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7. 3x + 6 = 3x + 10

8. 3x + 6 = x + 2(x + 3)

Step-by-step explanation:

7.

To write an equation with no solutions, we write an equation in which the x terms are eliminated, and we end up with a false statement when we try to solve the equation.

Start with

3x + 6 =

This means: take a number, x, multiply it by 3, then add 6 to it.

To make it into an impossible equation, take the same number x, multiply it by 3, and now add 10 instead of 6. The same number x, multiplied by 3 just like above, and now with 10 added to it cannot equal 3x + 6.

Let's write the equation and try to solve it.

3x + 6 = 3x + 10

Subtract 3x from both sides.

6 = 10

Since 6 = 10 is a false statement, there is no solution.

8.

Now we need an equation that is true for every value of x.

We start with 3x + 6 on the left side.

3x + 6 =

To make it true for every value of x, which means it has an infinite number of solutions, we write an expression on the right side that is the same as 3x + 6, just written in a different form.

For example, start with

3x + 6

Separate 3x into x + 2x, now you have

x + 2x + 6

Now factor 2 out of 2x + 6, so you get

x + 2(x + 3)

The expression x + 2(x + 3) is equal to the expression 3x + 6. Now we use x + 2(x + 3) on the right side of the equation we are writing. We get:

3x + 6 = x + 2(x + 3)

Let's solve this equation.

Distribute on the right side.

3x + 6 = x + 2x + 3

Combine like terms on the right side.

3x + 6 = 3x + 6

Subtract 6 from both sides.

3x = 3x

Subtract 3x from both sides.

0 = 0

0 = 0 is a true equation. Therefore, the equation 3x + 6 = x + 2(x + 3) has infinitely many solutions. in other words, every real number is a solution of the equation.

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