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Sec 1.7 Solve Absol= ute Value Equations and Inequalities

Slide 2

Goals

Geometric Definitio= n

"Equation"= ;

Solving an Absolute= Value

Algebraic Definitio= n

Example 1

Example 2

Extraneous Solution= s

Example 3

Slide 12

Absolute Value Inequalities

Absolute Value Inequalities

Absolute Value Inequalities

Absolute Value Inequalities

Example 4: Solve

Absolute Value Inequalities

Absolute Value Inequalities

Example 5: Solve

Slide 21

Example 6 Tolerance=

Example 6 Tolerance=

Example 6 Tolerance=

Example 6 Tolerance=

Example 6 Tolerance=

Example 6 Tolerance=

Example 7 Quality C= ontrol

Example 7 Quality C= ontrol

Example 7 Tolerance=

Example 7 Tolerance=

Example 7 Tolerance=

Example 7 Tolerance=

Closure

Assignment<= /font>

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Sec 1.7 Solve Absol= ute Value Equations and Inequalities

Slide 2

Goals
Goal 1: To solve ab= solute value equations and inequalities
Goal 2: To use abso= lute value equations and inequalities to solve real life problems

Geometric Definitio= n
The absolute value = of a number x, written |x|, is the distance the number is from zero on a number line.
Notice that the abs= olute value of a number is always nonnegative.

"Equation"= ;
Equation
Meaning
Graph
Solutions

Solving an Absolute= Value
To solve the absolu= te value               , use two equations.  One if the inside is positive or zero,                  .  The other if the inside is negative

Algebraic Definitio= n

Example 1
When inside is posi= tive, absolute value doesn't affect it.
When inside is nega= tive, absolute value takes the opposite of the inside.

Example 2
When inside is posi= tive, absolute value doesn't affect it.
When inside is nega= tive, absolute value takes the opposite of the inside.

Extraneous Solution= s

Example 3
When inside is posi= tive, absolute value doesn't affect it.
When inside is nega= tive, absolute value takes the opposite of the inside.

Slide 12

Absolute Value Inequalities
Graph the solution = to the following:
| x | < 3=

Absolute Value Inequalities
Graph the solution = to the following:
| x | ³ 2

Absolute Value Inequalities
This inequality is a conjunction.
When the inside (ax= +b) is negative, the absolute value will take the opposite of the inside<= /td>

Absolute Value Inequalities
These can be writte= n as a compound inequality.

Example 4: Solve

Absolute Value Inequalities
This inequality is a disjunction.
When the inside (ax= +b) is negative, the absolute value will take the opposite of the inside<= /td>

Absolute Value Inequalities
These can be writte= n as a compound inequality.

Example 5: Solve

Slide 21

Example 6 Tolerance=
A cereal manufactur= er has a tolerance of 0.75 ounce for a box of cereal that is supposed to weigh 20 ounces.  Write and solve an absol= ute value inequality that describes the acceptable weights for "20 ounce= " boxes of cereal.
We can show accepta= ble weights using a graph.

Example 6 Tolerance=
A cereal manufactur= er has a tolerance of 0.75 ounce for a box of cereal that is supposed to weigh 20 ounces.  Write and solve an absol= ute value inequality that describes the acceptable weights for "20 ounce= " boxes of cereal.
The amount that the= weight is off can be found by subtracting the ideal weight (20 oz) from the actual weight.
This difference sho= uld be less than the tolerance.

Example 6 Tolerance=
A cereal manufactur= er has a tolerance of 0.75 ounce for a box of cereal that is supposed to weigh 20 ounces.  Write and solve an absol= ute value inequality that describes the acceptable weights for "20 ounce= " boxes of cereal.
Why do we use the a= bsolute value bars here?

Example 6 Tolerance=
A cereal manufactur= er has a tolerance of 0.75 ounce for a box of cereal that is supposed to weigh 20 ounces.  Write and solve an absol= ute value inequality that describes the acceptable weights for "20 ounce= " boxes of cereal.
Give labels to our = values
Actual weight =3D x= (ounces)
Ideal weight =3D 20= (ounces)
Tolerance =3D 0.75 = (ounces)

Example 6 Tolerance=
A cereal manufactur= er has a tolerance of 0.75 ounce for a box of cereal that is supposed to weigh 20 ounces.  Write and solve an absol= ute value inequality that describes the acceptable weights for "20 ounce= " boxes of cereal.
Substitute labels i= n to verbal model to create algebraic model
| x - 20 | <= font face=3DSymbol size=3D2>£ 0.75<= /td>

Example 6 Tolerance=
Solve
| x - 20 | <= font face=3DSymbol size=3D2>£ 0.75<= /td>

Example 7 Quality C= ontrol
QUALITY CONTROL. Yo= u are a quality control inspector at a bowling pin company.  A regulation pin must weight between = 50 ounces and 58 ounces, inclusive.  Write an absolute value inequality describing the weights you should reject.
We can show accepta= ble weights using a graph.

Example 7 Quality C= ontrol
QUALITY CONTROL. Yo= u are a quality control inspector at a bowling pin company.  A regulation pin must weight between = 50 ounces and 58 ounces, inclusive.  Write an absolute value inequality describing the weights you should reject.
But we are looking = for weights we should reject. What would that graph look like?
(See green graph fr= om previous slide)

Example 7 Tolerance=
QUALITY CONTROL. Yo= u are a quality control inspector at a bowling pin company.  A regulation pin must weight between = 50 ounces and 58 ounces, inclusive.  Write an absolute value inequality describing the weights you should reject.
What do you think t= he “ideal” weight would be in this problem and how do you “calculate” it?
What is the toleran= ce then?
Write a verbal mode= l for this problem.

Example 7 Tolerance=
QUALITY CONTROL. Yo= u are a quality control inspector at a bowling pin company.  A regulation pin must weight between = 50 ounces and 58 ounces, inclusive.  Write an absolute value inequality describing the weights you should reject.
Give labels to the = parts of the verbal model.
Weight of pin =3D w= (ounces)
Average of extreme = weights (ideal weight) =3D (50+58)/2 =3D 54 (ounces)
Tolerance =3D 58-54= =3D 4 (ounces)

Example 7 Tolerance=
QUALITY CONTROL. Yo= u are a quality control inspector at a bowling pin company.  A regulation pin must weight between = 50 ounces and 58 ounces, inclusive.  Write an absolute value inequality describing the weights you should reject.
Create the algebrai= c model:
| w - 54 | > 4

Example 7 Tolerance=
Solve
| w - 54 | > 4

Closure
| ax + b | < c r= esults in a conjunction
What conjunction is= it?
| ax + b | >c re= sults in a disjunction
What disjunction is= it?

Assignment<= /font>
Sec 1.7
1-5, 9-37 LC, 43-59= LC, 74-77

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------=_NextPart_01C7FEB6.3DC28650 Content-Location: file:///C:/EC8262C1/Ch1Sec74A_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C7FEB6.3DC28650 Content-Location: file:///C:/EC8262C1/Ch1Sec74A_files/v3_slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Chapter 1 Sec 7
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Sec 1.7 Solve Absolute Value
Equations and Inequalities
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5nAG53E2Z6sKCAA7 ------=_NextPart_01C7FEB6.3DC28650 Content-Location: file:///C:/EC8262C1/Ch1Sec74A_files/v3_slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Chapter 1 Sec 7
Goals
<= /td> = <= /td> <= /td> <= /td>
n Goal 1: To solve absolute value
equations and inequalities
n Goal 2: To use absolute value
equations and inequalities to solve
real life problems
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Geometric Definition
<= /td> = <= /td> <= /td>
n Th= e absolute value of a number x,
wr= itten |x|, is the distance the number
is= from zero on a number line.
n No= tice that the absolute value of a
nu= mber is always nonnegative.
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<= /td> <= /td>
Equ= ation
Mea= ning
Graph
Sol= utions
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Solving an Absolute Value
<= /td> = = <= /td>
n To= solve the absolute value               , = use
two equations.  One if the inside i= s
po= sitive or zero,                  .  The other if
th= e inside is negative
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Algebraic Definition
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Example 1
<= /td> = <= /td>
n Wh= en inside is negative,
ab= solute value takes the
op= posite of the inside.
<= /td> <= /td> =
n Wh= en inside is positive,
ab= solute value doesn't
aff= ect it.
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Example 2
<= /td> = <= /td>
n Wh= en inside is negative,
ab= solute value takes the
op= posite of the inside.
<= /td> <= /td> =
n Wh= en inside is positive,
ab= solute value doesn't
aff= ect it.
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Extraneous Solutions
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Example 3
<= /td> = <= /td>
n Wh= en inside is negative,
ab= solute value takes the
op= posite of the inside.
<= /td> <= /td> =
n Wh= en inside is positive,
ab= solute value doesn't
aff= ect it.
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Absolute Value Inequalities
<= /td> <= /td> <= /td>
n Gr= aph the solution to the following:
= n | x= | < 3
<= /td> <= /td> <= /td>
n Do= es this give a conjunction or a disjunction?
= n Con= junction
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Absolute Value Inequalities
<= /td> <= /td> <= /td>
n Gr= aph the solution to the following:
= n | x= | 2
<= /td> <= /td> <= /td>
n Do= es this give a conjunction or a disjunction?
= n Dis= junction
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Absolute Value Inequalities
<= /td> = <= /td>
n Th= is inequality is a conjunction.
n Wh= en the inside (ax+b) is negative, the absolute
va= lue will take the opposite of the inside
<= /td> =
n Wh= en the inside (ax+b) is positive, the absolute
va= lue will not change the inside
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Absolute Value Inequalities
<= /td>
n Th= ese can be written as a compound inequality.
<= /td> =
n Ge= t the ax+b in the first inequality by itself by
mu= ltiplying both sides by -1.
<= /td> =
n Si= nce both inequalities are now in terms of
ax= +b, rewrite it without “and”
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Example 4: Solve
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Absolute Value Inequalities
<= /td> = <= /td>
n Th= is inequality is a disjunction.
n Wh= en the inside (ax+b) is negative, the absolute
va= lue will take the opposite of the inside
<= /td> =
n Wh= en the inside (ax+b) is positive, the absolute
va= lue will not change the inside
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Absolute Value Inequalities
<= /td>
n Th= ese can be written as a compound inequality.
<= /td> =
n Ge= t the ax+b in the first inequality by itself by
mu= ltiplying both sides by -1.
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Example 5: Solve
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Content-Type: text/html; charset="windows-1252" Chapter 1 Sec 7
Example 6 Tolerance
<= /td> = <= /td>
n A = cereal manufacturer has a tolerance of 0.75
ou= nce for a box of cereal that is supposed to
we= igh 20 ounces.  Write and solve an abs= olute
va= lue inequality that describes the acceptable
we= ights for "20 ounce" boxes of cereal.
n We= can show acceptable weights using a graph.
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Example 6 Tolerance
<= /td> = <= /td> = <= /td>
n A = cereal manufacturer has a tolerance of 0.75 ounce for a box of
ce= real that is supposed to weigh 20 ounces.  W= rite and solve an
ab= solute value inequality that describes the acceptable weights for
&q= uot;20 ounce" boxes of cereal.
n Th= e amount that the weight is off can be found
by= subtracting the ideal weight (20 oz) from the
ac= tual weight.
n Th= is difference should be less than the tolerance.
<= /td>
Actual
Weight
<= /td>
Weight
=
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Example 6 Tolerance
<= /td> = =
n A = cereal manufacturer has a tolerance of 0.75 ounce for a box of
ce= real that is supposed to weigh 20 ounces.  W= rite and solve an
ab= solute value inequality that describes the acceptable weights for
&q= uot;20 ounce" boxes of cereal.
n Wh= y do we use the absolute value bars here?
<= /td>
Actual
Weight
<= /td>
Weight
=
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