Question

Simplify $\frac{x+1}{3}+\frac{2-3x}{2}$. Express your answer as a single fraction.

Answer 1

Instead of $, use \( to start LaTeX.

Answer 2

ok

Answer 3

Then use \) to end LaTeX instead of $.

Answer 4

Also, I prefer dfrac for regular fractions and frac for fractions in exponents, numerators, and denominators.

\(\dfrac{x+1}{3}+\dfrac{2-3x}{2}\)

Answer 5

You are adding two fractions.
The denominators are 3 and 2.
What do you need to be able to add two fractions?

Answer 6

make the denominators the same

Answer 7

Correct.
What is the LCD of 3 and 2?

Answer 8

6

Answer 9

Correct.
The left fraction has a denominator of 3. We need 6, so we multiply it by 2/2.
The right fraction has a denominator of 2. We need 6, so we multiply it by 3/3.

Answer 10

\(\dfrac{2}{2} \cdot \dfrac{x+1}{3}+\dfrac{3}{3} \cdot \dfrac{2-3x}{2}\)

Do you follow so far?

Answer 11

I understand

Answer 12

Now we multiply the numerators together and denominators together in both fractions.

\(\dfrac{2(x+1)}{2 \times 3}+\dfrac{3(2-3x)}{3 \times2}\)

We use the distributive property in the numerators to get:

\(\dfrac{2x+2}{6}+\dfrac{6-9x}{6}\)

Now we have a common denominator, so we just add the fractions by adding the numerators and setting that over the common denominator.

Answer 13

\(\dfrac{2x+2+6-9x }{6}\)

Now we need to combine like terms.

Answer 14

\(\dfrac{-7x+8 }{6}\)