Transcript Solve

Chapter 1
Equations and Inequalities
Aim #1.1: How do we graph
and interpret information?
 Analytic Geometry this new branch of geometry
founded by Rene Descartes brings Algebra and
Geometry together.
 Key Terms
 Rectangular Coordinate System or Cartesian
Coordinate System
 X-axis,
 Y-axis
 Quadrants
Graphs of Equations

X
Y = 4 – x2
3
-5= 4 - 32
(X, Y)
(3, -5)
Solution of an equation that
satisfies the equation.
Using a Graphing Calculator
 Using your calculator Y = 4 – x2

y x
 Understanding the Viewing Rectangle
 [-10, 10, 1]
Minimum X-value, Maximum X- value and X scale
Intercepts
 X-intercept- is the point on the graph where it
intersects with the X- axis. Ex. (X, 0)
 Y-intercepts – is the point on the graph where it
intersects with the Y-axis. Ex, (0, Y)
 Look at Text- Identifying Intercepts
Interpreting Information
from Graphs
 Line Graph used to illustrate trends or data over
time.
 Look at Text
Summary:
Answer in complete sentences.
 What is the rectangular coordinate system?
 Explain why (5, -2) and (-2, 5) do not represent the
same point.
 What does [-20, 2, 1] by [ -4, 5, 0.5] viewing rectangle
mean?
 Determine whether the following is true or false.
Explain.
 If the product of a point’s coordinates is positive, the
point must be in quadrant I.
Warm-up: 8/23/2011
 Take out your homework and unit plan. Copy the questions and Answer.
 Explain how to graph an equation in the rectangular
coordinate system.
 Explain how to graph the point (7, -8) on the
rectangular coordinate system.
 Sketch a function that models the following
situation.
 As the blizzard got worse, the snow fell harder and
harder.
Aim #1.2: How do we solve
equations?
 What is a linear equation?
 A linear equation has one variable and can be
written in the form of y= mx +b, where a and b are
real numbers and a ≠ 0.
 How do we solve? 4x + 12= 0
Practice:
 4x + 5 = 29
 6x – 3 = 63
How do we solve other types
linear equations?
 Solve 2 (x – 3) – 17 = 13
 4( 2x + 1) = 29 + 3 (2x – 5)
Practice:
 5x – (2x – 10 ) = 35
 3x + 5 = 2x + 13
 13 x + 14 = 12 x – 5
How do we solve an equation w/ a
fraction?
x  2 x 1

2
4
3
 Steps:
 Multiply the entire equation
by a multiple of the
denominator. For this
example, it would be …
 This would get rid of all
denominators.
 Then combine like terms
and isolate the variable.
Guided Practice:
x3 5 x5
 
4
14
7
What is a Rational Equation?
 It is an equation containing
one or more rational
expressions.
 Solving:
1 1 3
 
x 5 2x
 Hint: Multiply
the entire
equation by the
LCD or multiple
of x, 5, and 2x.
Solving a Rational
Equation
x
3

9
x3 x3
Solving a Rational
Equation
x
2
2


x 2 x 2 3
Categorizing the Different Equations
Key Terms
1. Identity Equation is an equation that is true for
all values of x. Ex.: x + 3= x + 2 + 1
2. Conditional Equation is true for a particular
value of x.
Ex. 2x + 5 = 10
x = 2.5
3. Inconsistent Equation is an equation that is not
true for any value of x.
Ex. x = x + 7
Summary:
Answer in complete sentences.
1. What is a linear equation in one variable? Give an example.
What are some other types of linear equations?
2. Does the following make sense? Explain your reasoning.
 Although I can solve 3x + 1/5 = ¼ by first subtracting 1/5 on
both sides, I find it easier to multiply by 20, the least common
denominator, on both sides.
3. Is the equation (2x- 3)2 = 25 equivalent to 2x – 3 = 5? Explain.
Warm-up: 8/25/2011
 Take out your homework and unit plan. Copy the questions and Answer.
 Solve for X.
1. 7x – 5 = 72
2. 6x – 3 = 60
3. 13x + 14 = 12 x – 5
Warm-up: 8/25/2011
 Take out your homework and unit plan.
 Copy the questions and Answer.
 Solve for X.
2x x
a.   1
3 6
x 1 x 1
b.   
2 10 5 2
c.7 x  13  2(2 x  5)  3 x  23
Aim #1.3:How do we use linear
equations to model situations?
 Steps to solving Word Problems
1. Read the problem. Twice.
2. Define x =
3. Write your equation.
4. Solve.
5. Check your solution. Does it make sense?
Using the 5-step strategy and find
the number.
 When two times a number is decreased by 3, the
result is 11. What is the number?
 Let x= number
 Equation: 2x – 3 = 11 Solve for x.
 When a number is decreased by 30% of itself, the
result is 20. What is the number?
 Let x = the number
 Equation x • __ = ___
 Solve and Check your answer.
Solving a Formula for a
Variable
 2L + 2W = P Solve for l
 Your goal is to isolate L.
 Subtract 2W on both sides.
 Then divide by 2 on both
sides to isolate L.
 What’s your final answer?
 Note: 2W is like one term
because you are multiplying
W by 2.
Check for Understanding
 Solve the formula for W.
 2L + 2W = P
Solving a Formula for a Variable
that Occurs Twice
 Solve for P.
 A=P+Prt
 Factor out the P from P + P
rt.
 Then divide by the
expression in parenthesis to
isolate P.
Check for Understanding
 Solve the formula for C.
 P = C + MC
Summary:
Answer in complete sentences.
 Explain how to solve for P below and then solve.
 T= D + pm
 What does it mean to solve for a formula?
 Write an original word problem that can
solved using a linear equation. Then write
out all the steps for the solution.
Warm-up: 8/26/2011
 Take out your homework
and Unit Plan.
 What is the difference
between an identity,
conditional and inconsistent
equation?
 Solve for r.
1. A  r 2
2. A  4r
1
3. A  r  y
2
 Agenda
1. Go over hw.
2. Quiz
3. Review
Warm-up: 8-29-2011
 Take out your
homework and Unit
Plan.
 Reminder- Quiz FridayP.6B through- 1.5
 Note: We are skipping 1.4
 Solve.
at this time.
1.) 7( x  2)  4( x  1)  21
2.)  10  3(2 x  1)  8 x  1  0
2x  3 x  4 x 1
3)


4
2
4
Warm-up: 8/30/2011
 Take out your homework
and Unit Plan.
 Solve for x.
 x7
37 x  2


3
15
5
2x x
 4
5 3

 Reminder Quiz Friday P.6B
-1.5
Warm-up: 8-31 or 9-1
 Take out your homework and unit plan.
 First, write the value (s) that make the denominator
(s) zero. Then solve the equation.
x 6
x8
2
3x
x
 What type of equation is this?

Aim #1.5: How do we solve
quadratic equations?
 Definition of a Quadratic Function:
 A quadratic equation is an equation that can be
written in the general form of ax2 + bx + c = 0,
where a, b, and c are real numbers and a≠0.
i.e. Quadratics are second degree polynomial functions.
Solving Equations by the
Square Root Property
a. 3 x  15  0
 Notice in the examples to
b. 9 x  25  0
 Steps:
c.( x  2)  6
2. The find the square root of
both sides.
2
2
2
the left that there is no b
term.
1. Isolate the x2 term.
3. Simplify.
Check for Understanding:
 Solve by the Square Root Property.
1. 3 x  21  0
2
2. 5 x  45  0
2
3.( x  5)  11
2
Zero-Product Principle

If the product of two algebraic expressions is zero, then at least
one of the factors must equal to zero.
 If AB= 0 then A= 0 or B= 0

Example: x2 + 7x + 10 = 0
1. Factor.
2. Set each factor = 0
3. Solve.
(X + 5)(x+2) = 0
X+ 5 = 0 or x+2 = 0
x = -5 or x = -2
Solving Quadratic Equations by
Factoring
a. 4 x  2 x  0
2
b. 2 x  7 x  4
a. Is there a GCF, that can
factored out?
2
b. Subtract 4 on both sides and
set equation = 0. Now factor.
Solving Quadratics by Completing
the Square
 Completing the Square is a strategy to solve
quadratics when:
1. The Trinomial can not be factored
2. Zero Product property can not be used
Completing the square allows us to convert the
equation so that it can be solved using the square
root property.
 Completing the Square:
2
 If x + bx is a binomial, then by adding
b
 
 2 ,
which is
the square of half the coefficient of x,a perfect
square trinomial will result. That is,
2
b
b 
x  bx      x  
2
2 
2
2
Completing the Square

What term should be added to each binomial so
that it becomes a perfect square trinomial? Write
and factor the trinomial.
a. x2+ 8 x
b. Solution:
2
8
Add    4 2.
2
Add 16 to complete the square.
x  8 x  16  ( x  4)
2
2
Completing the Square
 What term should be added to each binomial so that
it becomes a perfect square trinomial? Write and
factor the trinomial.
b. x  7 x
2
3
c. x  x
5
2
Completing the Square
 What term should be added to each binomial so that
it becomes a perfect square trinomial? Write and
factor the trinomial.
a. x  6 x
2
b. x  5 x
2
2
c. x  x
3
2
Solving Quadratics using
Completing the Square
x  6x  4  0
2
 Steps:
1. Subtract 4 on both sides.
2. Take the b term and divide
by 2 and square it.
3. Now add it to both sides of
the equation.
4. Now you can express the left
hand side as a square.
5. Apply the square root
property and solve for x.
Solving Quadratics using
Completing the Square
 Guided Practice:
x2  4x 1  0
Solving Quadratics using
Completing the Square
9x  6x  4  0
2
 Steps:
1. Divide the entire equation
by 9, so that a = 1.
2. Add -4/9 to both sides.
3. Complete the square.
4. Then solve for x.
Solving Quadratics using
Completing the Square
 Guided Practice:
2 x  3x  4  0
2
What is the Quadratic
Formula?
 Quadratic formula: If ax2 + bx + c = 0 and a≠0
 b  b  4ac
x
2a
2
Using the Quadratic
Formula
 Solve x2 + 6 = 5x
 Steps:
1. Subtract 5x on both
sides, so the equation = 0.
2. Identify the values for a,
b, and c.
a= 1, b = -5, c = 6
3. Then substitute into the
x
b
b 2  4ac
2a
What is the discriminant?
Property of the Discriminant
For the equation ax2 + bx + c= 0, where
a ǂ 0, you can use the value of the
discriminant to determine that
number of solutions.
If b2 – 4ac > 0, there are two solutions.
If b2 – 4ac = 0, there is one solution.
If b2 – 4ac <0, there are no solutions
Using the Discriminant
 For each equation, compute the
discriminant. Then determine the
number and type of solutions.
a. 3 x 2  4 x  5  0
b. 9 x  6 x  1  0
2
c. 3 x 2  8 x  7  0
Summary:
Answer the following in complete
sentences.
1. What is a quadratic equation?
2. What are at least 3 different ways of solving
a quadratic?
3. When is using the square root property
helpful?
(Think of at least 2 ways.)
4. Solve using any method. Explain the method
you choose and why.
x  6x  8  0
2
Warm-up: 9-2-2011
 Take out your homework and Unit Plan.
 Find the x intercepts of the equation. (i.e.
Solve.)
y  x2  6x  2
2. 4( x  1)  3 x  (6  x)  0
x2 x 2
3. y 
 
3 2 3
 BE READY, YOU WILL BE CALLED TO THE
BOARD IN 10 MINUTES.
Warm-up: 9-6-011
 Take out your homework and Unit Plan.
 I will collect Chapter Review with work on TEST
DAY!!!
 Find the x-intercepts.
2x
6
28
a.

 2
x 3 x 3
x 9
2
b. y  x  6 x  7
c. 3x  60
2
BE READY, YOU WILL BE CALLED TO THE
BOARD IN 10 MINUTES.
Warm-up: 9-9-2011
 For the following equation, compute the
discriminant.
Then determine the number of solutions and type of
solutions:
a.) x  6 x  9  0
2
b.) 2 x  7 x  4  0
2
c.)3 x  2 x  4  0
2
BE READY, YOU WILL BE CALLED TO THE BOARD IN 8 MINUTES.
Aim#1.5B: How do we solve problems
modeled by quadratic equations?
 In a 25-inch television set,
the length of the screen’s
diagonal is 25 inches. If the
screen’s height is 15 inches,
what is its width?
 You need to use the
Pythagorean Theorem:
a2 + b2 = c2
 Sketch a figure.
 Substitute into the
equation what you
know and Solve.
 Write your final
answer in a sentence.
Problem:
 What is the width of a 15-inch television set whose
height is 9 inches?
Summary:
Answer in complete sentences.
 If you are given a quadratic equation, how do you
determine which method to use to solve it?
 Describe the relationship between the solutions of:
ax2 + bx + c =0 and the graph of y = ax2 + bx + c.
 Write a quadratic equation in general form whose
solution set is
 3, 5
Warm-up: 9-12-2011
 Take out your homework.- Do not copy problem. Show all
work and solve.
 Define your variables, write your equation, solve and express
final answer as a sentence.
 After a 20% reduction, you purchase a television for
$336. What was the television’s price before the
reduction?
 Including 5% sales tax, an inn charges $252 per night.
Find the inn’s nightly cost before the tax is added.
 Each side of a square is lengthened by 2 inches. The
area of this new, larger square is 36 sq. inches. Find
the length of a side of the original square.
Aim # 1.6: How do we solve
polynomial equations by factoring?
 A polynomial equation is the result of setting two
polynomials equal to each other.
 The equation is in general form if one side is 0. and the
polynomial on the other side is in descending powers of the
variable.
 The degree of a polynomial is the same as the degree of
any term in the equation. Here are some examples of
polynomial equation:
 3x + 5 = 14 Degree of 1 also linear
 2x2+7x =4 Degree of 2 also quadratic
 x3 + x2 = 4x + 4 Degree of 3 also cubic
Solving a Polynomial by Factoring
3x  27x
4
2
Steps:
1.Move all terms to one
side and set equation
= 0.
Subtract 27x2 on both
sides.
2. Factor the GCF.
3. Set each factor to 0
4. Solve.
Guided Practice:
 Solve by factoring:
4x4=12x2
Solving a Polynomial Equation
 x3 + x2 = 4x + 4
 Steps:
1. Move all terms to one
side and set equation
= 0.
2. Factor using the
grouping strategy.
3. Set each factor to zero
and solve.
Guided Practice:
 Solve by factoring:
2x3 + 3x2 = 8x + 12
Practice:
 Solve each polynomial equation by factoring
and then use zero-product principle.
1. 5x4- 20x2 = 0
2. 4x3 -12x2 = 9x – 27
3. x + 1 = 9x3 + 9x2
4. 9y3 + 8 = 4y + 18y2
How do we solve radical equations?
 A radical equation is an equation in which the
variable occurs in a square root, cube root or any
higher root. An example is:
x 9

Squaring both sides. Eliminates
radical sign.
Note:
 We solve radical equations with nth roots by raising
both sides of the equation to the nth power.
Unfortunately, if n is even, all the solutions of the
equation raised to the even power may not be
solutions of the original equation.
 For example:
x=4
 If we square both sides, we obtain x2= 16
 x = + √16 = + 4
This equation has two new
solutions, -4 and 4. By contrast only 4 is a solution to
the original equation.
 When raising both sides of an equation to an
even power, always check proposed solutions
in the original equation.
Solving Radical Equations Containing
nth Roots
1. If necessary, arrange terms so that one with the
radical is isolated on one side of the equation.
2. Raise both sides of the equation to the nth power to
eliminate the nth root
3. Solve the resulting equation. If there is still a
radical repeat step 1 and 2.
4. Check all proposed solutions.
 Extraneous solutions or extraneous roots are
solutions that do not satisfy the original equation.
Solving a Radical Equation
2x 1  2  x
 Steps:
1. Isolate radical on one side by
subtracting 2 on both sides.
2. Raise both sides to the nth
power. Because n, the index
is 2, we square both sides.
3. Solve the resulting equation.

Guided Practice:
 Solve:
x  3 3 x
How do we solving an equation that has
two radicals?
3x 1  x  4 1
 Steps:
1. Isolate a radical on one
side.
2. Square both sides.
3. Simplify.
4. Note the resulting equation
still has a radical sign so,
repeat steps 1 and 2.
5. Solve resulting equation.
Guided Practice:
x5 x32
Practice:
 Solve each radical equation. Check all
proposed solutions.
1. 20  8x  x
2. x  10  x  2

3. 6x  1  x 1
4.x  x  11  1
Summary:
Answer in complete sentences.
 Without actually solving the equation, give a general
description of how to solve x3 - 5x2 – x + 5 =0.
 In solving 3x  4  2x  4  2 why is it a good idea
to isolate a radical term? What if we don’t do this
and simply square each side? Describe what
happens.

 What is an extraneous solution to a radical equation?
Warm-up: 9-13-11
 Take out your homework and Unit Plan.
 Solve each polynomial equation by factoring.
Then use the zero product principle.
1.3x3 +2x2 = 12x + 8
2.2x4 = 16x
3.2x3 – x2 – 18 x + 9 = 0
BE READY, YOU WILL BE CALLED TO THE
BOARD IN 6 MINUTES.
Reminder: Quiz & Warm-up due
Friday!
Warm-up: 9-14-2011
 Take out your homework and Unit Plan.
 Solve each radical equation. Check all proposed
solutions.
1. x  8  x  4  2
2. x  2 x  5  5
3. 2 x  3  x  2  2
4. 3 x  1  3 x  5
Aim #1.6B: How do we solve
equations with rational exponents?
 We know that expressions with rational expressions
represent radicals:
m
n
a  ( a ).

n
m
SOLVING RATIONAL EQUATIONS
m
OF THE FORM n
x k
 Assume that m and n are positive integers, m/n is in
lowest terms, and k is a real number.
1. Isolate the expression with the rational exponent.

2. Raise both sides of the equation to the n/m power.
If m is even:
If m is odd:
m
n
m
n
x k
x k
n
m

n
 m 
x n   k m
 
x  k
n
m
n
m

n
 m 
x n   k m
 
xk
n
m
NOTE:
 It is incorrect to insert + symbol when the numerator
of the exponent is odd. An odd index has only ONE
root.
3. Check all proposed solutions in the original equation
to find out if they are actual solutions or extraneous
solutions.
Solving Equations Involving
Rational Exponents
 Steps:
 Solve:
3
4
 Goal is to isolate the
3x  6  0
expression with the rational
exponent.
 Undo the addition or
subtraction.
 Undo the Multiplication or

division.
 Raise both sides by the
reciprocal of the exponent.
 Note: exponent is odd so we do
not add +.
 Check proposed solution.
Guided Practice:
 Solve:
2
3
3
1
x  
4
2
Practice:
3
2
a. 5x  25  0
2
3
b. x  8  4
How do we solve equations that
are in quadratic form?
 An equation that is quadratic in form is one that
can be expressed using an appropriate substitution.
Solving an Equation in
Quadratic Form
x  8x  9  0
4
2
Steps:
1. Replace x2 with u.
2. Factor.
3. Apply the zero-product
principle.
4. Solve for u.
5. Then replace u with x2
and now solve for x.
Guided Practice:
 Solve:
x  5x  6  0
4
2
Solving an Equation in
Quadratic form
2
3
1
3
5 x  11x  2  0
Steps:
1. Replace
2. Factor.
x
1
3
with u.
3. Set each factor to 0.
4. Solve for u.
How do we solve equations
with Absolute Value?
 An absolute value is the distance a number is
from zero.
 An absolute value equation:
x  10
 What values of x would make the above
equation true?
Solving Absolute Value Equations
 Solve:
2 x  3  11
 Steps:
 Rewrite equation
without absolute value
bars.
 Write one equation = 11
and a second = -11
 Solve each equation.
 Check all proposed
solutions.
Solving Absolute Value Equations
NOTE:
51  4 x  15  0
 Before we can solve we need
to isolate the absolute value
expression.
Practice:
 Solve:
41  2 x  20  0
Summary:
Answer in complete sentences.
 Explain in words how to solve. Then show the work.
x  5x  4  0
4
2
 How do we solve an absolute value equation? Include
an example to support your answer.
 What do solving absolute value equations and
radical equations have in common?
 What other type of equation did we learn to solve in
this Aim 1B? Give an example.
Warm-up: 9/ 16/2011
 Take out your homework.
 Make the appropriate substitution, then solve.
1.( x  3)  7( x  3)  18  0
Solve :
2
2. x  2 x  36  12
2
3.x( x  1)  42( x  1)  0
3
2
Warm-up: 9-19-2011
 Solve each equation. Make the appropriate
substitution.
1. x  5 x  4  0
4
2
2. x  13 x  40  0
2
1
3. x  x  20  0
3
2
3
4
4. x  2 x  1  0
 Be ready, in 8 minutes you will be
called to the board.
Aim #1.7: What is interval notation
and how do we use it?
Goals of this Aim are:
 Use of Interval Notation
 Find Intersections and Unions of Intervals
 Solve Linear Inequalities
What is interval notation?
 Subsets of real numbers can be represented using
interval notation.
 Examples:
a. Open Interval: (a, b) represents the set between the
real numbers a and b and not including a and b.
(a, b) = {x /a < x < b}
b. Closed Interval: [a, b] represents the set between
the real numbers a and b including a and b.
[a, b] = {x /a < x < b}
More Examples of
Infinite Notation
 The infinite interval (a,) represents the set of real
numbers that are greater than a.

 The infinite interval(,
b] represents
the set of real
numbers that are less than or equal to b.


Remember:
Parentheses indicate endpoints not included in an
interval. Square brackets indicate that endpoints
that are included interval

Using Interval Notation
 Express each interval using set builder notation
and graph.
a. (-1, 4] = {x /-1< x < 4}
b. (2.5, 4] =
c. (-4,

How do we find the intersection
and union of intervals?
 Use graphs to find each set:
 Steps:
1. Graph each interval on a
a. (1,4)  [ 2,8]
b.(1,4)  [ 2,8]
number line.
2 a. To find intersection, take
the portion of the number
line that the two graphs
have in common.
b. To find the union, take the
portion of the number line
representing the total
collection of numbers in the
two graphs.
Practice:
 Use graphs to find each set:
a. 1,3  (2,6)
b. [1,3]  ( 2,6)
How do we solve Linear
Inequalities in One Variable?
 Solve and graph 3- 2x < 11.
 Try: 2-3x < 5
Solving a Linear Inequality
 Solve and graph.
 Solve and graph.
 -2x – 4 > x + 5
 3x + 1 > 7x - 15
How do we recognize inequalities
with unusual solution sets?
 Examples: x < x + 1
 The solution is all real numbers. Or using interval
notation its___.
 If you attempt to solve an inequality that has no
solution sets, you will eliminate the variable and
obtain a false statement such as 0 > 1.
 If you attempts to solve an inequality that is true for
all real numbers, you will eliminate the variable and
obtain a true statement such as 0 < 1.
Solving Linear Inequalities
 Solve each inequality.
 2 (x + 4)> 2x + 3
 x+7<x-2
Summary:
Answer in complete sentences.
 When graphing the solutions of an inequality,
what does a parenthesis signify? What does a
bracket signify?
 Describe the ways that solving a linear
inequality is similar to solving a linear
equation.
 Describe ways that solving a linear inequality
is different from solving a linear equation.
Warm-up: 9-20-2011
 Take out your homework and Unit Plan.
 Solve:
3
a.) 4 1  x  7  10
4


3
4
b. x  x  4  2  6
2
 Solve. Express answer in interval notation and
graph.
 C. 18x + 45 < 12x - 8
d. -4 (x + 2) > 3x + 20
Warm-up: 9-21or 9-22
 Take out your homework and Unit Plan.
 Solve each linear inequality.
1.4( x  1)  2  3x  6
x 3 x
2.    1
4 2 2
x4 x2 5
3.


6
9
18
 Be READY you will be CALLED in 8 minutes
to the board.
Aim #1.7B: How do we solve other
types of inequalities?
 Two inequalities, such as :
-3 < 2x + 1 and 2x + 1 < 3
That can written as a compound inequality.
-3< 2x + 1 < 3
How do we Solve
a Compound Inequality?
 Solve and graph.
-3 < 2x + 1 < 3
 Goal is to isolate variable.
 Subtract from all parts.
 Simplify.
 Divide each part by 2.
 Simplify.
Now you try: 1< 2x + 3 < 11
How do we Solve Inequalities
with Absolute Value?
 Solving an Absolute Value Inequality
 If X is an algebraic expression and c is a
positive number.
x

c
 The solutions of
are the numbers that
satisfy –c< X < c.
 The solutions of
x c
satisfy X < -c or X > c.
are the numbers that
 These rules are valid if < replaced by <,
 And > is replaced by >.
Solving an Absolute Value
Inequality
 Given the inequality:
x  4  3 means  3  x  4  3
 Solve the compound inequality.
 The solution set written in interval notation is:
 And the graph is …
Solving an Absolute Value
Inequality
 Solve and graph the solution on a number line:
x2 5
Solving an Absolute Value
Inequality
 Steps:
 Remember to isolate Absolute Value Expression
 Then before you can rewrite without bars
 2 3x  5  7  13
Solving an Absolute Value
Inequality
 Solve and graph on a number line:
 3 5x  2  20  19
Solving an Absolute Value
Inequality
 Solve and graph the solution set on a number line:
7  5  2 x is the same as 5  2 x  7
 Solve and graph on a number line:
18  6  3x
Applications:
 Acme car rental agency
charges $ 4 a day plus
$0.15 per mile. Interstate
rental agency charges $20
a day and $0.05 per mile.
How many miles must be
driven to make the daily
cost of an Acme rental a
better deal than an
Interstate rental?
 First define your variable:
 Let x =
 Represent both quantities in
terms of x.
 Which inequality symbol do
we use and why?
 Write and solve the
inequality.
 Check your proposed
solution.
Practice:
 A car can be rented from Basic Rental for $260 per
week with no extra charge for mileage. Continental
charges $80 per week plus 25 cents for each mile
driven to rent the same car. How many miles must
be driven in a week to make this rental cost for Basic
Rental a better deal than Continental’s?
Summary:
Answer in complete sentences.
 Describe how to solve an absolute value inequality
involving this symbol <. Give an example.
 Describe the solution set of
x  4
 What’s wrong with this argument? Suppose x and y
represent two real numbers, where x > y:
Warm-up: 9-23-2011
 Take out your homework and Unit Plan.
 Solve each inequality.
2
1.  3  x  5  1
3
2. 2 x  6  8
2x  2
3.
2
4
4 .3 x  1  2  8
Please Read
DO NOW: 9-26-2011
 Take out your Review, Work and Unit
Plan.
 Turn in last week’s warm-up in the
tray on table.
 Complete Today’s Warm-up in your
notebook.
 Reminder: Crossword and Unit Plan
due Tuesday, Sept. 27th
 Test Thursday