution

Use the distributive law:

        0.3x + (0.3)(4) - 0.1x - (0.1)(2)  =  0.3x + 1.2 - 0.1x - 0.2

        =  0.2x + 1

 

Problem 2  Evaluate the following expression when x = 2, y = -1, and z = 3

     x3y - 4y2 +2xz -3z + 2

Solution

We have

        (2)3(-1) - 4(-1)2 + 2(2)(3) - 3(3) + 2

        =  (8)(-1) - (4)(1) + 12 - 9 + 2

        =  -8 - 4 + 12 - 9 + 2

        =  -12 +12 - 9 + 2

        =  0 - 9 + 2

        =  -9 + 2

        =  -7

 

Problem 3  Give the name of the property that the following identity uses.

A.  (2x + 3) + y  =  2x + (3 + y)

Solution

This is the associative property of addition, since we are just regrouping.

 

B.  z + (3 - x)  =  (3 - x) + z

Solution

This is the commutative property of addition since we are changing the order of the terms.

 

Problem 4

Solve the following equations.  Then identify each as a conditional equation, an inconsistent equation, or an identity.

A.  3x - 5  =  4x

Solution

We subtract 3x from both sides

        3x - 5  =  4x
        -3x          -3x
________________
            -5  =  x

        x = -5  This is a Conditional Equation

B.  5(x - 3) - (x + 5)  =  4x - 1

Solution

First distribute

        5x - 15 - x - 5  =  4x - 1

Combine like terms

        4x - 20  =   4x - 1
        -4x + 20    -4x + 20
__________________________
                0  =  19

This is an inconsistent equation (No Solution)

 

       

                x             2x
C.  3  +          =            
                3              5

Solution

Multiply all three terms by the common denominator 15

                    (15)  x             (15)2x
    (15)  3  +               =                    
                      3                       5

        45 + 5x  =  6x
                -5x    -5x
  _________________
             45   =  x

        x  =  45

This is a conditional equation.

 

              x            x              x
D.  1 -           +            =               + 1
              4            2              4

Solution

Multiply all five terms by the common denominator 8

                (8)x           (8)x           (8) x
    (8)1 -              +               =                 + (8)1
                 4                2                4

        8 - 2x + 4x  =  2x + 8

        8 + 2x  =  2x + 8

This is an identity.

 

Problem 5

Translate each verbal expression into an algebraic expression with one variable:

A.  3 less than twice a number

Solution

Twice a number can be represented by 2x.  Three less than this is

        2x - 3

B.  The area of a rectangle given that the width is 5 meters more than the length.

Solution

If L is the length of the rectangle, then the width is 5 + L.  The area is the product of the length and the width. 

    Area  =  L(5 + L)

C.  The product of two consecutive even numbers is 168.

Solution

If the first number is x then the next even number is x + 2.  their product is x(x + 2).  We can now write

        x(x + 2)  =  168

Problem 6

Determine whether the given number is a solution to the equation following it

A.   -6, -x + 5 = 11

Solution

We plug in (-6) where we see an x:

        -(-6) + 5 =? 11

        6 + 5  =? 11

    Yes x = -6 is a solution to the equation.

             x - 4
B.  2,                =  3
             x + 1

Solution

We plug in (2) where we see an x:

        (2) - 4
                       =?  3
        (2) + 1

         -2
                  =?  3
          3

Since the left hand side does not equal the right hand side, we can conclude that x = 2 is not a solution to the equation.

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