Math  /  Algebra

Question(2) 3x232x+14=\frac{3 x-2}{3}-\frac{2 x+1}{4}=

Studdy Solution

STEP 1

What is this asking? We're asked to simplify an expression with two fractions involving xx. Watch out! Don't forget the order of operations and be careful with distributing the negative sign!

STEP 2

1. Find a Common Denominator
2. Combine the Fractions
3. Simplify

STEP 3

Alright, let's **get this party started**!
We've got two fractions with different denominators: **3** and **4**.
To combine them, we need a common denominator.
The **least common multiple** of 3 and 4 is **12**.
So, we'll rewrite both fractions with a denominator of 12.

STEP 4

For the first fraction, we multiply the top and bottom by **4**: 3x2344=4(3x2)12=12x812 \frac{3x - 2}{3} \cdot \frac{4}{4} = \frac{4 \cdot (3x - 2)}{12} = \frac{12x - 8}{12} Remember, multiplying by 44\frac{4}{4} is the same as multiplying by **1**, so we're not changing the value of the fraction, just its appearance!

STEP 5

For the second fraction, we multiply the top and bottom by **3**: 2x+1433=3(2x+1)12=6x+312 \frac{2x + 1}{4} \cdot \frac{3}{3} = \frac{3 \cdot (2x + 1)}{12} = \frac{6x + 3}{12} Again, multiplying by 33\frac{3}{3} is just like multiplying by **1**!

STEP 6

Now that both fractions have the same denominator, we can **combine them**: 12x8126x+312 \frac{12x - 8}{12} - \frac{6x + 3}{12} Remember, we're subtracting the second fraction from the first!

STEP 7

Let's carefully distribute the negative sign to the terms in the second fraction: 12x812+6x312 \frac{12x - 8}{12} + \frac{-6x - 3}{12} This is like adding the opposite!

STEP 8

Now, we combine the like terms in the numerators: (12x6x)+(83)12=6x1112 \frac{(12x - 6x) + (-8 - 3)}{12} = \frac{6x - 11}{12} Boom! We've combined the fractions!

STEP 9

Let's check if we can simplify the fraction 6x1112\frac{6x - 11}{12} further.
Since there are no common factors between **6**, **11**, and **12**, this fraction is already in its **simplest form**!

STEP 10

Our **final, simplified expression** is: 6x1112 \frac{6x - 11}{12}

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