Rational-Equations-Inequalities-and-Functions_Intro.pptx

ANNOU NCEMEN TS & REMIN DERS
• Letter for 4P’s Beneficiaries for the
Meeting on Friday
• Beautification (Plants and Pictures)
• Uniform and Haircut
• Cleaning Assignments
• Proper Excuse Letter Format

August 15, 2024
Dear Ma’am/Sir,
Please excuse me for being absent today because I have a fever. I
will try my best to cope up with the lessons and missed activities
when I get back. Hoping for your kind consideration.
Yours truly,
Juan Dela Cruz
Noted:
Juana Dela Cruz
Parent/Guardian
DATE
NAME OF
PARENT/GUARDIAN
WITH SIGNATURE
NAME OF STUDENT
CLARIFY YOUR
REASON

GUESS THE WORD.
THE LOSER WILL
ANSWER ANOTHER
QUESTION.

I AM
Two of the most powerful
words. Because what you
put after them shapes
your reality.

SOLVING RATIONAL
EQUATIONS AND
INEQUALITIES

REVIEW
1.What is a function?
2. What are the ways of representing
function?
4. What is a rational expression?
3. Differentiate equation from
inequality.

DRILL
1. Given: and Determine as function or not
function.
2. Given: Determine as function or not function.
3. Given: and. Determine as equation or
inequality.

PROBLEM
The local barangay received a budget of
P100,000 to provide medical checkups for the
children in the barangay. The amount is to be
allotted equally among all the children in the
barangay. Write an equation representing the
relationship of the allotted amount per child (y-
variable) versus the total number of children (x-
variable).
Answer: , graph the function

Basing on the details from the
problem, complete the table
No. of children 10 20 50 100
Allocated
amount
10,000 5,000 2,000 1,000
No. of children 200 300 500 1000
Allocated
amount
500 333.33 200 100

1.
2.
3.
4.
Determine whether the given is a rational function,
a rational equation, a rational inequality, or none of
these.
rational
function
rational
equation
none of these
rational
inequality

IMPORTANT CONCEPT
Rational expression is
an expression that can
be written as a ratio of
two polynomials.

29
Rational expression
because it is a ratio of two
polynomials.
Rational expression
because the numerator 1 is
a polynomial. (of degree 0)
Not a rational expression

30
Rational expression
because the
expression is equal
to .
Rational expression
which is also a
polynomial.

IMPORTANT CONCEPT
Rational equation
is an equation
involving rational
expressions.

EXAMPLE OF A RATIONAL EQUATION

IMPORTANT CONCEPT
Rational inequality is an
inequality involving
rational expressions.

6 𝑥 −
5
𝑥 +3
≥ 0
EXAMPLE OF A RATIONAL INEQUALITY

IMPORTANT CONCEPT
Rational function is a function of
the form where and are
polynomial and is not zero
function.
A rational function can be
represented through its equation,
table values, or graphs.

37
Rational
Equation
Rational Inequality Rational Function
Definition An equation
involving
rational
expressions
An inequality
involving rational
expressions
A function of the
form of where p(x)
and q(x) are
polynomials, and q(x)
is not the zero
function.
Example

ACTIVITY
Determine whether the given is a
rational function, a rational
equation,
a rational inequality, or none of
these.

rational function, a rational equation,
a rational inequality, or none of these.
1. 3

Determine whether the given is a rational
function, a rational equation,
2.

3.

4.

5.

Answer: Rational Function
1.

Answer: Rational Equation
2.

Answer: None of these
3.

Answer: Rational Equation
4.

Answer: None of these
5.

Examples of Rational Equations
51

Steps:
52
1. Eliminate denominators by multiplying
each term of the equation by the least
common denominator (LCD)
2. Check the solutions of the transformed
equations with the original equation.

Let’s solve these equations:
53

ACTIVITY
Solve the following rational equations:
1.

Make sure that your activity
notebooks have your name.
Afterwards, pass it to the front.

Rational inequalities are easier to
solve if their denominators are
eliminated.
Remember that the sense of an
inequality is unchanged if the same
real number is added to, or subtracted
Rational inequality

•Moreover, the sense of an inequality
remains if both sides of the
inequality is multiplied by, or divided
by the same positive real number.
•But the sense of an inequality is
reversed if both sides of the
inequality is multiplied by, or divided
Rational inequality

1. Rewrite the inequality as a single
rational expression on one side of the
inequality and 0 on the other side.
2. Determine over what intervals the
rational expression takes on the
positive and negative values.
Solving rational inequality

D E T E R M I N E O V E R W H AT I N T E R V A L S T H E
R AT I O N A L E X P R E S S I O N TA K E S O N T H E P O S I T I V E
A N D N E G AT I V E V A L U E S .
Locate the x-values for which the rational expression is zero or
undefined (factoring the numerator and denominator is a
useful strategy).
Mark the numbers found in (i) on a number line. Use a shaded
circle to indicate that the value is included in the solution set,
and a hollow circle to indicate that the value is excluded. These
numbers partition the number line into intervals.
Select a test point within the interior of each interval in (ii). The
sign of the rational expression at this test point is also the sign
of the rational expression at each interior point in the
aforementioned interval.

S O L V E T H E I N E Q U A L I T Y
2 𝑥
𝑥 +1
≥ 1
2 𝑥
𝑥 +1
− 1≥ 0
2𝑥−(𝑥+1)
𝑥+1
≥0
𝑥 −1
𝑥+ 1
≥ 0
is part of the solution
while is not.
Why?
−1 1

−1 1
Interval
Test
point
Table
of
signs
𝑥=−2
−
−
+¿
𝑥=0
−
+¿
+¿
−
−
𝑥=2
+¿
+¿
+¿
+¿
| or )

Solve the following, determine its domain
and range, then draw its graph.
1.
3.
2.
4.
3
𝑥 −2
<
1
𝑥

SOL VE T HE FOL L OWING AND
DE TE RM INE I TS DOM AIN AND RANGE
1.
2.
2 𝑥
𝑥 +1
+
5
2 𝑥
=2
3.
4.

SOLVE TH E RAT IO NAL EQUATION
1.
2.
3.

SOLVE TH E RAT IO NAL INEQUALITY
1.
2.
3.

Rational-Equations-Inequalities-and-Functions_Intro.pptx

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