QuestionSimplify and State Restrictions $\frac{4}{x-2}-\frac{5 x-6}{3 x^{2}-11 x+10}$

Studdy Solution

STEP 1

What is this asking? We've got to subtract two rational expressions and then figure out any sneaky x values that would cause a divide-by-zero catastrophe! Watch out! Don't forget to find the common denominator and be super careful with those negative signs when subtracting.
Also, stating the restrictions is super important!

STEP 2

1. Factor the Denominators
2. Find the Least Common Denominator (LCD)
3. Rewrite with the LCD
4. Subtract the Rational Expressions
5. Simplify the Numerator
6. State the Restrictions

STEP 3

Let's factor that second denominator, $3x^2 - 11x + 10$ .
We're looking for two numbers that multiply to $3 \cdot 10 = 30$ and add up to $-11$ .
Those numbers are $-5$ and $-6$ .

STEP 4

So, we can rewrite the expression as $3x^2 - 5x - 6x + 10$ .
Now, let's factor by grouping: $x(3x - 5) - 2(3x - 5)$ .
This gives us $(x - 2)(3x - 5)$ .
Nice!

STEP 5

Now, our denominators are $(x - 2)$ and $(x - 2)(3x - 5)$ .
The LCD is $(x - 2)(3x - 5)$ .
It's like finding the ultimate ingredient list that covers both fractions!

STEP 6

The first fraction needs the $(3x - 5)$ part, so we multiply the top and bottom by $(3x - 5)$ : $\frac{4(3x - 5)}{(x - 2)(3x - 5)}$ .
The second fraction already has the LCD, so we leave it as is: $\frac{5x - 6}{(x - 2)(3x - 5)}$ .

STEP 7

Now, we can subtract! $\frac{4(3x - 5) - (5x - 6)}{(x - 2)(3x - 5)}$ .
Remember to distribute that negative sign to both terms in the second numerator.
Super important!

STEP 8

Let's distribute the $4$ in the first part of the numerator: $4(3x - 5) = 12x - 20$ .
Then, distribute the negative sign in the second part: $-(5x - 6) = -5x + 6$ .

STEP 9

Now, combine like terms: $12x - 5x = 7x$ and $-20 + 6 = -14$ .
So, our numerator becomes $7x - 14$ .

STEP 10

We can factor a $7$ out of the numerator: $7(x - 2)$ .
Our fraction now looks like $\frac{7(x - 2)}{(x - 2)(3x - 5)}$ .

STEP 11

Since $(x - 2)$ is in both the numerator and denominator, we can divide to one: $\frac{7(x - 2)}{(x - 2)(3x - 5)} = \frac{7}{3x - 5}$ .
Boom! Simplified!

STEP 12

Remember, we can't divide by zero!
So, we set each original factor of the denominators equal to zero and solve for x.

STEP 13

From $x - 2 = 0$ , we get $x = 2$ .
From $3x - 5 = 0$ , we get $3x = 5$ , so $x = \frac{5}{3}$ .

STEP 14

Therefore, our restrictions are $x \ne 2$ and $x \ne \frac{5}{3}$ .

STEP 15

Our simplified expression is $\frac{7}{3x - 5}$ , with restrictions $x \ne 2$ and $x \ne \frac{5}{3}$ .

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