Solve for w 1/2*|2w-1|-3=1

$\frac{1}{2} \cdot | 2 w - 1 | - 3 = 1$

Step 1

Combine $\frac{1}{2}$ and $| 2 w - 1 |$ .

$\frac{| 2 w - 1 |}{2} - 3 = 1$

Step 2

Move all terms not containing $| 2 w - 1 |$ to the right side of the equation.

Add $3$ to both sides of the equation.

$\frac{| 2 w - 1 |}{2} = 1 + 3$

Add $1$ and $3$ .

$\frac{| 2 w - 1 |}{2} = 4$

Step 3

Multiply both sides of the equation by $2$ .

$2 \frac{| 2 w - 1 |}{2} = 2 \cdot 4$

Step 4

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{| 2 w - 1 |}{2} = 2 \cdot 4$

Rewrite the expression.

$| 2 w - 1 | = 2 \cdot 4$

Simplify the right side.

Multiply $2$ by $4$ .

$| 2 w - 1 | = 8$

Step 5

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 w - 1 = \pm 8$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$2 w - 1 = 8$

Move all terms not containing $w$ to the right side of the equation.

Add $1$ to both sides of the equation.

$2 w = 8 + 1$

Add $8$ and $1$ .

$2 w = 9$

Divide each term in $2 w = 9$ by $2$ and simplify.

Divide each term in $2 w = 9$ by $2$ .

$\frac{2 w}{2} = \frac{9}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 w}{2} = \frac{9}{2}$

Divide $w$ by $1$ .

$w = \frac{9}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$2 w - 1 = - 8$

Move all terms not containing $w$ to the right side of the equation.

Add $1$ to both sides of the equation.

$2 w = - 8 + 1$

Add $- 8$ and $1$ .

$2 w = - 7$

Divide each term in $2 w = - 7$ by $2$ and simplify.

Divide each term in $2 w = - 7$ by $2$ .

$\frac{2 w}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 w}{2} = \frac{- 7}{2}$

Divide $w$ by $1$ .

$w = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$w = - \frac{7}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$w = \frac{9}{2}, - \frac{7}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$w = \frac{9}{2}, - \frac{7}{2}$

Decimal Form:

$w = 4.5, - 3.5$

Mixed Number Form:

$w = 4 \frac{1}{2}, - 3 \frac{1}{2}$

Solve for w 1/2*|2w-1|-3=1

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