QuestionWhich expression is equivalent to $-\frac{2}{3}(6 x+15) ?$ $4 x+10$ $4 x-10$ $-4 x-10$ $-4 x+10$

Studdy Solution

STEP 1

What is this asking? We need to simplify the expression $-\frac{2}{3}(6x + 15)$ and find the equivalent expression among the given options. Watch out! Don't forget to distribute the $-\frac{2}{3}$ to both terms inside the parentheses, and be careful with the signs!

STEP 2

1. Distribute the Fraction
2. Simplify the Terms

STEP 3

Alright, let's distribute that $-\frac{2}{3}$ to both the $6x$ and the $15$ inside the parentheses.
Remember, distributing means multiplying the term outside the parentheses by each term inside!
Why? Because the parentheses tell us that the entire expression inside is being multiplied by the fraction outside.

STEP 4

So, first we have $-\frac{2}{3} \cdot 6x$ .
Let's rewrite $6$ as $\frac{6}{1}$ so we can multiply the fractions easily: $-\frac{2}{3} \cdot \frac{6}{1}x$ .
Multiply the numerators together and the denominators together: $-\frac{2 \cdot 6}{3 \cdot 1}x = -\frac{12}{3}x$ .
Now, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3: $-\frac{12}{3}x = -\frac{12 \div 3}{3 \div 3}x = -\frac{4}{1}x = -4x$ .

STEP 5

Next, we distribute $-\frac{2}{3}$ to $15$ .
Again, rewrite $15$ as $\frac{15}{1}$ : $-\frac{2}{3} \cdot \frac{15}{1}$ .
Multiply the numerators and denominators: $-\frac{2 \cdot 15}{3 \cdot 1} = -\frac{30}{3}$ .
Simplify the fraction by dividing both the numerator and denominator by 3: $-\frac{30}{3} = -\frac{30 \div 3}{3 \div 3} = -\frac{10}{1} = -10$ .

STEP 6

Now, let's put it all together!
We found that $-\frac{2}{3} \cdot 6x = -4x$ and $-\frac{2}{3} \cdot 15 = -10$ .
So, our simplified expression is $-4x - 10$ .
Awesome!

STEP 7

The equivalent expression is $-4x - 10$ .

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