Download - Solving Inequalities (Algebra 2)
Solving Inequalities Solving Inequalities
1) set-builder notation2) interval notation
Solve inequalities.
For any two real numbers, a and b, exactly one of the following statements istrue.
Solving Inequalities Solving Inequalities
For any two real numbers, a and b, exactly one of the following statements istrue.
ba
Solving Inequalities Solving Inequalities
For any two real numbers, a and b, exactly one of the following statements istrue.
ba ba
Solving Inequalities Solving Inequalities
For any two real numbers, a and b, exactly one of the following statements istrue.
ba ba ba
Solving Inequalities Solving Inequalities
For any two real numbers, a and b, exactly one of the following statements istrue.
ba ba ba
This is known as the Trichotomy Property
Solving Inequalities Solving Inequalities
For any two real numbers, a and b, exactly one of the following statements istrue.
ba ba ba
This is known as the Trichotomy Property or the property of order.
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
then, If ba
ab
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
ab a+cb+c
c
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
ab a+cb+c
c
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
ab a+cb+c
Solving Inequalities Solving Inequalities
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
ab a+cb+c
Solving Inequalities Solving Inequalities
then, If Likewise, ba
Adding the same number to, or subtracting the same number from, each side of aninequality does not change the truth of the inequality.
Addition Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
ab a+cb+c
Solving Inequalities Solving Inequalities
then, If Likewise, ba cbca
Subtraction Property of Inequality
For any real numbers a, b, and c:
Solving Inequalities Solving Inequalities
Subtraction Property of Inequality
For any real numbers a, b, and c:
then, If ba
ab
Solving Inequalities Solving Inequalities
Subtraction Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
a-cb-c ab
c
Solving Inequalities Solving Inequalities
Subtraction Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
a-cb-c ab
c
Solving Inequalities Solving Inequalities
Subtraction Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
a-cb-c ab
Solving Inequalities Solving Inequalities
Subtraction Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
a-cb-c ab
Solving Inequalities Solving Inequalities
then, If Likewise, ba
Subtraction Property of Inequality
For any real numbers a, b, and c:
then, If ba cbca
a-cb-c ab
Solving Inequalities Solving Inequalities
then, If Likewise, ba cbca
a
ax
Solving Inequalities Solving Inequalities
a
ax
Solving Inequalities Solving Inequalities
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
a
ax
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
a
ax
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
a
ax
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
We use a closed circle (dot) to indicate that a IS part of the solution set.
a
ax
Solving Inequalities Solving Inequalities
a
ax
Solving Inequalities Solving Inequalities
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
a
ax
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
a
ax
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
a
ax
a
ax
Solving Inequalities Solving Inequalities
We use an open circle (dot) to indicate that a is NOT part of the solution set.
We use a closed circle (dot) to indicate that a IS part of the solution set.
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
c is positive:
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
then, If ba bcac
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
c is positive:
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
then, If ba bcac
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
then, If ba bcac c is positive:
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
then, If ba bcac
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
then, If ba bcac c is positive:
c is negative:
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
then, If ba bcac
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
then, If ba bcac c is positive:
then, If ba bcac c is negative:
Solving Inequalities Solving Inequalities
Multiplying or dividing each side of an inequality by a positive number does not changethe truth of the inequality.
Multiplication Property of Inequality
For any real numbers a, b, and c:
then, If ba bcac
However, multiplying or dividing each side of an inequality by a negative numberrequires that the order of the inequality be reversed.
then, If ba bcac c is positive:
then, If ba bcac then, If ba bcac
c is negative:
Solving Inequalities Solving Inequalities
Division Property of Inequality
Most books run us through the “rules” for division. Why is this not necessary?
Solving Inequalities Solving Inequalities
Division Property of Inequality
Most books run us through the “rules” for division. Why is this not necessary?
HINT: axax
by divided
is the same as
Solving Inequalities Solving Inequalities
Division Property of Inequality
Most books run us through the “rules” for division. Why is this not necessary?
axa
x of reciprocal by the multiplied
11
HINT: axax
by divided
is the same as
Solving Inequalities Solving Inequalities
Division Property of Inequality
Most books run us through the “rules” for division. Why is this not necessary?
axa
x of reciprocal by the multiplied
11
HINT: axax
by divided
is the same as
So, see rules for multiplication!
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using set-builder notation.
4
4x
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using set-builder notation.
4
4x
4| xx
set-builder notation
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using set-builder notation.
4
4x
4| xx
set-builder notation
Solving Inequalities Solving Inequalities
Read: { x “such that” x is less than 4 }
The solution set of an inequality can also be described by using set-builder notation.
4
4x
4| xx
set-builder notation
Solving Inequalities Solving Inequalities
Read: { x “such that” x is less than 4 }
Identify the variable used
The solution set of an inequality can also be described by using set-builder notation.
4
4x
4| xx
set-builder notation
Solving Inequalities Solving Inequalities
Read: { x “such that” x is less than 4 }
Identify the variable used Describe the limitations or
boundary of the variable
The solution set of an inequality can also be described by using set-builder notation.
-7
7x
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using set-builder notation.
-7
7x
7| xxset-builder notation
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using set-builder notation.
-7
7x
7| xxset-builder notation
Solving Inequalities Solving Inequalities
Read: { x “such that” x is greater than or equal to negative 7 }
The solution set of an inequality can also be described by using set-builder notation.
-7
7x
7| xxset-builder notation
Solving Inequalities Solving Inequalities
Read: { x “such that” x is greater than or equal to negative 7 }
Identify the
variable used
The solution set of an inequality can also be described by using set-builder notation.
-7
7x
7| xxset-builder notation
Solving Inequalities Solving Inequalities
Read: { x “such that” x is greater than or equal to negative 7 }
Identify the
variable used Describe the limitations or
boundary of the variable
The solution set of an inequality can also be described by using interval notation.
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used.
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used.
4
4x
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used.
4
4x
4,interval notation
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used.
4
4x
4,interval notation
To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used.
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used.
4
4x
4,interval notation
To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used.
-7
7x
Solving Inequalities Solving Inequalities
The solution set of an inequality can also be described by using interval notation.
The infinity symbols and are used to indicate that a set is unbounded inthe positive or negative direction, respectively.
To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used.
4
4x
4,interval notation
To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used.
-7
7x
,7interval notation
Solving Inequalities Solving Inequalities
Solving Inequalities Solving Inequalities
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Using Glencoe’s Algebra 2 text,© 2005
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