Solved on Feb 14, 2024

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

STEP 1

Assumptions
1. We are given the equation $x = \square - 2(4x + 1) = 2x - 2(2x + 6)$ .
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 $-2(4x + 1) = 2x - 2(2x + 6)$ .
3. We will solve for $x$ by simplifying and rearranging the equation.
4. We will check our solution by substituting the value of $x$ 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: $-2(4x + 1) = -8x - 2$
For the right side of the equation: $2x - 2(2x + 6) = 2x - 4x - 12$

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

First, simplify the left side of the check equation: $-2(4 \cdot \frac{5}{3} + 1) = -2(\frac{20}{3} + 1) = -2(\frac{20}{3} + \frac{3}{3}) = -2 \cdot \frac{23}{3} = -\frac{46}{3}$

STEP 14

Now, simplify the right side of the check equation: $2 \cdot \frac{5}{3} - 2(2 \cdot \frac{5}{3} + 6) = \frac{10}{3} - 2(\frac{10}{3} + \frac{18}{3}) = \frac{10}{3} - 2 \cdot \frac{28}{3} = \frac{10}{3} - \frac{56}{3} = -\frac{46}{3}$

STEP 15

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

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

STEP 1

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

First, simplify the left side of the check equation: $-2(4 \cdot \frac{5}{3} + 1) = -2(\frac{20}{3} + 1) = -2(\frac{20}{3} + \frac{3}{3}) = -2 \cdot \frac{23}{3} = -\frac{46}{3}$

STEP 14

Now, simplify the right side of the check equation: $2 \cdot \frac{5}{3} - 2(2 \cdot \frac{5}{3} + 6) = \frac{10}{3} - 2(\frac{10}{3} + \frac{18}{3}) = \frac{10}{3} - 2 \cdot \frac{28}{3} = \frac{10}{3} - \frac{56}{3} = -\frac{46}{3}$

STEP 15

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

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

STEP 1

STEP 2

STEP 3

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

STEP 5

Next, we want to get all the x x x-terms on one side and the constants on the other side. Let's add 8x 8x 8x to both sides to move the x x x-terms to the right side: −8x+8x−2=−2x+8x−12 -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 −2=6x−12

STEP 7

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

STEP 8

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

STEP 9

To solve for x x x, divide both sides by 6 6 6: 106=6x6 \frac{10}{6} = \frac{6x}{6} 610​=66x​

STEP 10

Simplify the fraction on the left side and the equation: 53=x \frac{5}{3} = x 35​=x

STEP 11

The solution to the equation is: x=53 x = \frac{5}{3} x=35​

STEP 12

STEP 13

STEP 14

STEP 15

Since the left side and the right side of the check equation are equal, our solution x=53 x = \frac{5}{3} x=35​ is correct.The solution to the equation is x=53 x = \frac{5}{3} x=35​.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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

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

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

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

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

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

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

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

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

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