SM015 - 2.3 (Notes Solutions)

2.3 Absolute Values

Friday, July 10, 2020 11:31 AM

At the end of the lesson, students should be able to
(a) State the properties of absolute values.
(b) Solve absolute equations.
(c) Solve absolute inequalities .

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Absolute Values

 The absolute value represents the distance of a point on the
number line from the origin.

Definition I: Definition II:

 Properties of Absolute Values
(a)
(b)
(c)
(d)
(e)

(f)

Remarks:

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Example 1

Thursday, June 11, 2020 8:37 PM

Write the following without the absolute value symbol:
Solution:

Solution:

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Equations (I)

Thursday, June 11, 2020 10:03 PM

 For the equation in the form of
,

solve it based on definition,

.

i.e. , OR
OR
can be defined as
Proof:

For For
OR
OR

Procedures to solve:

(I) Write the equation in the corresponding general form.

(II) Define the equation into 2 cases with "OR".

(III) Solve the equation in each of the case separately.

(IV) Verify the solutions obtained if necessary.

 No verification required for where is a positive constant.

(V) Combine the answers obtained from both case.

(VI) Write the answer in the set notation if required by the question.

Example 2(a)

Solve the equation .

Solution: Remarks: NO verification required as where is a positive constant

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Example 2(b) & 3(a)

Thursday, June 11, 2020 8:37 PM

Example 2(b)

Solve the equation .

Solution: Remarks: NO verification required as where is a positive constant

Example 3(a) . Give the answer in set notation.

Solve the equation is NOT neccessarily positive or zero.

Solution: Remarks: Verification required as

Verification: [rejected]

LHS of equation RHS of equation

is NOT the solution because

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Equations (II)

Thursday, June 11, 2020 10:03 PM

 For the equation in the form of
,

solve it by squaring both sides for simplicity.

Procedures to solve:

(I) Write the equation in the general form.

(II) Square both sides of the equation to get rid of modulus sign.

[This technique is only valid when both sides of the equation is surely NON negative.]

(III) Solve the linear or quadratic equation obtained.

 No verification required for .

(IV) Write the answer in the notation required by the question.

[interval notation or set notation]

Remarks: , it can also be solved by definition,
For the case OR

can be defined as OR
Proof:

For For
OR

For For For For
OR OR OR
OR OR OR

OR

Example 3 (b)

Solve the equation . Give the answer in set notation.

Solution: Remarks: NO verification required as both sides of is positve or zero

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Inequalities (I) OR

Thursday, June 11, 2020 10:03 PM OR
OR
 Based on definition,

can be defined as
Proof:

For For
OR
OR

Remarks: Similar proof for the case

AND

AND

can be defined as AND
Proof:

For For
AND
AND

Remarks: Similar proof for the case

Procedures to solve:
(I) Write the inequality in the corresponding general form.

(II) Define the inequality into 2 cases with AND / OR
correspondingly.

(III) Solve the inequality in each of the case separately.

(IV) Combine the answers obtained from both cases with by
using number line. [Union for AND / Intersection for OR]

(V) Write the answer in the notation required by the
question. [interval notation or set notation]

Remarks:
Note that it is also valid for the following:

However, they are useful only if y is a linear function & k is a constant.

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Example 4 & 5 Alternative method:

Thursday, June 11, 2020 8:37 PM Can just apply

Example 4 when is a positive constant

Solve the following inequality
Solution:

AND

Set Notation:
Interval Notation:

Set Notation:
Interval Notation:

Example 5

Solve the following inequality
Solution:

OR

Set Notation:
Interval Notation:

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Example 7 (a) [ToS]

Thursday, June 11, 2020 8:37 PM

Solve the following inequalities:

Solution:

AND

Interval Interval

Conclusion Conclusion

Set Notation:
Interval Notation:

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Example 7 (a) [NLoS]

Thursday, June 11, 2020 8:37 PM

Solve the following inequalities:

Solution:

AND

Let Let

, ,

,

Set Notation:
Interval Notation:

2.3 Absolute Values Page 9

Example 7 (b) [ToS]

Thursday, June 11, 2020 8:37 PM

Solve the following inequalities:

Solution:

OR

Interval Interval

Conclusion Conclusion

Set Notation:
Interval Notation:

2.3 Absolute Values Page 10

Example 7 (b) [NLoS]

Thursday, June 11, 2020 8:37 PM

Solve the following inequalities:

Solution:

OR

Let Let

, ,

,

Set Notation:
Interval Notation:

2.3 Absolute Values Page 11

Example 8 .
AND
Thursday, June 11, 2020 8:37 PM

Solve the inequality

Solution:

Set Notation:
Interval Notation:

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Inequalities (II)

Thursday, June 11, 2020 10:03 PM

 For the inequalities in the following form:
•
•
•
•

Solve them by squaring both sides for simplicity.

Procedures to solve:
(I) Write the inequality in the general form.

(II) Square both sides of the inequality to get rid of modulus sign.

[This technique is only valid when both sides of the equation is confirmed NON negative.]

(III) Write the quadratic inequality in the general form.

[If obtained a linear inequality, then just solve it by rearranging the terms.]

(IV) Apply graphical method to solve.

[can also use algebraic method such as real number line & table of signs to solve but the
steps will longer]

(V) Write the answer in the notation required by the question.
[interval notation or set notation]

Example 6 .

Solve the inequality

Solution: Remarks: NO verification required as both sides of the is positve or zero

Set Notation:
Interval Notation:

2.3 Absolute Values Page 13