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

The problem presented requires solving for the variable $x$ in a linear equation. The equation contains terms with the variable on both sides, as well as constants. The equation will need to be simplified by combining like terms and isolating the variable on one side to find the value of $x$ that satisfies the equation.

$- 2 x - 5 = 2 - 4 x - (x - 1)$

Answer

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Solution:

Step 1:

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

Step 2:

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

Step 2.1:

Break down the expression term by term.

Step 2.1.1:

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

Step 2.1.2:

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

Step 2.2:

Combine like terms.

Step 2.2.1:

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

Step 2.2.2:

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

Step 3:

Isolate terms with $x$ on one side.

Step 3.1:

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

Step 3.2:

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

Step 4:

Shift terms without $x$ to the opposite side.

Step 4.1:

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

Step 4.2:

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

Step 5:

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

Step 5.1:

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

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. $\frac{- 3 x}{- 3} = \frac{- 8}{- 3}$

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

Knowledge Notes:

Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. For example, $a (b + c) = a b + a c$ .
Combining Like Terms: When simplifying expressions, we combine terms that have the same variable raised to the same power. For instance, $- 4 x - x$ becomes $- 5 x$ .
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.
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.
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.