Math  /  Algebra

Questionwand check the linear equation. (2) 2(x1)=x+1(x5)-2(x-1)=x+1-(x-5)
What is the solution? A. The solution set is \square

Studdy Solution

STEP 1

What is this asking? We're asked to solve a linear equation, which means we need to find the value of xx that makes the equation true! Watch out! Be super careful with those negative signs when distributing; they can be sneaky little tricksters!

STEP 2

1. Simplify both sides of the equation.
2. Isolate the variable xx.
3. Verify the solution.

STEP 3

Let's **distribute** that 2-2 on the left side.
Remember, distributing means multiplying 2-2 by *both* terms inside the parentheses.
So, 2(x1)-2(x-1) becomes 2x+(2)(1)-2 \cdot x + (-2) \cdot (-1), which simplifies to 2x+2-2x + 2.
We're doing this to get rid of those parentheses and make the equation easier to work with!

STEP 4

Now, let's tidy up the right side.
We have x+1(x5)x + 1 - (x - 5).
Distribute the negative sign in front of the parentheses: x+1x+5x + 1 - x + 5.
Notice how the sign of 5 changes from negative to positive.
Now, combine like terms: xx+1+5x - x + 1 + 5.
This simplifies to 66.
See how much cleaner that looks?

STEP 5

After simplifying both sides, our equation now looks like this: 2x+2=6-2x + 2 = 6.
Much better, right?

STEP 6

To get xx by itself, we need to move that +2+2 to the other side.
We can do this by subtracting 22 from *both* sides of the equation: 2x+22=62-2x + 2 - 2 = 6 - 2.
This simplifies to 2x=4-2x = 4.
Remember, what we do to one side, we *must* do to the other to keep the equation balanced.

STEP 7

Now, we're *so* close!
We have 2x=4-2x = 4.
To isolate xx, we need to divide both sides by 2-2: 2x2=42\frac{-2x}{-2} = \frac{4}{-2}.
This gives us x=2x = -2.
We divided by 2-2 to make the coefficient of xx equal to one, which isolates xx.

STEP 8

Let's double-check our work to make sure 2-2 is the correct solution.
Substitute 2-2 for xx in the original equation: 2(21)=2+1(25)-2(-2 - 1) = -2 + 1 - (-2 - 5).

STEP 9

Simplify the left side: 2(3)=6-2(-3) = 6.
Simplify the right side: 1(7)=1+7=6-1 - (-7) = -1 + 7 = 6.
Both sides equal 66, so our solution is correct!

STEP 10

The solution set is {2}\{-2\}.

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