Solve for x -2x-5=2-4x-(x-1)

Solution:

Step 1:

Reposition the equation to have $x$ on the left side by swapping both sides. $2 - 4x - (x - 1) = -2x - 5$

Step 2:

Begin simplifying $2 - 4x - (x - 1)$.

Step 2.1:

Break down the expression term by term.

Step 2.1.1:

Utilize the distributive property to expand. $2 - 4x - x + 1 = -2x - 5$

Step 2.1.2:

Multiply $-1$ by $1$ to get $2 - 4x - x + 1 = -2x - 5$

Step 2.2:

Combine like terms.

Step 2.2.1:

Combine $2$ and $1$ to get $-4x - x + 3 = -2x - 5$

Step 2.2.2:

Combine $-4x$ and $-x$ to get $-5x + 3 = -2x - 5$

Step 3:

Isolate terms with $x$ on one side.

Step 3.1:

Add $2x$ to both sides. $-5x + 3 + 2x = -5$

Step 3.2:

Combine $-5x$ and $2x$ to get $-3x + 3 = -5$

Step 4:

Shift terms without $x$ to the opposite side.

Step 4.1:

Subtract $3$ from both sides. $-3x = -5 - 3$

Step 4.2:

Combine $-5$ and $-3$ to get $-3x = -8$

Step 5:

Divide each term in the equation $-3x = -8$ by $-3$.

Step 5.1:

Divide $-3x$ and $-8$ by $-3$. $\frac{-3x}{-3} = \frac{-8}{-3}$

Step 5.2:

Simplify the left side.

Step 5.2.1:

Eliminate the common factor of $-3$.

Step 5.2.1.1:

Cancel out the common factor. $\frac{\cancel{-3} x}{\cancel{-3}} = \frac{-8}{-3}$

Step 5.2.1.2:

Divide $x$ by $1$ to get $x = \frac{-8}{-3}$

Step 5.3:

Simplify the right side.

Step 5.3.1:

Dividing two negatives yields a positive. $x = \frac{8}{3}$

Step 6:

Present the solution in various forms.

Exact Form: $x = \frac{8}{3}$

Decimal Form: $x \approx 2.67$

Mixed Number Form: $x = 2 \frac{2}{3}$

Solution:"The solution to the equation $-2x - 5 = 2 - 4x - (x - 1)$ is $x = \frac{8}{3}$, which can also be expressed as $x \approx 2.67$ or $x = 2 \frac{2}{3}$."

Knowledge Notes:

1. Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. For example, $a(b + c) = ab + ac$.

2. Combining Like Terms: When simplifying expressions, we combine terms that have the same variable raised to the same power. For instance, $-4x - x$ becomes $-5x$.

3. Isolating the Variable: To solve for $x$, we move all terms with $x$ to one side and constants to the other, maintaining balance by performing the same operations on both sides.

4. Dividing by a Negative: When we divide both sides of an equation by a negative number, the inequality direction remains the same, and the signs of the terms change.

5. Equivalent Forms of a Number: A number can be expressed in different forms, such as an exact fraction, a decimal approximation, or a mixed number.