Adding and Subtracting Rational Expressions With The Same Denominators

Summary

This video tutorial offers a clear guide on simplifying rational expressions, especially when dealing with like denominators. It demonstrates the process through several examples, starting with combining fractions with a common denominator and simplifying them by finding a common factor. The video also covers more complex cases, such as subtracting fractions with the same denominator and simplifying expressions with unlike terms, emphasizing the importance of distributing negative signs and identifying the greatest common factor for simplification. The examples provided are practical and illustrate the step-by-step approach to solving such problems, making the content accessible and informative for viewers.

Takeaways

• 📚 Simplifying rational expressions involves combining fractions with like denominators into a single fraction.
• 🔍 When adding or subtracting fractions with the same denominator, the numerators are combined while the denominator remains the same.
• 📝 Example provided: (3x/5) + (4x + 7/5) simplifies to (7x + 7)/5, which further simplifies to (x + 1) after factoring out the greatest common factor (GCF).
• 🧩 The GCF can be factored out from the numerator to simplify the expression further, if applicable.
• ➖ Subtracting fractions with the same denominator involves combining the numerators while keeping the denominator constant, as shown in the example (7/x) - (11/x).
• 🔢 Negative results in the numerator, such as in the subtraction example, lead to a simplified expression with a negative numerator over the common denominator.
• 📐 The process of combining like terms is crucial when simplifying complex rational expressions, as demonstrated in the third example.
• 👉 It's important to distribute the negative sign correctly when combining terms with different signs, as in the fourth example with (3x - 10) - (5x - 12) over (x - 1).
• ✂️ Simplification may involve canceling out common factors in the numerator and denominator, which can lead to a more simplified form.
• 📉 Negative signs in the numerator and denominator can sometimes cancel each other out, resulting in a positive value, as in the final example.
• 🎯 The final answer in the last example is simplified to -2, demonstrating the importance of careful handling of signs and terms during simplification.

Q & A

What is the main topic of the video?

The main topic of the video is simplifying rational expressions when adding or subtracting fractions with like denominators.

What is the first example given in the video for simplifying rational expressions?

The first example is simplifying the expression 3x/5 + (4x + 7)/5.

How are the two fractions in the first example combined?

The two fractions are combined by adding the numerators since they share the same denominator, resulting in (3x + 4x + 7)/5.

What is the greatest common factor (GCF) of the numerator in the first example, and how is it used?

The GCF of the numerator 7x + 7 is 7, which is factored out, simplifying the expression to x + 1 over 5.

What is the second example presented in the video, and what is the result?

The second example is 7/x - 11/x, which simplifies to -4/x after combining the fractions.

How does the video handle the third example with the expression (7x + 4)/(x + 2) + (5x - 7)/(x + 2)?

The video combines the numerators and simplifies the expression to 4x - 1 over x + 2.

What is the fourth example in the video, and what is the final simplified result?

The fourth example is (3x - 10)/(x - 1) - (5x - 12)/(x - 1), which simplifies to -2 after canceling out the common denominator.

What is the significance of the negative sign in the fourth example, and how does it affect the simplification process?

The negative sign in the fourth example applies to both the 5x and -12, turning the expression into (3x - 5x - 10 + 12)/(x - 1), which simplifies to -2x + 2 over x - 1.

How does the video suggest simplifying the expression (3x - 10)/(x - 1) - (5x - 12)/(x - 1)?

The video suggests combining the numerators with the negative sign distributed, resulting in -2x + 2 over x - 1, and then canceling out the common denominator x - 1 to get -2.

What is the common denominator in the fourth example, and why can it be canceled out?

The common denominator in the fourth example is (x - 1), and it can be canceled out because it appears in both the numerator and the denominator, simplifying the expression to -2.

What is the key takeaway from the video regarding the simplification of rational expressions?

The key takeaway is that when adding or subtracting rational expressions with like denominators, you can combine the numerators and then simplify the result by factoring out the greatest common factor and canceling out common factors in the numerator and denominator.

Outlines

00:00

📚 Simplifying Rational Expressions with Like Denominators

This paragraph introduces the concept of simplifying rational expressions when adding or subtracting fractions that have the same denominator. The example given is (3x/5) + (4x + 7)/5, which simplifies to (7x + 7)/5, then further to x + 1 after factoring out the greatest common factor (GCF) of 7. The explanation emphasizes the process of combining like terms over a common denominator and simplifying the expression.

🔢 Subtracting Fractions with a Common Denominator

The second paragraph demonstrates the subtraction of fractions with a common denominator, using the example (7/x) - (11/x). It simplifies to -4/x by combining the numerators while keeping the common denominator. This example highlights the straightforward process of subtracting fractions when they share the same denominator.

🧩 Combining Unlike Terms in Rational Expressions

This paragraph tackles the combination of unlike terms in rational expressions, exemplified by (7x + 4)/(x + 2) + (5x - 7)/(x + 2). The process involves combining like terms (7x + 5x and 4 - 7) over the common denominator (x + 2), resulting in (12x - 3)/(x + 2), which simplifies to 4x - 1/(x + 2) after factoring out the GCF of 3.

⚠️ Caution with Mixed Operations in Rational Expressions

The final paragraph warns about the careful handling of mixed operations in rational expressions, illustrated by (3x - 10)/(x - 1) - (5x - 12)/(x - 1). The solution involves distributing the negative sign and combining like terms, which results in -2x + 2. The greatest common factor of -2 is factored out, and the (x - 1) in the numerator and denominator cancels out, leading to the final answer of -2.

Keywords

💡Rational Expressions

Rational expressions are mathematical expressions that consist of a numerator and a denominator, where the denominator is not equal to zero. They are the focus of the video, which discusses how to simplify them. In the script, rational expressions are simplified by combining like terms and finding the greatest common factor (GCF), as seen in the first example where '3x/5 + (4x + 7)/5' simplifies to '7x + 7/5'.

💡Like Denominators

Like denominators refer to fractions that share the same denominator, which allows for the addition or subtraction of the numerators while keeping the common denominator. The script demonstrates this concept in the first example, where two fractions with the denominator '5' are combined into a single fraction '(3x + 4x + 7)/5'.

💡Combining Fractions

Combining fractions is the process of adding or subtracting fractions with like denominators by adding or subtracting their numerators. The script illustrates this with the example '3x/5 + (4x + 7)/5', which is combined into '(3x + 4x + 7)/5'.

💡Greatest Common Factor (GCF)

The GCF is the largest number that divides two or more numbers without leaving a remainder. In the context of the video, the GCF is used to simplify expressions by factoring it out, as shown in the first example where '7x + 7' is factored to '7(x + 1)/5'.

💡Simplification

Simplification in mathematics involves making an expression easier to understand or work with by reducing it to its simplest form. The entire video is dedicated to simplifying rational expressions, as demonstrated in each example provided.

💡Numerator

The numerator is the top part of a fraction and represents the number of parts being considered. In the script, numerators are combined or modified during the simplification process, such as in '7x + 5x' which becomes '12x'.

💡Denominator

The denominator is the bottom part of a fraction and represents the total number of equal parts into which the whole is divided. The video discusses how to handle like denominators and how they remain constant when combining fractions, as in '3x/5 + 4x/5'.

💡Distributive Property

The distributive property is a fundamental algebraic principle that allows for the multiplication of a term by a sum or difference. Although not explicitly mentioned, the script implicitly uses this property when combining like terms under a common denominator, as seen in the last example where '3x - 10 - 5x + 12' is simplified.

💡Negative Signs

Negative signs indicate the subtraction of a quantity in mathematics. The script discusses how to handle negative signs when combining terms, such as in the example '7x + 4 - 7' which simplifies to '12x - 3'.

💡Cancellation

Cancellation in the context of fractions occurs when a common factor in the numerator and denominator is divided out, simplifying the expression. The script mentions this in the last example, where 'x - 1' in the numerator and denominator cancels out, leading to the final answer of '-2'.

Highlights

The video discusses simplifying rational expressions when adding or subtracting fractions with like denominators.

Example given: Simplifying 3x/5 + (4x + 7)/5 by combining like denominators.

Result of first example is (3x + 4x + 7)/5, which simplifies to 7x + 7/5.

Greatest common factor (GCF) of 7 can be factored out to simplify further to x + 1.

Second example: Simplifying 7/x - 11/x by combining to get -4/x.

Third example involves combining 7x + 4/(x + 2) + 5x - 7/(x + 2).

Combining like terms results in 12x - 3/(x + 2).

Factoring out 3 gives 4x - 1/(x + 2).

Fourth example combines 3x - 10/(x - 1) - (5x - 12)/(x - 1).

Care must be taken to distribute the negative sign correctly in the fourth example.

Resulting expression is (3x - 5x - 10 + 12)/(x - 1) which simplifies to -2x + 2/(x - 1).

GCF of -2 can be factored out to get x - 1/(-2).

The (x - 1) terms cancel out, leaving the final answer of -2.

The video provides step-by-step instructions for simplifying rational expressions with like denominators.

Each example demonstrates the process of combining like terms and factoring out common factors.

The importance of correctly distributing negative signs is emphasized in the examples.

The final simplified form of each expression is clearly presented.

Transcripts

in this video we're going to talk about

how to

simplify rational expressions when

adding or subtracting them whenever they

have like denominators

so let's start with this example 3x over

5

plus

four x plus seven over five

because these two fractions share the

same denominator

we can combine it as a single fraction

so we can write it as three x

plus four x plus seven

all divided by the common denominator

which is five

three x plus four x is seven x

and if we want to we can take out the

gcf which is seven

leaving behind x plus one

so that's the final answer for this

example

let's try another one

7 over x

minus 11 over x

so we can combine it as a single

fraction

seven minus eleven over x

and seven minus eleven is negative four

so the answer is going to be negative

four divided by x

try this one

seven x plus four

divided by x plus two

plus five x

minus seven over x plus two

so let's write it as a single fraction

seven x

plus five x i'm gonna put the like terms

together

plus four minus seven

all divided by x plus two

seven x plus five x is twelve x

four minus seven is negative three

so we can take out a three if we want so

it's going to be four x

minus one over x plus two

and so that's it

let's try one more example

three x minus ten

divided by

x minus one

minus

five x minus twelve

over x minus one

now be careful with this one go ahead

and try it

so first let's write it as a single

fraction

so it's going to be three x minus ten

now this negative sign applies to the 5x

and

negative 12.

so initially i'm going to write it using

parentheses

before you combine like terms distribute

the negative sign

so it's going to be 3x minus 10

minus 5x

plus 12.

3x minus 5x is negative 2x

negative 10 plus 12

is positive 2.

now let's take out the gcf which is

negative two negative two x divided by

negative two

that's positive x

and positive two divided by negative two

is negative one

so notice that we can cancel x minus one

which means the final answer

is simply minus two