Real Numbers

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Chapter 1

• Real Numbers

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1.5 Equations

• Sections 1.1 1.4 are review. You will not be
  tested over these sections.
• All of 1.5 is fair game for tests. The linear
  equation section is YOUR responsibility.

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Quadratic Equations

• Definition
• A quadratic equation has the form

where a, b, and c are real numbers with
Zero Product Property
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Solving Using the Zero Product Property
1. Set equal to zero.
2. Factor.
3. Set each factor to zero.
4. Solve.
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Extracting Roots

• Solving Simple Quadratic Equations

Example
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Consider the following
it neither factors nor is it a simple
quadratic equation.
We solve an equation like this using a process
called Completing the Square.
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Step 1 Separate the variable terms and the
constant term.
Step 2 Divide the coefficient of x by 2 and add
its square to both sides of the equation.
Step 3 Factor the left hand side, and simplify
the right hand side.
Step 4 Take square root of both sides.
Step 5 Solve.
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Step 1 Separate the variable terms and the
constant term.
Step 2 Divide the coefficient of x by 2 and add
its square to both sides of the equation.
Step 3 Factor the left hand side, and simplify
the right hand side.
Step 4 Take square root of both sides.
Step 5 Solve.
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Step 1 Separate the variable terms and the
constant term.
Step 2 Divide each term by 2.
Step 3 Divide the coefficient of x by 2 and add
its square to both sides of the equation.
Step 3 Factor the left hand side, and simplify
the right hand side.
Step 4 Take square root of both sides.
Step 5 Solve.
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The Quadratic Formula

• The solutions of are given by
• Where does it come from?
• Completing the square!

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The Discriminant

• The radicand in the quadratic formula is called
  the discriminant. The discriminant shows the
  nature of solutions.
• If , then there are two different real number
  solutions.
• If , then there is one solution.
• If , then there are two different complex
  solutions.

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Solving Rational Equations
Step 1 Multiply each term by the LCD of all the
denominators to eliminate the denominators.
Step 2 Solve the equation.
Step 3 If a variable is in the denominator of
the original equation, it is imperative that you
check your answers.
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Step 1 When a variable is in the denominator,
first note any restricted values.
Step 2 Multiply each term by the LCD of all the
denominators to eliminate the denominator.
Step 3 Solve the equation.
Step 4 If a variable is in the denominator of
the original equation, it is imperative that you
check your answers.
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Radical Equations
Step1 Isolate a radical.
Step 2 Square (or cube) both sides. Remember to
FOIL.
Step 3 Repeat 1 and 2 if radical still exists
Step 4 Solve.
Step 5 Check.
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Step1 Isolate a radical.
Step 2 Square (or cube) both sides. Remember to
FOIL.
Step 3 Repeat 1 and 2 if radical still exists
Step 4 Solve.
Step 5 Check.
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Equations Reducible to Quadratic

• Some equations are quadratic in form and can be
  solved as a quadratic equation. Consider
• Let and substitute into the
  equation.Now the equation is quadratic. Solve
  the resulting equation.

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1.6 Modeling With Equations

• Word Problems!!
• Follow five steps when solving word problems.

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Five Steps for Problem Solving

1. Read the problem. Identify what you are looking
   for.
2. Define variables.
3. Set up the equation.
4. Solve the equation.
5. Check. Is your answer reasonable? Write the
   answer using a complete sentence.

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1.7 Inequalities

• Linear Inequalities Review!
• Remember to change the direction of the
  inequality symbol when multiplying or dividing by
  a negative number.

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Two Part Inequalities

• Should be review.
• Be able to solve an inequality, graph the
  solution on a number line, and express the
  solution using interval notation.

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Compound Inequalities

• Conjunction and
• Three part inequality
• Be sure to add, subtract, multiply, divide in all
  three sections.

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Absolute Value Inequalities

• Consider
• Two solutions exist. x 5 or x -5.
• What about ?
• Any x, , will have an absolute value less
  than 5.

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Absolute Value Inequalities

• When solving an absolute value inequality of the
  form , we form the compound inequality of
  the form .
• Consider
• Set up the inequality and solve.

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Absolute Value Inequalities
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Absolute Value Inequalities

• Consider .
• Any x, ,will have an absolute value
  greater than 5.
• To solve, set up two inequalities joined with an
  or.

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Absolute Value Inequalities

• Consider
• Set up the two inequalities and solve.

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Nonlinear Inequalities
Step 1 Find the zeros.
Step 2 Divide the number line into three
regions using the zeros.
2
3
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Step 3 Construct a sign table
Interval



Sign of
Positive
Positive
Negative
Sign of
Positive
Negative
Negative
Sign of
Positive
Negative
Positive
Note Even number of negative factors, the
product is positive. Odd number of negative
factors, the product is negative.
Solution
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1.8 Coordinate Geometry
Derivation of the Distance Formula
y
x
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DISTANCE FORMULA
The distance between the points and
in the plane is
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MIDPOINT FORMULA
The midpoint of the line segment from to is
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Intercepts of a Graph

• X-Intercepts
• Set y0 and solve for x.
• The x-intercepts occur where the graph of a
  function crosses the x-axis.

• Y-Intercepts
• Set x0 and solve for y.
• The y-intercepts occur where the graph of a
  function crosses the y-axis.

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CIRCLES
The standard form equation of a circle with
center (h,k) and radius r is
If the center of the circle is the origin (0,0),
then the equation is
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Symmetry in Graphs
Symmetry with respect to the x-axis
y
(x, y)
Equation is unchanged when y is replaced by y.
x
(x, -y)
Graph is unchanged when reflected in the x-axis.
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y
Symmetry with respect to the y-axis.
(-x, y)
(x, y)
Equation is unchanged when x is replaced by x.
x
Graph is unchanged when reflected in the y-axis.
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Symmetry with respect to the origin.
y
(x, y)
Equation is unchanged when x is replaced by x
and y by y.
x
(-x, -y)
Graph is unchanged when rotated 180 degrees about
the origin.
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1.10 Lines

• Review!
• Be able to graph a line.
• Be able to find the equation of a line.

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Slope
The slope m of a line containing points and is
given by
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Point-Slope Equation
The point-slope equation of the line with slope m
passing through is
Example Find an equation of the line containing
the point (1/2, -1) and with slope 5.
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Slope-Intercept Form
The slope-intercept equation is given bywhere
m is the slope, and (0,b) is the y-intercept
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Example Find the slope and y intercept of the
line with equation
Step 1 Separate the y term from the other
terms.
Step 2 Divide each term by -6.
Step 3 Simplify.
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Horizontal and Vertical Lines

• An equation of the horizontal line through the
  point (a, b) is are given by equations of the
  type . The slope is 0.
• An equation of the vertical line through the
  point (a, b) is . The slope is undefined.

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Parallel and Perpendicular Lines

• Two lines are parallel if and only if they have
  the same slope
• Vertical lines are parallel. Horizontal lines
  are parallel.
• Two lines are perpendicular if and only if the
  product of their slopes is 1.
• A vertical line and a horizontal line are
  perpendicular.