Solve the equation. 2 x + 4 x − 1 = 5 \frac{2x+4}{x-1}=5

Hey, everyone. Let's take a look at this practice problem. So here I have 2x+4 over x minus 1 is equal to 5, and I wanna solve this rational equation. So my very first step is going to be to determine my restriction by setting my denominator equal to 0. So the denominator that I have with a variable is x minus 1. So if I take this x minus 1 and set it equal to 0 and add 1 to both sides, I'm left with x equals 1. This tells me that if I plug 1 back in for my denominator, then I'm gonna get a 0, which is what I don't want. So I know that x cannot be equal to 1. Now if you just wanna eyeball this, denominator and see that, oh, if I plugged 1 in here, that would make my denominator 0, that's totally fine too, but you can always set your denominator equal to 0 to find this as well. So we are done with step 1. Step 2 is gonna be to multiply by our least common denominator in order to eliminate fractions. So I'm gonna rewrite my equation over here. I have 2x+xminusoneequals5. So I have this denominator, x minus 1, and I don't have anything else. So that means that this denominator is my least common denominator. So I can go ahead and take my whole equation and multiply it by x minus 1. Now remember when we're multiplying by our LCD, we want to make sure to distribute it to every term in our equation. So here I'm going to get x minus 1 times my first term, 2x+4 over x minus 1 is equal to x minus 1 at times 5. Now simplifying this a little bit, I'm gonna see that this x minus 1 just cancels, leaving me with just 2x+4, and that's equal to I'm just gonna pull this 5 over to the front to make it easier, 5 times x minus 1. So we have completed step 2, and we can go ahead and move on to step 3 because I now just have a linear equation that I can solve. So I'm gonna go ahead and distribute this 5. The left side of my equation is gonna stay the same, 2x+4, and that's equal to 5x minus 5 with that 5 distributed. Okay. So our next step is gonna be to combine any like terms, which I don't have any like terms here. So I can go ahead and move on to the next, which is to take all of my x terms to one side, all of my constants to the other. So I'm gonna go ahead and move this 2 over here and move my negative 5 to the other side. So in order to do that, I need to subtract 2 x to cancel it. Whatever I do to one side, I have to do to the other. And then I'm gonna add 5 to both sides. So 4 +5 is going to give me a 9, and then I have 5x minus 2x x, which is going to give me positive 3x. Now my very last step is gonna be to isolate x, which I can do by dividing both sides by 3. So that 3 will cancel, and I'm left with my solution of x equals 3 because 9 divided by 3 gives me 3. So this is my solution. I've completed step number 3. And then my very last step is, of course, to check my solution with my restriction. I said that x couldn't be equal to 1 and for my solution I got x equals 3 which is not 1, so I'm good to go and my solution is x equals 3. That's all for this one. I'll see you in the next one.

8. Rational Expressions and Equations

Rational Equations

Beginning Algebra

8. Rational Expressions and Equations

Rational Equations

Struggling with Beginning Algebra?

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Multiple Choice

Solve the equation.
$\frac{2 x + 4}{x - 1} = 5$

$x=3$

$x=1$

$x=5$

No solution

Verified step by step guidance

Start with the given equation: $\frac{2x+4}{x-1} = 5$.

To eliminate the fraction, multiply both sides of the equation by the denominator $(x-1)$ to get rid of the fraction: $(x-1) \times \frac{2x+4}{x-1} = 5 \times (x-1)$.

Simplify both sides: the left side becomes \$2x + 4$, and the right side becomes $5(x - 1)$, so the equation is now $2x + 4 = 5(x - 1)$.

Distribute the 5 on the right side: \$2x + 4 = 5x - 5$.

Next, get all terms involving $x$ on one side and constants on the other side by subtracting \$2x$ from both sides and adding 5 to both sides: $4 + 5 = 5x - 2x$, which simplifies to $9 = 3x$. Then solve for $x$ by dividing both sides by 3.

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Solve the equation. 2x+4x−1=5\frac{2x+4}{x-1}=5

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Solve the equation.
$\frac{2 x + 4}{x - 1} = 5$