Solve the equation. 9 2 + 1 4 ( x + 2 ) = 3 4 x \frac92+\frac14\left(x+2\right)=\frac34x 2 9 + 4 1 ( x + 2 ) = 4 3 x

Hey, everyone. Let's take a look at this practice problem. So we're gonna solve this linear equation, 9 halves plus 1 fourth times x plus 2 is equal to 3 fourths times x. Now looking at this equation, I definitely have some fractions in it. So that means that I wanna start with step 0, which is to multiply by my least common denominator to eliminate those fractions. Now I have my I have 3 different denominators going on here. I have 2, 4, and then another 4. So the least common denominator of those 3 is going to be 4. So I'm gonna multiply my entire equation by 4. Remember, we need to distribute this least common denominator to every single term in our equation to make sure we get rid of all of those fractions. So let's go ahead and expand this out a little bit more. I'm gonna have 4 times 9, which is 36 divided by 2 plus 4 over 4 times x plus 2 is equal to 3 times 4 is going to give me 12 divided by 4 times x. Now I can go ahead and simplify this a little bit more. So 36 divided by 2 is going to give me 18 plus 4 over 4 is just 1. So that x plus 2 is just by itself now and that's equal to 12 divided by 4, which is 3 times x. So we have completed step 0. We have gotten rid of our fractions, and now we can move on solving as normal. So step number 1 is to distribute our constants. Now we had a constant of 1 in front of that x plus 2. We have, like, an invisible one right there. So we can go ahead and take that out of our parentheses, and this just becomes x plus 2. I don't have any other constants that need to get distributed, so step 1 is done. Now step 2 is to combine like terms, and I can do that here. So on the left side of my equation, I have this 18 and this 2 that I can go ahead and combine and that will combine to 20. So I'm left with 20 plus x and then there are no like terms to combine on that right side, so I still have 3 x. And step 2 is done. Now step 3 is to group my terms with x on one side with constants on the other. And to do that, all I need to do is move this x over to my right side. So to do that, I need to go ahead and subtract x from both sides. It will cancel out, and I will be left with 20 equals 3x minus x will give me 2x. And step 3 is done. Step 4 is to go ahead and isolate our x value. And here, I can do that by dividing both sides by 2. So my 2 will cancel on that right side. I'm left with 20 divided by 2, which is 10, is equal to x. Now we can flip that around if you like seeing your answer the other way, and our solution is x equals 10. Now whenever we have fractions, it can be really difficult to plug it back in and just be a little time consuming to do that step number 5. So since it's optional, I'm gonna leave it out here, but remember, you can always check your answer that way. That's all for this one. Let me know if you have questions.

1. Equations and Inequalities

Linear Equations

Precalculus

1. Equations and inequalities

Linear Equations

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Multiple Choice

Solve the equation.
$\frac92+\frac14\left(x+2\right)=\frac34x$

$x=10$

$x=4$

$x=8$

$x=-1$

Verified step by step guidance

Start by simplifying the equation: $ \frac{9}{2} + \frac{1}{4}(x+2) = \frac{3}{4}x $.

Distribute $ \frac{1}{4} $ into $ (x+2) $ to get $ \frac{1}{4}x + \frac{1}{2} $.

Rewrite the equation as $ \frac{9}{2} + \frac{1}{4}x + \frac{1}{2} = \frac{3}{4}x $.

Combine like terms on the left side: $ \frac{9}{2} + \frac{1}{2} = 5 $, so the equation becomes $ 5 + \frac{1}{4}x = \frac{3}{4}x $.

Subtract $ \frac{1}{4}x $ from both sides to isolate $ x $ terms on one side: $ 5 = \frac{3}{4}x - \frac{1}{4}x $, which simplifies to $ 5 = \frac{1}{2}x $.