Solving Rational Inequalities | TAGALOG-ENGLISH

Summary

This 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

00:00

📘 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.

05:03

🔍 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.

10:09

📐 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.

15:10

🔎 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.

20:11

📐 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.

25:13

🔍 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.

Keywords

💡Rational Inequality

A rational inequality involves an inequality that contains a rational expression, which is a fraction with polynomials in both the numerator and denominator. In the video, the instructor explains how to solve rational inequalities by finding the zeros of the numerator and evaluating the values of x that make the inequality true or false.

💡Numerator

The numerator is the top part of a fraction. In the context of rational inequalities, solving involves setting the numerator equal to zero to find the critical points. For example, in the script, the numerator 'x + 4' is set to zero to find where the rational expression might equal zero.

💡Denominator

The denominator is the bottom part of a fraction. In rational inequalities, it is essential to determine where the denominator is zero because division by zero is undefined. The script shows how the value 'x - 1' in the denominator is set to zero to find critical points that make the expression undefined.

💡Zero of the function

A zero of a function is a point where the function's value equals zero. In rational inequalities, zeros help divide the number line into intervals where the inequality might change from true to false. The script demonstrates how finding the zero of the numerator (e.g., x = -4) helps determine when the inequality equals zero.

💡Interval Testing

Interval testing is the method of dividing the number line into intervals based on critical points (where the numerator or denominator equals zero) and testing values within those intervals to determine if the inequality holds true. The script explains how to choose test points like -5, 0, and 4 to evaluate which intervals satisfy the inequality.

💡Undefined Expression

An undefined expression occurs when a function involves dividing by zero, which is not mathematically valid. In the video, the instructor mentions how at x = 1, the denominator becomes zero, making the rational expression undefined, and this point must be excluded from the solution.

💡Critical Points

Critical points are values of x where the numerator or denominator equals zero, which divides the number line into intervals. These points are crucial in solving rational inequalities as they represent potential boundaries where the inequality might change from true to false.

💡Greater than or equal to (≥)

The 'greater than or equal to' symbol (≥) in inequalities signifies that the expression on one side of the inequality is either greater than or equal to the expression on the other side. In the script, the instructor uses this to check whether the rational expression's value at specific points is greater than or equal to zero.

💡Less than or equal to (≤)

The 'less than or equal to' symbol (≤) is used to compare two expressions, stating that one is smaller than or equal to the other. The script involves testing whether fractions, such as 0 or negative values, are less than or equal to zero in the context of solving inequalities.

💡Positive and Negative Numbers

In interval testing, the instructor checks whether test points in the intervals produce positive or negative results. The script explains how dividing negative by negative gives a positive result and how this affects whether the inequality is true or false in different intervals.

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