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Question Solve the equation x=□−2(4x+1)=2x−2(2x+6)x = \square - 2(4x + 1) = 2x - 2(2x + 6) and verify the solution.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation x=□−2(4x+1)=2x−2(2x+6) x = \square - 2(4x + 1) = 2x - 2(2x + 6) .
2. The symbol □ \square seems to be a placeholder for an unknown term or a mistake in the equation. Since it is not a standard mathematical symbol and no value is given for it, we will ignore it and focus on the equation −2(4x+1)=2x−2(2x+6) -2(4x + 1) = 2x - 2(2x + 6) .
3. We will solve for x x by simplifying and rearranging the equation.
4. We will check our solution by substituting the value of x x back into the original equation.

STEP 2

First, let's simplify both sides of the equation by distributing the multiplication over addition.
For the left side of the equation: −2(4x+1)=−8x−2 -2(4x + 1) = -8x - 2
For the right side of the equation: 2x−2(2x+6)=2x−4x−12 2x - 2(2x + 6) = 2x - 4x - 12

STEP 3

Now, rewrite the equation with the simplified terms: −8x−2=2x−4x−12 -8x - 2 = 2x - 4x - 12

STEP 4

Combine like terms on the right side of the equation: −8x−2=−2x−12 -8x - 2 = -2x - 12

STEP 5

Next, we want to get all the x x -terms on one side and the constants on the other side. Let's add 8x 8x to both sides to move the x x -terms to the right side: −8x+8x−2=−2x+8x−12 -8x + 8x - 2 = -2x + 8x - 12

STEP 6

Simplify the equation by combining like terms: −2=6x−12 -2 = 6x - 12

STEP 7

Now, add 12 12 to both sides to isolate the x x -term on the right side: −2+12=6x−12+12 -2 + 12 = 6x - 12 + 12

STEP 8

Simplify the equation by combining the constants: 10=6x 10 = 6x

STEP 9

To solve for x x , divide both sides by 6 6 : 106=6x6 \frac{10}{6} = \frac{6x}{6}

STEP 10

Simplify the fraction on the left side and the equation: 53=x \frac{5}{3} = x

STEP 11

The solution to the equation is: x=53 x = \frac{5}{3}

STEP 12

To check the solution, substitute x=53 x = \frac{5}{3} back into the original equation (ignoring the □ \square symbol): 53=−2(4⋅53+1)=2⋅53−2(2⋅53+6) \frac{5}{3} = -2(4 \cdot \frac{5}{3} + 1) = 2 \cdot \frac{5}{3} - 2(2 \cdot \frac{5}{3} + 6)

STEP 13

First, simplify the left side of the check equation: −2(4⋅53+1)=−2(203+1)=−2(203+33)=−2⋅233=−463 -2(4 \cdot \frac{5}{3} + 1) = -2(\frac{20}{3} + 1) = -2(\frac{20}{3} + \frac{3}{3}) = -2 \cdot \frac{23}{3} = -\frac{46}{3}

STEP 14

Now, simplify the right side of the check equation: 2⋅53−2(2⋅53+6)=103−2(103+183)=103−2⋅283=103−563=−463 2 \cdot \frac{5}{3} - 2(2 \cdot \frac{5}{3} + 6) = \frac{10}{3} - 2(\frac{10}{3} + \frac{18}{3}) = \frac{10}{3} - 2 \cdot \frac{28}{3} = \frac{10}{3} - \frac{56}{3} = -\frac{46}{3}

STEP 15

Since the left side and the right side of the check equation are equal, our solution x=53 x = \frac{5}{3} is correct.
The solution to the equation is x=53 x = \frac{5}{3} .

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