The document provides information about Philippine Normal University Visayas including:
- It is located in Cadiz City, Negros Occidental and serves as the National Center for Teacher Education and Environment and Green Technology Education Hub.
- The document then discusses solving equations and inequalities, properties of equations and inequalities, and examples of solving different types of equations and inequalities.
Patient Counselling. Definition of patient counseling; steps involved in pati...
ย
Lesson 1.2 NT (Equation and Inequalities).pdf
1. PHILIPPINE NORMAL UNIVERSITY VISAYAS
The National Center for Teacher Education
The Environment and Green Technology Education Hub
Cadiz City, Negros Occidental
2. PHILIPPINE NORMAL UNIVERSITY VISAYAS
The National Center for Teacher Education
The Environment and Green Technology Education Hub
Cadiz City, Negros Occidental
4. An equation is a statement that two expressions are equal.
Example 1.4 The following are examples of equation.
2๐ฅ โ 6 = 0 ๐ฅ2
โ 5๐ฅ + 6 = 0
To solve an equation means to find all the values of the variables
that will make the statement true. The values of the variables that
make the statement true are called solutions or roots of the
equation.
Example 1.5 The equation 2๐ฅ โ 6 = 0 is a true statement for
๐ฅ = 3 but it is false for any other number. The root or solution
of this equation is 3.
5. Properties of Equations
Let ๐, ๐, ๐ โ โ.
1. Reflexive Property
๐ = ๐
Example 2 = 2
2. Symmetric Property (or Replacement Property)
If ๐ = ๐, then ๐ = ๐.
If ๐ = ๐, then ๐ can replace ๐ in any instance.
Example If 5 = ๐ฅ, then ๐ฅ = 5.
6. 3. Transitive Property
If ๐ = ๐, and ๐ = ๐, then ๐ = ๐.
Example If ๐ฅ = ๐ฆ and ๐ฆ = 5, then ๐ฅ = 5.
4. Addition Property
If ๐ = ๐, then ๐ + ๐ = ๐ + ๐.
Example If ๐ฅ = ๐ฆ then ๐ฅ + 3 = ๐ฆ + 3.
5. Multiplication Property
If ๐ = ๐, then ๐๐ = ๐๐.
Example If ๐ฅ = ๐ฆ, then 5๐ฅ = 5๐ฆ.
8. Solve the equation
2
3
๐ฅ โ 1 = ๐ฅ โ 3.
Solution: ๐ฅ = 7
To check, using replacement property,
2
3
๐ฅ โ 1 = ๐ฅ โ 3
2
3
7 โ 1 = 7 โ 3;
4 = 4 which is true by reflexive property of equality.
9. An equation that is true for all values of the variable is called an identity.
An equation that is true for some values of the variables but not true for other
values is called a conditional equation. An equation that has no solutions is
called a contradiction.
Example 1.7
Classify each equation as an identity, a conditional equation, or a
contradiction.
a) ๐ฅ2
+ 2๐ฅ = ๐ฅ(๐ฅ + 2)
b) 5๐ฅ โ 2 = ๐ฅ โ 10
c) 3๐ฅ + 2 = 1 + 3๐ฅ
10. Classify each equation as an identity, a conditional equation, or a
contradiction.
a) ๐ฅ2
+ 2๐ฅ = ๐ฅ(๐ฅ + 2)
Solution
Getting the product of the right side of ๐ฅ2
+ 2๐ฅ = ๐ฅ(๐ฅ + 2),
we have
๐ฅ2
+ 2๐ฅ = ๐ฅ2
+ 2๐ฅ
Since the left and right expressions are the same, the equation will
hold true for whatever value of ๐ฅ, thus we have an identity.
11. Classify each equation as an identity, a conditional equation, or a
contradiction.
๐. 5๐ฅ โ 2 = ๐ฅ โ 10
Solution
Adding both sides of 5๐ฅ โ 2 = ๐ฅ โ 10 by (2 โ ๐ฅ),
we have
5๐ฅ โ 2 + 2 โ ๐ฅ = ๐ฅ โ 10 + (2 โ ๐ฅ)
4๐ฅ = โ8.
Multiplying both sides of 4๐ฅ = โ8 by
1
4
,
we get ๐ฅ = โ2
The original equation is satisfied by ๐ฅ = โ2 but not by other values.
Thus, the equation is a conditional equation.
12. Classify each equation as an identity, a conditional equation, or a
contradiction.
๐. 3๐ฅ + 2 = 1 + 3๐ฅ
Solution
Subtracting 3๐ฅ from both sides of the equation
3๐ฅ + 2 โ 3๐ฅ = 1 + 3๐ฅ โ 3๐ฅ
we get 2 = 1
which is a false statement.
The equation is a false statement no matter what the value of ๐ฅ is
and thus, it has no solution. Therefore, it is a contradiction.
13. Real numbers are shown on the number line with larger
numbers written to the right. For any two real numbers, the
one to the left is less than the one to the right. The symbol <
means โis less thanโ and the symbol > means โis greater
thanโ.
A statement that one quantity is greater than or less than
another quantity is called an inequality.
14. โช If ๐ and ๐ are real numbers,
๐ is greater than ๐
or ๐ > ๐
means ๐ is to the right of ๐ on the number line,
and ๐ โ ๐ is positive.
โช If ๐ is less than ๐,
or ๐ < ๐,
this means ๐ is to the left of ๐ on the number line,
and ๐ โ ๐ is negative
15. Note the following:
๐ > ๐ and ๐ < ๐ have the same meaning.
๐ > 0 means that ๐ is positive,
๐ < 0 means that ๐ is negative.
17. The following statements are also inequalities:
๐ โค ๐ means โ๐ is less than or equal to ๐โ
๐ โฅ ๐ means โ๐ is greater than or equal to ๐
If ๐ โค ๐ and ๐ โฅ ๐ hold simultaneously, then ๐ = ๐.
For any real number ๐,
๐ โค 0 means ๐ is not positive, and
๐ โฅ 0 means ๐ is not negative.
18. Example 1.9 The following are examples of inequalities:
a) 2๐ฅ + 3 < 12
b) ๐ฅ2
โ 2๐ฅ โ 8 โค 0
c) ๐ฅ + 2๐ฆ > 1 + ๐ฆ
19. Properties of Inequalities
Let ๐, ๐, ๐ โ โ.
1. Trichotomy Property. If ๐, ๐ โ โ, then one and only one of the following
relations holds: ๐ < ๐, ๐ = ๐, ๐๐ ๐ > ๐.
2. Transitive Property. If ๐, ๐, ๐ โ โ, such that ๐ > ๐ and ๐ > ๐, then ๐ > ๐.
3. Addition Property. If ๐, ๐, ๐ โ โ, such that ๐ > ๐ , then ๐ + ๐ > ๐ + ๐.
4. Multiplication Property. Let ๐, ๐, ๐ โ โ.
๐. If ๐ > ๐ and ๐ > 0, then ๐๐ > ๐๐.
๐๐. If ๐ > ๐ and ๐ < 0, then ๐๐ < ๐๐.
21. Writing the solution in a set,
๐ฅศ๐ฅ โค 7
Read: The set of all elements ๐ฅ such that ๐ฅ is less than or equal to 7.
To illustrate the solution in a number line,
To check, choose one value in the solution, say ๐ฅ = โ2 .
Substituting this value for ๐ฅ,
2
3
โ2 โ 1 โฅ โ2 โ 3
2
3
โ3 โฅ โ5 and
โ2 โฅ โ5 which is true.
22. Compound Inequalities
A compound inequality is formed by joining two
inequalities with the connective word and or or.
The solution set of a compound inequality with the
connective word ๐๐ is the union of the solution sets of the two
inequalities.
The solution set of a compound inequality with the
connective word ๐๐๐ is the intersection of the solution sets of the
two inequalities.
23. Example 1.11 Solve the following of compound inequalities:
a) 2๐ฅ < 8 ๐๐ ๐ฅ โ 3 > 4
b) ๐ฅ + 3 > 5 ๐๐๐ 3๐ฅ โ 7 < 14
Solution
a) 2๐ฅ < 8 ๐๐ ๐ฅ โ 3 < 4
Solving each inequality, we have
2๐ฅ < 8 ๐ฅ โ 3 > 4
๐ฅ < 4 ๐ฅ > 7
๐ฅศ๐ฅ < 4 ๐ฅศ๐ฅ > 7
Write the union of the solution sets.
๐ฅศ๐ฅ < 4 โช ๐ฅศ๐ฅ > 7 = ๐ฅศ๐ฅ < 4 ๐๐ ๐ฅ > 7
24. Illustrating the solution in a number line,
๐ฅศ๐ฅ < 4 โช ๐ฅศ๐ฅ > 7 = ๐ฅศ๐ฅ < 4 ๐๐ ๐ฅ > 7
25. b) ๐ฅ + 3 > 5 ๐๐๐ 3๐ฅ โ 10 > 14
Solving each inequality, we have
๐ฅ + 3 > 5
๐ฅ > 2
๐ฅศ๐ฅ > 2
3๐ฅ โ 10 > 14
3๐ฅ > 24
๐ฅ > 8
๐ฅศ๐ฅ > 8
Write the intersection of the solution sets.
๐ฅศ๐ฅ > 2 โฉ ๐ฅศ๐ฅ > 8 = ๐ฅศ ๐ฅ > 8 .
26. The inequality given by ๐ < ๐ < ๐, is equivalent to the compound
inequality ๐ < ๐ ๐๐๐ ๐ < ๐.
Example 1.12 Solve the inequality 5 < 2๐ฅ โ 3 < 13.
Solution. This can be solved by either of the following methods.
Method 1 5 < 2๐ฅ โ 3 < 13 is equivalent to 5 < 2๐ฅ โ 3 ๐๐๐ 2๐ฅ โ 3 < 13.
5 < 2๐ฅ โ 3 ๐๐๐ 2๐ฅ โ 3 < 13
8 < 2๐ฅ 2๐ฅ < 16
4 < ๐ฅ ๐๐ ๐ฅ > 4 ๐ฅ < 8
๐ฅศ๐ฅ > 4 โฉ ศ
๐ฅ ๐ฅ < 8 = { ศ
๐ฅ 4 < ๐ฅ < 8}
27. Method 2.
Solve 5 < 2๐ฅ โ 3 < 13
5 < 2๐ฅ โ 3 < 13 Add 3 to all parts of the inequality.
8 < 2๐ฅ < 16 Multiply the parts by ยฝ.
4 < ๐ฅ < 8
Thus, the solution set is
{ ศ
๐ฅ 4 < ๐ฅ < 8}.
28. Example 1.13 Solve the inequality 2๐ฅ โ 5 < 9.
Solution
The definition of the absolute value of a number ๐, denoted by
๐ , is the real number such that
๐ = ๐ when ๐ โฅ 0.
๐ = โ๐ when ๐ < 0.
From the definition, we have
2๐ฅ โ 5 < 9 and โ 2๐ฅ โ 5 < 9
Note that โ 2๐ฅ โ 5 < 9 when multiplied by โ1, becomes
2๐ฅ โ 5 > โ9
Thus, we can write the inequalities as
2๐ฅ โ 5 < 9 and 2๐ฅ โ 5 > โ9
29. Thus, we can write the inequalities as
2๐ฅ โ 5 < 9 and 2๐ฅ โ 5 > โ9
Or
โ9 < 2๐ฅ โ 5 < 9
Add 5,
โ4 < 2๐ฅ < 14
Multiply by
1
2
(or divide by 2),
โ2 < ๐ฅ < 7
Thus, the solution set is { ศ
๐ฅ โ 2 < ๐ฅ < 7}.
30. Solving the two inequalities,
2๐ฅ โ 5 < 9 and โ 2๐ฅ โ 5 < 9 ๐๐ 2๐ฅ โ 5 > โ9. Or
โ9 < 2๐ฅ โ 5 < 9 Add 5.
โ4 < 2๐ฅ < 14 Multiply the parts by ยฝ.
โ2 < ๐ฅ < 7. Thus, the solution set is { ศ
๐ฅ โ
2 < ๐ฅ < 7}.
Solving the two inequalities,
2๐ฅ โ 5 < 9 and โ 2๐ฅ โ 5 < 9 ๐๐ 2๐ฅ โ 5 > โ9. Or
โ9 < 2๐ฅ โ 5 < 9 Add 5.
โ4 < 2๐ฅ < 14 Multiply the parts by ยฝ.
โ2 < ๐ฅ < 7. Thus, the solution set is { ศ
๐ฅ โ
2 < ๐ฅ < 7}.
31. An inequality that is true for all values of the variable is
called an absolute inequality.
An inequality that is true for some values of the variables
but not true for other values is called a conditional inequality.
32. Example 1.14
The inequality ๐ฅ + 5 < 8 is true only if ๐ฅ < 3 and thus, is
a conditional inequality.
In the inequality ๐ฅ2
+ 1 > 0, the lowest possible value of
๐ฅ2
is 0, that is, when ๐ฅ = 0. For all other real values of ๐ฅ, ๐ฅ2
will always be positive. Thus, ๐ฅ2
+ 1 > 0 is true for all real
values of ๐ฅ and is an absolute inequality.