Solving Rational Inequalities | TAGALOG-ENGLISH
Summary
TLDRThis video tutorial guides viewers on solving rational inequalities, a mathematical concept involving fractions where the numerator and denominator are polynomials. The presenter explains the process of finding zeros of both the numerator and denominator, and then tests intervals to determine where the inequality holds true. Examples are provided to illustrate the steps, including handling undefined expressions and ensuring the denominator is not zero. The tutorial aims to make complex mathematical problems more approachable.
Takeaways
- 📐 **Identify Critical Points**: Solve the numerator and denominator separately to find critical points where the expression is undefined or zero.
- 🔍 **Rational Inequality Form**: Rational inequalities are presented as fractions with inequality symbols, representing the relationship between the numerator and the denominator.
- 🎯 **Zero Solutions**: Set the numerator equal to zero to find the zeros, which are critical for determining the intervals to test.
- ✂️ **Interval Division**: Divide the number line into intervals based on the critical points to test the inequality within each segment.
- 📉 **Test Intervals**: Substitute test values from each interval into the inequality to determine where it holds true.
- 🚫 **Exclusion of Undefined Points**: Points that make the expression undefined (like division by zero) are excluded from the solution set.
- 🔄 **Sign Analysis**: Analyze the sign of the expression within each interval to determine if it satisfies the inequality.
- 🔢 **Substitution of Values**: Substitute specific values within each interval to test the inequality and refine the solution set.
- 🔄 **Reduction to Simpler Form**: Simplify the inequality by reducing fractions or finding the greatest common factor (GCF) to make it easier to solve.
- 📋 **Final Solution as Intervals**: Present the final solution as a set of intervals where the inequality is true, including or excluding endpoints as necessary.
Q & A
What is a rational inequality?
-A rational inequality is an inequality that involves rational expressions, which are fractions where both the numerator and the denominator are polynomials.
How do you solve a rational inequality like x + 4 over x - 1 ≤ 0?
-To solve the inequality x + 4 over x - 1 ≤ 0, you find the zeros of the numerator (x + 4 = 0) and the undefined points of the denominator (x - 1 ≠ 0), then test intervals between these points to see where the inequality holds true.
What are the zeros of the numerator and the undefined points of the denominator for the inequality x + 4 over x - 1 ≤ 0?
-The zero of the numerator is x = -4, and the undefined point of the denominator is x = 1.
How do you test the intervals for the inequality x + 4 over x - 1 ≤ 0?
-You test the intervals (-∞, -4], (-4, 1), and (1, ∞) by substituting values from each interval into the inequality and checking if it holds true.
What is the solution to the inequality x + 4 over x - 1 ≤ 0?
-The solution to the inequality x + 4 over x - 1 ≤ 0 is the interval [-4, 1), where -4 is included and 1 is not included.
How do you handle a rational inequality where the inequality sign is '>' instead of '≤'?
-When the inequality sign is '>', you look for intervals where the rational expression is positive instead of non-negative.
What is the process of finding the solution to a rational inequality with a quadratic numerator?
-For a rational inequality with a quadratic numerator, you find the zeros of both the numerator and the denominator, consider the undefined points, and test the resulting intervals to find where the inequality holds.
Can you give an example of solving a rational inequality with a quadratic numerator?
-Sure, for the inequality x^2 + 3x over 2x - 1 > 0, you would find the zeros of x^2 + 3x = 0 and 2x - 1 ≠ 0, then test the intervals between these points to find where the expression is positive.
What are the steps to solve a rational inequality with a compound expression like 1/(x - 3) ≤ 5/(x - 3)?
-First, simplify the compound expression to a single rational inequality, then find the zeros and undefined points, and test the intervals to find where the inequality holds.
How do you determine the intervals to test for a rational inequality?
-The intervals to test are determined by the zeros of the numerator and the values that make the denominator zero or undefined. These points divide the number line into intervals to be tested.
What is the significance of including or excluding certain points in the solution set of a rational inequality?
-Points are included in the solution set if the inequality holds true at those points, and excluded if the inequality does not hold. This is determined by testing values within the intervals and at the boundaries.
Outlines
📘 Introduction to Solving Rational Inequalities
The paragraph introduces the process of solving rational inequalities. It presents an example with the inequality x + 4 / (x - 1) ≤ 0. The narrator explains the importance of finding the zeros of the numerator and denominator to determine the intervals where the inequality holds true. The zeros are found by setting the numerator and denominator to zero and solving for x. The zeros for the given example are x = -4 and x = 1. The narrator then evaluates the inequality in the intervals determined by these zeros and concludes that the inequality is true for x values between -4 and 1, including -4 but not including 1.
🔍 Testing Intervals for Rational Inequalities
This paragraph continues the discussion on solving rational inequalities by testing the intervals identified in the previous paragraph. The narrator tests the inequality x + 4 / (x - 1) ≤ 0 with specific values from the intervals: negative five, zero, four, and one. The results show that the inequality holds true for the interval between -4 and 1, but not for the other intervals. The paragraph emphasizes the importance of testing each interval to determine where the inequality is satisfied.
📐 Solving a Quadratic Rational Inequality
The paragraph demonstrates how to solve a quadratic rational inequality using the example x^2 + 3x / (2x - 1) > 0. The narrator explains the process of finding the zeros of the numerator and denominator, which are x = 0 and x = 1/2, respectively. The zeros are then used to define intervals that are tested to see if the inequality holds true. The testing reveals that the inequality is true for x values between 0 and 1/2, and also for values less than 0 and greater than 1/2.
🔎 Evaluating Rational Inequalities with Different Intervals
This paragraph further explores solving rational inequalities by testing different intervals with the inequality x^2 + 3x / (2x - 1) > 0. The narrator tests values such as negative three, one half, negative four, and one. The results confirm that the inequality is true for the interval between 0 and 1/2, and also for values less than 0 and greater than 1/2. The paragraph concludes with the final answer for the intervals where the inequality holds true.
📐 Solving Another Rational Inequality Example
The paragraph presents another rational inequality example: 1 / (x - 3) ≤ 5 / (x - 3). The narrator simplifies the inequality to -4 / (x - 3) ≤ 0 and then finds the zeros by setting the denominator to zero, which gives x = 3. The intervals are then tested, and it is found that the inequality holds true for all x values except x = 3, where the expression is undefined.
🔍 Solving a Rational Inequality with a Common Denominator
The final paragraph discusses solving a rational inequality with a common denominator: x - 2 / (x + 2) ≥ 0. The narrator multiplies both sides by the common denominator to simplify the inequality. The zeros are found by setting the numerator to zero, which gives x = 2. The intervals are then tested, and the inequality is found to be true for all x values except x = -2, where the expression is undefined. The paragraph concludes with the final answer for the intervals where the inequality holds true.
Mindmap
Keywords
💡Rational Inequality
💡Numerator
💡Denominator
💡Zero of the function
💡Interval Testing
💡Undefined Expression
💡Critical Points
💡Greater than or equal to (≥)
💡Less than or equal to (≤)
💡Positive and Negative Numbers
Highlights
Introduction to solving rational inequalities.
Explanation of rational inequality in the form of a fraction or ratio.
The importance of finding zeros of the numerator and denominator.
How to determine the zeros by setting the numerator and denominator to zero.
Evaluating the inequality at the zeros to find valid intervals.
The concept of testing intervals to determine where the inequality holds true.
The process of solving the first example inequality x + 4 / (x - 1) ≤ 0.
Explanation of why zero divided by any number is zero and its significance in the inequality.
The method of testing intervals for the inequality x + 4 / (x - 1) ≤ 0.
Conclusion of the first example with the solution interval [-4, 1).
Introduction to the second example with a quadratic numerator.
How to handle the inequality x^2 + 3x / (2x - 1) > 0.
The process of finding zeros for the quadratic numerator and linear denominator.
Testing the intervals for the inequality x^2 + 3x / (2x - 1) > 0.
Final answer for the second example with the solution interval (-∞, -3) ∪ (0, ∞).
Introduction to the third example with a different form of rational inequality.
Solving the inequality 1 / (x - 3) - 5 / (x - 3) ≤ 0 by combining terms.
Determining the intervals for the third example and testing them.
Final solution for the third example indicating the valid intervals.
Introduction to the fourth and final example of the video.
Solving the inequality x - 2 / (x + 2) ≥ 0 by factoring and simplifying.
Testing the intervals for the fourth example and finding the solution.
Final answer for the fourth example with the solution interval (-∞, -2] ∪ [0, ∞).
Summary of how to evaluate rational inequalities through the examples provided.
Transcripts
in this video i am going to show you
how to solve a rational inequality
so let us take x plus 4 all over x minus
1
is less than or equal to 0.
rational inequality in equality rational
inequality
as you can see in a form of fraction or
ratio okay
inequality symbol or initial
[Music]
rational inequality is you have to get
the zeros of the numerator and then that
denominator
numerator denominator so numerator
i x plus four and then on the
denominator not an
i x minus 1 okay so to get the zeros you
will
equate this to zero and then hana pinoy
on volume
x so in this case lipato is a positive
force
negative four plus four over
negative four minus one so basically
um zeros numerator okay and then so
you evaluate nothing to a negative four
plus four that is zero
over negative five okay negative four
minus one is negative five
is less than or equal to zero zero
divided by negative five
is zero any number divided by i mean
zero divided by any number is zero so
less than or equal to zero
at a statement that guys is true because
of the equal sign here
zero is equal to zero statement
numerator
okay and then the next one is positive
one denominator so
input value x so i will have
one plus four over one minus one
less than or equal to zero so this one
is five over zero
less than or equal to zero alumni then
any number
divided by zero is undefined
which will be undefined so we which will
make the rational inequality undefined
so anger
nothing hindi included c one
okay so
so we have three intervals this interval
this one and then this two afternoon
it is interval okay so miligao numbers
numbers between negative infinity hang
on negative four union first interval
mi miliken number between negative four
and one that is the second interval
and mamma militant number between one
and positive
infinity so that that is the third
interval so the top of the nothing
is negative negative seven to negative
one million puede guys
okay first interval or not intervals
than you
said
okay so between here one up to positive
infinity and people
in four and then i will test each
of the intervals so this is a negative
five it is
x plus four okay over x minus one
is less than or equal to zero rational
inequality nothing
so i will have negative five plus four
over negative five minus one is less
than or equal to
zero okay so now this one would be
negative one
negative five plus four is negative one
over negative 5 minus 1 is negative
6 less than or equal to 0. now this one
is negative
over negative so once i got a positive 1
over
6. okay so
guys falsto positive one over six is not
less than or equal to zero zero
so false okay
and then the next one would be zero
interval i mean the second interval in
appealing
is zero so yanami x so
zero plus four over zero minus one
less than or equal to zero so this one
is four over
negative one less than or equal to zero
so four over negative one
is negative four less than or equal to
zero indeed negative four
is less than or equal to zero less than
zero and guys at
all okay so it makes a b and this one is
true
okay and then the last one would be
um four i thought a number
so i will put that to the value of x so
four plus four
over four minus one is less than or
equal to zero
so four plus four is eight over three
okay this one is false again because any
positive number
that is greater than zero so eight
thirds less than zero falls
on eight thirds
between negative four and
one
um
okay so we will have negative four
to one but negative four is included
and then positive one is not included so
a toyota
final answer nothing
okay let's have a
second example so let us have
x squared plus 3x all over
2x minus 1 is greater than zero
okay so as you can see quadratic
numerator
numerator and denominator so uncommon
terminal
is x
x times x plus three okay over
two x minus one is greater than zero
so cabin so x times x is x squared
and then x times three is three x so
so now we will get the value of um
the zeros the numerator
[Music]
numerator and then denominator two x
minus one
equals zero so li pattern making two x
plus one sorry equals positive one so
divide both sides by two
so on zero and denominator i
[Music]
zero squared plus three times zero
over two times zero minus one greater
than zero
so obviously this one is zero over
negative one okay so zero over negative
one is zero greater than zero
zero is not greater than zero equals
so this one is false okay
guys
is not greater than zero indian
zero equals okay
next one is a negative three so that
would be
um negative three squared
plus three times negative three
over two times negative three minus one
greater than zero so negative three
squared is nine
okay then three times negative three is
negative nine
over negative six minus one greater than
zero
so this one is zero over negative seven
again
zero greater than zero falls than a
manhattan
okay
indeed
x is raised to one i mean x is equal to
one half
so i will put that at the value of x so
x squared
so i have x squared and my gig one half
squared plus three times one half all
over
two times one half minus one greater
than zero
okay so one half raised to two is one
fourth
plus three halves all over so it will
cancel
one minus one greater than zero again
this one
is a number right in the united state
it will make the denominator equal to
one half ayan and then
mammalita you know number so therefore
it
let's have one fourth okay or zero point
um zero point one now i am paramount
so zero point one and then the last one
so one half
is um let's have
two yeah one half
actually putting one so one along
okay so test negative four so this one
would be
negative four squared plus 3 times
negative 4
all over 2 times negative 4 minus 1
greater than 0. so negative 4 squared is
16
minus 12 over negative 8 minus 1
greater than 0 so 16 minus 12 is 4
over negative 9 so obviously guys
negative angle
negative number is greater than 0
0 negative
i will have negative 1 squared plus 3
times negative 1
all over 2 times negative 1 minus 1
greater than 0
so 1 minus 3 over negative 2 minus 1.
okay so this one would be negative 2
over negative 3
negative divided by negative that will
give you a positive answer
two-thirds so this one is true
okay two-thirds is greater than zero
okay
satisfying
0.1 so 0.1 is
the third interval 0.1 squared plus
3 times 0.1 all over 2 times
0.1 minus 1 greater than 0.
okay so i'm gonna be nothing guys
so
0.1 squared
okay plus 3 times 0.1
over 2 times zero point
one minus one
so analogous a negative number so
negative 31
over 80 greater than zero so this one is
false
and then let us test the last interval
which is one so
one squared plus three times one over
2 times 1 minus 1 greater than 0. so 1
squared is 1
plus 3 over 2 minus 1
so this one is 4 over 1 again 4 is
greater than 4
so positive number is always greater
than
zero guys so
so a final answer nothing rational
inequality
inequality nato is so an interval number
three
negative three to zero so again
okay so that is our final answer
now let us have the third example
okay let's have one over x minus three
less than or equal to five over x minus
three okay so
guys
um
x minus three minus five over
x minus three so from positive making
negative
[Music]
and one minus five so
this one is four over x minus three
less than or equal to zero a negative
four sorry
again one minus five is negative four so
as you can see
okay so x minus three equations zero so
that would be positive three okay and
shortcuts are because
you guys
okay so i have two intervals negative
infinity to positive three
positive three two positive infinity so
pili aho
nito pipilli um zero
d2i5 ayan so
x minus three is zero minus three so
okay less than or equal to zero so
negative four
over negative three less than or equal
to zero so this one is positive four
thirds
okay so fosto
a positive number case is a zero so
fosto
hey testament so that would be negative
four
over five minus three
less than or equal to zero so this one
would be negative four
over positive two okay so this one is
negative two less than or equal to
zero this one is true so
an interval and a true is three
positive infinity so positive or
negative infinity
symbol i parenthesis so the final answer
is
this one okay let us have the last
example
of evaluating rational inequalities or
solving rational
inequality so eto guys
[Music]
x and that would be minus na oh minus 2
all over x plus 2 is greater than or
equal to zero
so again fractions
you will get the gcf
they have the same denominator
like i have one half plus one half yes
atta
x times x plus two and that would be
our new denominator so i'm again
so both sides we will multiply by x and
then x
plus two and deter n x
times x plus two so atom first term when
i multiplied x times x plus 2
by 1 over x omega and see x
1 x 2 so 1
x plus 2 okay second term naman
so negative two times x okay so i'm
getting
formula nothing i mean i'm gigging
rational inequality not n i
x plus 2 minus 2 x all over
x times x plus 2.
expanded
to get the zeros okay so simplified on
your numerator
so x minus 2x is negative x
so negative x plus 2 all over x
times x plus 2 greater than or equal to
zero ayan so
negative two m it's a positive two
negative two
divide both sides by negative one to get
the value of x so on x not indeed to i
positive two
okay and then denominator naman i have
we have two
x plus two
ayan
so remember
ascending order so i have negative 2
0 so not c 2 okay so
i'm denominator and 0 denominator
0 is greater than or equal to 0 because
of the equal sign
[Music]
okay so we have one two three four
intervals
space a positive infinity okay so
nothing interval is negative infinity to
negative two
so not negative two to zero so not zero
to two
and then two to positive infinity okay
so pilia number angus whole number due
to the first interval
and negative three in along a negative
one
detail a positive
omega i haven't gained the long negative
sign okay so plus two
all over negative three times negative
three plus two
is greater than or equal to zero so this
one is positive three now
plus two over negative three times
negative one
greater than or equal to zero so five
over
positive three is greater than or equal
to zero through yan
let's say five thirds is greater than
zero positive say
x so negative times negative 1 plus 2
over negative 1 times negative 1 plus 2
is greater than or equal to 0
so positive 1 plus 2 over negative 1
times 1
greater than or equal to zero so this
one is one over negative hey sorry
three paletto guys three over negative
one
okay so that is negative three greater
than or equal to zero that
is false
negative 3 is not greater than 1 i mean
not greater than 0. 0 is a negative
number okay so let's have the last two
intervals
one so negative then x a one plus two
over one times one plus two greater than
or equal to zero so this one is one
over one times three okay so one over
three is greater than zero
that is true one third is greater than
zero
okay so
okay and then the last one is three in
testing value nothing
so negative three plus two over negative
sorry three times three plus two
greater than or equal to zero so your
numerator a negative one
over three times five okay so negative
one
over 15 false yan because a negative 1
over 15
is not greater than zero so this one is
false
so annoying interval not a true so
my interval not true is so
infinity i parenthesis not included
so parenthesis guys
positive and negative infinity included
points
okay and then union
not included therefore two is included
okay so the final answer is this one
ayan as a good nathan
so that is how you evaluate rational
inequalities
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