Solving Rational Equations | General Mathematics

Summary

This educational video tutorial guides viewers through solving rational equations, which are equations containing rational expressions, or quotients of two polynomials. The presenter methodically demonstrates the process using three examples, starting with finding the least common denominator (LCD), multiplying through by the LCD, simplifying, and solving for the variable. Each solution is checked for extraneous roots. The video is interactive, encouraging viewers to ask questions in the comments section, and the presenter ensures clarity by checking solutions in the original equations.

Takeaways

• 📘 The video is a tutorial on solving rational equations, which are equations involving rational expressions.
• 🔢 A rational expression is defined as a quotient of two polynomials.
• ❌ The instructor emphasizes that the denominator should not be equal to zero.
• 📝 The first example involves solving the equation 2/x - 3/2x = 1/5 by finding the least common denominator (LCD).
• 🔄 The process of solving involves multiplying each term by the LCD and simplifying the equation.
• 🔍 After simplifying, the example leads to solving for x, which results in x = 5/2.
• 🔄 The solution x = 5/2 is checked by substituting it back into the original equation to confirm its validity.
• 📐 The second example demonstrates solving a rational equation with binomials in the denominator.
• 🔢 The LCD for the second example is the product of the binomials (x + 1)(x - 3), and the solution is found to be x = 11.
• 🔄 The solution x = 11 is verified by substitution, confirming it is a valid solution to the equation.
• 📘 The third example is a more complex rational equation that simplifies into a quadratic equation, leading to two potential solutions, x = -7 and x = 2.
• 🔄 The solutions are checked, and it's found that x = -7 is valid while x = 2 is an extraneous solution due to making the denominator zero.

Q & A

What is a rational equation according to the video?

A rational equation is an equation involving rational expressions, which are expressions that can be written as a quotient of two polynomials.

Why is it important to not set the LCD to zero when solving rational equations?

Setting the LCD (Least Common Denominator) to zero would eliminate the denominators, which is not allowed in mathematics as it can lead to undefined expressions and loss of potential solutions.

What is the first step in solving the example equation 2/(x-3) + 3/2 = 1/5?

The first step is to find the LCD, which in this case is 10x, by multiplying the denominators x, 2, and 5.

How is the equation 2/(x-3) + 3/2 = 1/5 transformed after multiplying by the LCD?

After multiplying by the LCD (10x), the equation becomes 20x/x + 30x/2x = 10x/5.

What is the simplified form of the equation after the first example's transformation?

The simplified form is 20x = 15, which simplifies further to x = 5/2 after combining like terms and solving for x.

How do you check if the solution x = 5/2 is valid for the first example?

You substitute x = 5/2 back into the original equation and simplify to ensure both sides of the equation are equal.

What is the LCD for the second example equation 3/x + 1 = 2/(x-3)?

The LCD for the second example is the product of the two binomials (x + 1) and (x - 3).

How is the equation 3/x + 1 = 2/(x-3) simplified after multiplying by the LCD?

After multiplying by the LCD, the equation simplifies to 3(x - 3) = 2(x + 1), which further simplifies to 3x - 9 = 2x + 2.

What is the solution for x in the second example after simplification?

After simplifying and solving for x, the solution is x = 11.

How do you verify the solution x = 11 in the second example?

You substitute x = 11 back into the original equation and check if both sides are equal, confirming it is a valid solution.

What is the LCD for the third example equation x^2 - 4x / (x - 2) = 14 - 9x / (x - 2)?

The LCD for the third example is x - 2, as it is the common denominator in the equation.

What type of equation results after multiplying the third example by the LCD?

After multiplying by the LCD, the resulting equation is a quadratic equation: x^2 - 4x = 14 - 9x.

What are the solutions for x in the third example after solving the quadratic equation?

The solutions for x are x = -7 and x = 2, found by factoring the quadratic equation.

How do you determine if x = 2 is an extraneous solution in the third example?

By substituting x = 2 into the original equation, you find that the denominator becomes zero, making it an extraneous solution.

Outlines

00:00

📘 Introduction to Solving Rational Equations

The video begins with a welcome to the channel and an introduction to rational equations. The instructor explains that a rational equation involves a rational expression, which is a quotient of two polynomials. The importance of not having a zero denominator is emphasized. The first example involves solving the equation 2/(x-3) + 3/2x = 1/5. The process includes finding the least common denominator (LCD), multiplying each term by the LCD, simplifying, and solving for x. The solution x = 5/2 is found and verified by substituting back into the original equation. The instructor also cautions about the possibility of extraneous roots.

05:02

🔍 Detailed Walkthrough of Rational Equation Example

The second paragraph delves deeper into solving a rational equation with the example 3/(x-3) + 1 = 2/(x+1) - 3. The instructor demonstrates the process of finding the LCD, which in this case is (x+1)(x-3), and then multiplying each term by the LCD. The equation is then simplified to 3(x+1) - 3(x-3) = 2(x+1) - 6(x-3). After simplifying, the equation is solved for x, yielding x = 11. The solution is checked by substituting x = 11 back into the original equation, confirming that it is indeed a solution.

10:02

📐 Advanced Rational Equation Solving Techniques

The final paragraph introduces a more complex rational equation, x^2 - 4x / (x-2) = (14 - 9x) / (x-2). The LCD is identified as (x-2), and the equation is multiplied by this LCD, resulting in a quadratic equation. The instructor then rearranges and simplifies the equation to x^2 - 4x + 5x - 14 = 0, which simplifies further to x^2 + x - 14 = 0. Factoring the quadratic equation yields (x+7)(x-2) = 0, leading to two potential solutions: x = -7 and x = 2. The instructor checks both solutions and confirms that x = -7 is a valid solution while x = 2 is an extraneous solution due to it making the denominator zero. The video concludes with an invitation for questions and a farewell.

Keywords

💡Rational Equation

A rational equation is an equation that includes one or more rational expressions. In the context of the video, rational equations are the main focus, and the video aims to teach viewers how to solve them. An example from the script illustrates this: 'solve for x two over x minus three over two x equals one over five.' This equation involves a quotient of two polynomials, which is the definition of a rational expression.

💡Rational Expression

A rational expression is defined as an expression that can be written as a quotient of two polynomials. The video script explains that these expressions are the building blocks of rational equations. An example used in the video is 'two over x minus three over two,' which is a rational expression because it is the division of two polynomials.

💡LCD (Least Common Denominator)

The Least Common Denominator is a key concept when solving rational equations. It is the smallest multiple that all the denominators can divide into without leaving a remainder. In the video, the LCD is used to clear the denominators from the equation, making it easier to solve. The script mentions finding the LCD as the first step in solving an example equation: 'so first step not in detail is we, need to find the, lcd.'

💡Quotient

A quotient is the result of division, which is fundamental to understanding rational expressions and equations. The video script refers to quotients when discussing the division of polynomials to form rational expressions, such as 'two over x minus three over two x equals one over five,' where each term is a quotient of polynomials.

💡Multiplying by LCD

Multiplying each term of a rational equation by the LCD is a common step in solving such equations. This process eliminates the denominators, allowing for simplification and solving of the equation. The video script describes this step: 'then step two we need to multiply, each term of the equation, by the lcd.'

💡Simplifying

Simplification is a crucial step in solving equations, where like terms are combined, and expressions are reduced to their simplest form. The video script includes this step after multiplying by the LCD: 'so we have 20 x over x omega0, is 20 minus 30 x divided by 2x we have, 15.'

💡Like Terms

Like terms are terms in a polynomial that have the same variables raised to the same power. The video script refers to combining like terms when simplifying the equation: 'combining like terms, 20 minus 15 we have 5 equals, 2x.'

💡Extraneous Root

An extraneous root is a solution that arises from the process of solving an equation but does not satisfy the original equation. The video script emphasizes the importance of checking for extraneous roots after solving a rational equation: 'so after nothing must muslim cx we need to, check, out or not indeed, extraneous root.'

💡Substitution

Substitution is a method used to check the validity of a solution by inserting it back into the original equation. The video script uses substitution to verify the solution: 'so to check this is a substitute not in, your 5 over to the unsatin, original equation.'

💡Quadratic Equation

A quadratic equation is a polynomial equation of degree two. The video script includes an example where the resulting equation after clearing the denominators is a quadratic equation: 'then i'm acting resulting equation is a, quadratic equation, x squared this is a left side so paramus.'

Highlights

Introduction to solving rational equations.

Definition of a rational expression as a quotient of two polynomials.

Explanation of the importance of not setting the denominator to zero.

Step-by-step guide to solving the first example equation.

Finding the least common denominator (LCD) for a rational equation.

Multiplying each term by the LCD to eliminate denominators.

Simplification of the equation after multiplying by the LCD.

Combining like terms to simplify the equation further.

Solving for x by isolating the variable.

Checking the solution by substituting back into the original equation.

Verification that the solution does not make any denominator zero.

Introduction to the second example with binomials in the denominator.

Multiplying each term by the LCD to simplify the equation.

Simplification and solving for x in the second example.

Checking the solution for the second example.

Introduction to the third example with a quadratic equation.

Identifying the LCD for the third example and multiplying through.

Transforming the resulting equation into a standard quadratic form.

Solving the quadratic equation for x.

Checking the solutions for the third example and identifying extraneous roots.

Conclusion of the lesson and invitation for questions or clarifications.

Transcripts

hello class welcome to my channel

in this video i will show you how to

solve rational

equations so rational equation is an

equation involving rational expression

so rational expression is an expression

that can be written

as a quotient of two polynomials

so okay so the starting uh quotient

you queued up but not in detail should

not be equal to

zero let's try example number one

solve for x two over x minus three over

two

x equals one over five

so one first step not in detail is we

need to find the

lcd okay so dito

merlin denominators we have x

two x and five so two and five alumni

lcd

okay then step two we need to multiply

each term of the equation

by the lcd so nito

distribute nothing on 10x don't starting

equation okay

so we have 10x times 2 that

is 20x over x

minus 10x times 3

that is 30x over 2x

equals 10 x times one

that is ten x over five

next is we need to simplify

so 20 x over x omega0

is 20 minus 30 x divided by 2x we have

15

then young adding x

equals 10x divided by 5

that is 2x then combining like terms

20 minus 15 we have 5 equals

2x then solving for x we need to divide

both sides of the equation

by two so that we can cancel this one

then x is equals to five

over two okay so

after nothing must muslim cx we need to

check

out or not indeed

extraneous root okay

so to check this is a substitute not in

your 5 over to the unsatin

original equation so we have 2 over

x which is 5 over 2

minus 3 over two

times five over two

equals one over five

okay so we have two over five over two

so d two paramount simplifying

which is two over five

minus determine three

over hands so we have three over

five equals one over five

so two times two that is four over five

minus three over five equals

one over five so four minus three

that is one then capital nothing in the

denominator

then we have one over five

equals one over five so therefore

five over two is the solution of two

over x

minus three over two x equals

one over five so next let's have

example number two three over x

plus one equals two over x

minus three so our first step is we need

to find

that lcd so in our case we have two

binomials

x plus one and x minus three

so normally pagoda you adding

denominators

i'm adding lcd i x plus one and

x minus three and then okay

so next we need to multiply each term

of our equation by the lcd

so 2 over x minus 3. so multiply that

into

by x plus one

and x minus three

so first multiplication as you can see

x plus one so um

3 and x minus 3

okay so we have 3 times

x minus 3 equals

it's the second multiplication not in

this right side

america hansel namant is the x minus 3

then my a1 is 2 times

x plus 1 next after that after now

multiplication

we need to simplify our equation so by

distribution

we have three times x that is three x

then three times negative three we have

negative nine

then for two x number one f four two

multiply nothing and we have

two x plus two

okay the next letting my x

really but not in the left side then lat

no

constants are right side so we have 3x

so in 2x pagoda but we have negative 2x

then say negative 9 transpose nothing

done and

that is positive 9. combining like terms

we have x

equals 11.

okay then after nothing what is cx

we need to check kunta lagabang solution

c11

then rational

equation so by substitution we have 3

over 11 plus 1

equals 2 over x which is eleven

minus three so we have

three over twelve

equals two over eight

then simplifying that n so three over 12

that is equivalent

to 1 4 then 2 over 8 is also equivalent

to

1 4. so therefore

11 is the solution

duns equation okay

next let's have example number three

number three x squared minus four x all

over

x minus 2 equals 14 minus 9 x

all over x minus 2. so again i'm adding

first

first step is we need to identify the

lcd

so as you notice on denominators nothing

ipad

so meaning our lcd is x

minus two okay then we need to multiply

each term of our equation by

the lcd so dito

see 14 minus nine x over x minus two

is multiply not in guys by

x minus two okay

lcd so we have x squared minus 4x

equals 14 minus 9x

then i'm acting resulting equation is a

quadratic equation

x squared this is a left side so paramus

also we need to transpose

everything the unselect

left side okay so we have x squared

minus four x

minus four x so alpha but nothing though

we have negative 14

then your negative nine x figure but we

have positive

nine x equals zero

then combining like terms we have x

squared

then negative four x plus nine x that is

five x

minus fourteen equals zero

so power of magnifying sulfide i think

quadratic equation

is if a factor or nothing since factor

volume

quadratic equation so we have x

and x so negative 14

factorial negative 14 napa peanut at nut

and we have positive five so

in this case and i and i is equal

numbers i

positive seven and negative two

dial seven times negative two we have

negative fourteen

then seven minus two that is

positive five so tamayo atom factor

after nothing macho in factors equate

not

in both factors by zero

so we have x plus seven equals zero

and x minus two equals zero

okay then solving for x we have two

values of x

x equals negative seven and x

equals positive two okay

so parenthesis that we need to check

your negative seven bar and two i put a

solution

equation so checking dial

so checking unit is negative seven

so negative seven squared

minus 4 times negative 7

over negative 7

minus 2 equals

14 minus nine times negative seven

all over negative seven minus two

so simplifying nothing in the numerator

negative seven squared that

is 49 then negative 4 times negative 2

negative 7 is positive 28

all over negative 9

equals 14

then negative 9 times negative 7 that is

positive 63 over

negative 9 okay

then 49 plus 28 that is

77 over negative 9

equals 14 plus 63 is

77 over negative 9.

so therefore c negative 7 is a

solution non-acting equation

okay next trinomial not in c

positive two so we have

two squared minus 4

times 2 over 2

minus 2 equals

14 minus 9 times 2

over two minus two so these

guys as you notice in denominator

nothing

not two minus two is making zero so

meaning

undefined so therefore

hindi kung

is a extraneous

solution

okay so ditonic data posts and

in lesson guys if you have questions

or clarification by

section below so thank you guys for

watching

see you in our next video bye

you