Intermediate Algebra: A Graphing Approach

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Intermediate Algebra A Graphing Approach

• 1.3 Operations on Real Numbers
• 1.6 An Introduction to Problem Solving
• 1.7 Numerical Approach Modeling with Tables
• 1.8 Formulas and Problem Solving

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Section 1.3

• Adding 2 numbers with same sign
• Add their absolute values.
• Use common sign as sign of sum.
• Adding 2 numbers with different signs
• Take difference of absolute values (smaller
  subtracted from larger).
• Use sign of larger absolute value as sign of sum.

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Add the following numbers. (-3) 6 (-5)
-2
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• Note that to evaluate expressions with a
  calculator, we sometimes need to insert
  parentheses that may not be shown in the
  expression.
• This is especially true when entering a
  fraction whose numerator or denominator contains
  more than one term.
• The fraction bar notation when writing
  fractions tells us to calculate the numerator and
  denominator first.
• We have to include parentheses when inputting
  these values to a calculator, where all the
  values will be input on a single line of
  information.

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• Subtracting real numbers
• Substitute the opposite of the number being
  subtracted
• Then add
• a b a (-b)

Subtract the following numbers. (-5) 6 (-3)

(-5) (-6) 3
-8
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• Multiplying or dividing 2 real numbers with same
  sign
• Result is a positive number
• Multiplying or dividing 2 real numbers with
  different signs
• Result is a negative number

Find each of the following products. 4 (-2)
3
-24
(-4) (-5)
20
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• Note that when inputting fractions for
  computations in a calculator, the results will
  often be displayed in decimal form.
• However, there is often a method for converting
  the answer to fraction form, or for having the
  calculator display the result in fraction form
  directly.
• You should determine how your calculator will
  convert these results (perhaps by reading the
  users manual).

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• If b is a real number, 0 b b 0 0.
• Reciprocals are two numbers whose product is 1.
• Quotient of any real number and 0 is undefined.

• Quotient of 0 and any real number 0.

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• Note that if you input a quotient that involves
  dividing by 0 into your calculator, you will get
  some type of error message.
• Sometimes this will involve a choice between
  stopping the calculation, or going back to fix
  the denominator.

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If a and b are real numbers, and b ? 0,
Simplify the following.
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• Exponential notation for 35
• Base is 3
• Exponent is 5
• The notation means 3 3 3 3 3

Evaluate 35. 3 3 3 3 3 243
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• Opposite of squaring a number is taking the
  square root of a number.
• A number b is a square root of a number a if b2
  a.
• In order to find a square root of a, you need a
  that, when squared, equals a.

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• The principal (positive) square root is noted as

The negative square root is noted as
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• The cube root of a real number a

Note a is not restricted to non-negative
numbers for cubes.
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Note that while graphing utilities will usually
contain a square root key (often as a secondary
function of the squaring key), they will contain
other root options under a menu, such as a math
menu.
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• Order of operations
• Grouping symbols
• Parentheses, absolute values, brackets, braces,
  fraction bars
• Exponential expressions left to right
• Multiplication and division left to right
• Addition and subtraction left to right

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Simplify the following expressions.
Remember that when inputting this second example
into a graphing utility, you will have to insert
parentheses around the numerator and denominator
to get it to follow the proper order of
operations that you want for this problem.
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• We can evaluate an algebraic expression by
  assigning specific values to any variables that
  might be in the expression.

Evaluate 3x2 2y 5 when x -2 and y
-4. 3(-2)2 2(-4) 5 34 8 5
12 8 5 25
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Note that to evaluate an algebraic expression
(such as the last example) in a graphing utility,
you can either 1) enter the numerical values
in the expression and compute, or 2) store the
numerical value into the variable location, then
enter and evaluate the algebraic expression.
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Section 1.6

• General strategy for problem solving
• Understand the problem.
• Read and reread the problem.
• Choose a variable to represent the unknown.
• Construct a drawing, whenever possible.
• Propose a solution and check.
• Translate the problem into an equation.
• Solve the equation.
• Interpret the result.
• Check proposed solution in problem.
• State your conclusion.

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The product of twice a number and three is the
same as the difference of five times the number
and ¾. Find the number.
Read and reread the problem. If we let x
the unknown number, then twice a number
translates to 2x, the product of twice a
number and three translates to 2x 3,
five times the number translates to 5x, and
the difference of five times the number and ¾
translates to 5x - ¾.
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2x 3 5x ¾
6x 5x ¾ (simplify left side)
x - ¾ (simplify both sides)
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Check Replace number in the original
statement of the problem with -¾. The product
of twice -¾ and 3 is 2(-¾)(3) -4.5. The
difference of five times -¾ and ¾ is 5(-¾) ¾
-4.5. We get the same results for both
portions. State The number is -¾.
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A car rental agency advertised renting a Buick
Century for 24.95 per day and 0.29 per mile.
If you rent this car for 2 days, how many whole
miles can you drive on a 100 budget?
Read and reread the problem. Lets propose that
we drive a total of 100 miles over the 2 days.
Then we need to take twice the daily rate and add
the fee for mileage to get 2(24.95) 0.29(100)
49.90 29 78.90. This gives us an idea of how
the cost is calculated, and also we know that
the number of miles will be greater than 100. If
we let x the number of whole miles driven,
then 0.29x the cost for mileage driven
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2(24.95) 0.29x 100
49.90 0.29x 100 (simplify left side)
0.29x 50.10 (simplify both sides)
x ? 172.75 (simplify both sides)
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Check Recall that the original statement of the
problem asked for a whole number of miles. If
we replace number of miles in the problem with
173, then 49.90 0.29(173) 100.07, which is
over our budget. However, 49.90 0.29(172)
99.78, which is within the budget. State The
maximum number of whole number miles is 172.
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Section 1.7

• In the two previous sections, we have used an
  algebraic approach to problem solving.
• In this section, we will introduce a numerical
  approach that uses tables.

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• To generate tables on a graphing utility, it is
  necessary to use x as the independent variable
  (the variable to be input to the algebraic
  expression).
• We enter the expression to be evaluated into y1
  (or another subscripted value of y) using the Y
  key.
• Then we need to specify the minimum x-value to
  start with, and the increment or change in the
  x-value under the table setup menu. (In many
  calculators, the Greek letter delta, ?, is used
  to represent the change in x.)
• Now we can display the table.

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• Suppose the cost of tuition at the local
  community college is 52.50 per credit unit for
  in-state students and 212.50 per credit unit for
  out-of-state students.
• Write an algebraic representation, y1, for the
  cost of in-state tuition dependent on x, the
  number of credit units.
• Write an algebraic representation, y2, for the
  cost of out-of-state tuition dependent on x, the
  number of credit units.
• Complete a table that evaluates the tuition costs
  for both in-state and out-of-state students
  taking from 6 to 12 credit units, inclusive.
• What would be the tuition costs for an
  out-of-state student taking 15 credit units, if a
  5 computer lab fee must be included?

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a) To model the problem in words,
Now we translate to get

• In a similar fashion to the above work, we would
  get

y2 212.50x
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c) For the table, we enter both the y1 and y2
algebraic expressions, a start value of 6, and
increment of 1.
6 315 1275 7
367.5 1487.5 8 420 1700
9 472.5 1912.5 10 525
2125 11 577.5 2337.5 12
630 2550
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• Although the tuition costs for 15 credit units
  does not appear in our table, we can generate the
  formula (that includes the added computer lab
  fee) for our calculator from the work we have
  already completed.
• y 212.50(15) 5 3192.50

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Baskets, Inc., is planning to introduce a new
woven basket. The company estimates that 640
worth of new equipment will be needed to
manufacture this new type of basket and that it
will cost 15 per basket to manufacture. The
company also estimates that the revenue from each
basket will be 31. Use a table to find the
break-even point, when the cost of manufacturing
baskets will equal the revenue from selling those
baskets.
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Read and reread the problem. The break-even
point will occur when the cost of manufacturing
the baskets is the same as the revenue that is
generated from the baskets.
We can define the cost of the equipment, cost per
basket, and revenue per basket made in terms of
the number of baskets that are made and sold.
So, we let x number of baskets made and sold
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So we need to find when C(x) R(x)
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For the table, we enter both the y1 C(x) and y2
R(x) algebraic expressions, a start value of 0,
and increment of 10.
0 640 0 10 790
310 20 940 620 30 1090
930 40 1240 1240 50
1390 1550 60 1540 1860
The table values are equal when x 40.
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Check Since the table entries for both Cost and
Revenue are the same when x 40, we do not need
to substitute into the equations again. State
The sale and production of 40 baskets will
produce the break-even point.
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In addition to linear models, we can use
quadratic models with the methodology of tables.
A firecracker rocket is fired from ground level.
Neglecting air resistance, the height of the
rocket at time x seconds is given by the equation
y -16x2 150x, where y is measured in
feet. Using a table, determine to the nearest
tenth of a second when the rocket hits the ground
(assuming it doesnt explode before hitting).
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0 0 1 134 2
236 3 306 4 344 5
350 6 324 7 266 8
176 9 54 10 -100
First we need to determine between which two
whole number seconds the rocket will hit the
ground. (Note that the height above the ground
will be 0 when it hits.) Input y1 -16x2 150x,
start value of 0 and increment of 1. Note you
will have to scroll with the down arrow to access
all the data values you need.
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We now know the rocket will hit the ground
between the 9th and 10th second. We can use a
table again, using the same y1, with start of 9,
and increment of 0.1
9 54 9.1 40.04 9.2
25.76 9.3 11.16 9.4 -3.76
9.5 -19 9.6 -34.56
Since 3.75 is closer to 0 than 11.16, we would
use the value of 9.3 seconds as when the rocket
would hit the ground.
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Section 1.8

• A formula is an equation that states a known
  relationship among multiple quantities (has more
  than one variable in it).

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A lw (Area of a rectangle length
width) I PRT (Simple Interest
Principal Rate Time) P a b c
(Perimeter of a triangle side a side b side
c) d rt (distance rate time) V lwh
(Volume of a rectangular solid length
width height) C 2?r (Circumference of a
circle 2 ? radius)
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• Steps for solving formulas
• Multiply to clear fractions.
• Use distributive to remove grouping symbols.
• Combine like terms to simply each side.
• Get all terms containing specified variable on
  the same time, other terms on opposite side.
• Isolate the specified variable.

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A flower bed is in the shape of a triangle with
one side twice the length of the shortest side,
and the third side is 30 feet more than the
length of the shortest side. Find the dimensions
if the perimeter is 102 feet.
Read and reread the problem. Recall that the
formula for the perimeter of a triangle is P a
b c. If we let x the length of the
shortest side, then 2x the length of the
second side, and x 30 the length of the
third side
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Formula P a b c
Substitute 102 x 2x x 30
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102 x 2x x 30
102 4x 30 (simplify right side)
72 4x (simplify both sides)
18 x (simplify both sides)
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Check If the shortest side of the triangle is
18 feet, then the second side is 2(18) 36 feet,
and the third side is 18 30 48 feet. This
gives a perimeter of P 18 36 48 102 feet,
the correct perimeter. State The three sides of
the triangle have a length of 18 feet, 36 feet,
and 48 feet.