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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