Add the following expressions and simplify if possible: 4 x − 5 + 3 5 − x \frac{4}{x-5}+\frac{3}{5-x}

problem asks us to add the following expressions and simplify, if possible. Now notice here we have four over x minus five plus three over five minus x. Now in order to get these to be added together, these rational expressions, we have to get a common denominator. Now the denominators actually look pretty close, but they're different because we have x minus five, and here we have five minus x. When it comes to subtraction, we can't just change around the order when we want to. But there is something that I can do. I can choose one of these two fractions that I want to change here. And I personally am going to choose the second fraction, five minus x. You could do this with the first fraction and still get a legitimate answer to this question. So we'll keep four over x minus five. Everything stays the same for the first fraction, but the second fraction is going to change. So what I'm going to do here is we'll have plus, that's going to be three over, and I'm just gonna switch the order, but I have to keep that negative sign on the x. So it's gonna be negative x plus five. Negative x plus five is the same thing as five minus x. I can't change switch these order and keep the subtraction in the same place. But if I put a negative sign on the x, then that allows me to switch the order. So notice we have a negative x plus five in the denominator. But now it's still not quite where we want to get it because even though I've changed the order here, I still don't have the same denominator that I have for the first fraction. So I'm going to continue to keep this first fraction the same. But what I'm going to do is factor a negative one out of the denominator. Now to do that, we'll have a negative here. Then we're going to have x, and I have to change the sign here, so it's going to be minus five. Now we're really close to getting a common denominator here, but notice I have this negative sign up front. But that's actually okay because I can take this negative sign and just make the entire expression on the right side negative. But If I make this negative, that's the same thing as just subtracting this expression. So really, this becomes four over x minus five minus three over x minus five. And we have a common denominator, so now we just need to subtract the numbers in the numerator. So it's gonna be four minus three, which is just one, over x minus five, and that right there is the solution to the problem.

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Different Denominators

Beginning Algebra

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Different Denominators

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Multiple Choice

Add the following expressions and simplify if possible:
$\frac{4}{x - 5} + \frac{3}{5 - x}$

$\frac{1}{x + 5}$

$\frac{x}{x - 5}$

$\frac{1}{5 - x}$

$\frac{1}{x - 5}$

Verified step by step guidance

Identify the expressions to be added: $\frac{4}{x-5} + \frac{3}{5-x} + \frac{1}{x+5} + \frac{x}{x-5} + \frac{1}{5-x}$.

Notice that \$5 - x$ is the negative of $x - 5$, so rewrite fractions with denominator $5 - x$ in terms of $x - 5$ by factoring out a negative sign: $\frac{3}{5-x} = -\frac{3}{x-5}$ and $\frac{1}{5-x} = -\frac{1}{x-5}$.

Rewrite the entire expression using a common denominator of $x - 5$: $\frac{4}{x-5} - \frac{3}{x-5} + \frac{1}{x+5} + \frac{x}{x-5} - \frac{1}{x-5}$.

Combine the fractions with denominator $x - 5$ by adding their numerators: $\frac{4 - 3 + x - 1}{x-5} + \frac{1}{x+5}$.

Simplify the numerator of the combined fraction and then consider adding the remaining fraction $\frac{1}{x+5}$ by finding a common denominator if needed, or recognize if further simplification leads to the final simplified form.

8:03m

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Add the following expressions and simplify if possible:
$\frac{4}{x - 5} + \frac{3}{5 - x}$