Solve for x -(2x+2)-1=-x-(x+3)

The given problem is a linear equation that requires you to find the value of the variable x. You would need to perform a series of algebraic steps like combining like terms, using the distributive property to remove parentheses, and isolating the variable on one side of the equation in order to solve for x. The solution would provide the value of x that makes the equation true.

$- (2 x + 2) - 1 = - x - (x + 3)$

Answer

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Solution:

Step:1

Reposition the equation so that $x$ appears on the left side. $- x - (x + 3) = - (2 x + 2) - 1$

Step:2

Begin simplifying $- x - (x + 3)$ .

Step:2.1

Break down the expression term by term.

Step:2.1.1

Utilize the distributive property to expand. $- x - x - 1 \cdot 3 = - (2 x + 2) - 1$

Step:2.1.2

Execute the multiplication of $- 1$ and $3$ . $- x - x - 3 = - (2 x + 2) - 1$

Step:2.2

Combine like terms by subtracting $x$ from $- x$ . $- 2 x - 3 = - (2 x + 2) - 1$

Step:3

Now, simplify $- (2 x + 2) - 1$ .

Step:3.1

Dissect each component of the expression.

Step:3.1.1

Apply the distributive property again. $- 2 x - 3 = - (2 x) - 1 \cdot 2 - 1$

Step:3.1.2

Carry out the multiplication of $- 1$ and $2$ . $- 2 x - 3 = - 2 x - 1 \cdot 2 - 1$

Step:3.1.3

Perform the multiplication of $- 1$ by $2$ . $- 2 x - 3 = - 2 x - 2 - 1$

Step:3.2

Subtract $1$ from $- 2$ . $- 2 x - 3 = - 2 x - 3$

Step:4

Isolate all terms with $x$ on one side of the equation.

Step:4.1

Add $2 x$ to both sides to eliminate the $x$ terms. $- 2 x - 3 + 2 x = - 3$

Step:4.2

Cancel out the terms that are additive inverses.

Step:4.2.1

Sum up $- 2 x$ and $2 x$ . $0 - 3 = - 3$

Step:4.2.2

Subtract $3$ from $0$ . $- 3 = - 3$

Step:5

Since $- 3 = - 3$ holds true, the equation is valid for all $x$ values. Hence, the solution is all real numbers.

Step:6

The solution can be expressed in various forms, including interval notation. Interval Notation: $(- \infty, \infty)$

Knowledge Notes:

To solve the given linear equation, we follow a systematic approach:

Rearranging the Equation: We start by moving all terms containing the variable $x$ to one side to make it easier to simplify.
Simplification: We simplify both sides of the equation by expanding brackets and combining like terms. This often involves using the distributive property, which states that $a (b + c) = a b + a c$ .
Combining Like Terms: Terms that contain the same variable to the same power are combined by addition or subtraction.
Isolating the Variable: If the variable terms cancel each other out and we are left with a true statement (like $- 3 = - 3$ ), it means that the original equation is an identity, and the solution is all real numbers.
Interval Notation: When the solution set includes all real numbers, it is represented in interval notation as $(- \infty, \infty)$ , indicating that there are no boundaries to the set of solutions.

This problem demonstrates that not all linear equations have a single unique solution; some may have infinitely many solutions, as in this case, where any real number can satisfy the equation.