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

Solution:

Step:1

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

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. $-x - x - 1 \cdot 3 = -(2x + 2) - 1$

Step:2.1.2

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

Step:2.2

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

Step:3

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

Step:3.1

Dissect each component of the expression.

Step:3.1.1

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

Step:3.1.2

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

Step:3.1.3

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

Step:3.2

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

Step:4

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

Step:4.1

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

Step:4.2

Cancel out the terms that are additive inverses.

Step:4.2.1

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

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:

1. Rearranging the Equation: We start by moving all terms containing the variable $x$ to one side to make it easier to simplify.

2. 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 $a(b + c) = ab + ac$.

3. Combining Like Terms: Terms that contain the same variable to the same power are combined by addition or subtraction.

4. 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.

5. 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.