Adding and Subtracting Rational Expressions With The Same Denominators
Summary
TLDRThis 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
📚 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.
Mindmap
Keywords
💡Rational Expressions
💡Like Denominators
💡Combining Fractions
💡Greatest Common Factor (GCF)
💡Simplification
💡Numerator
💡Denominator
💡Distributive Property
💡Negative Signs
💡Cancellation
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
Weitere ähnliche Videos ansehen
5.0 / 5 (0 votes)