Solve the Inequality for x 8-|2x-1|>=6

$8 - | 2 x - 1 | \geq 6$

Write $8 - | 2 x - 1 | \geq 6$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$2 x - 1 \geq 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Divide each term in $2 x \geq 1$ by $2$ and simplify.

Divide each term in $2 x \geq 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{1}{2}$

Divide $x$ by $1$ .

$x \geq \frac{1}{2}$

In the piece where $2 x - 1$ is non-negative, remove the absolute value.

$8 - (2 x - 1) \geq 6$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$2 x - 1 < 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 x < 1$

Divide each term in $2 x < 1$ by $2$ and simplify.

Divide each term in $2 x < 1$ by $2$ .

$\frac{2 x}{2} < \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{1}{2}$

Divide $x$ by $1$ .

$x < \frac{1}{2}$

In the piece where $2 x - 1$ is negative, remove the absolute value and multiply by $- 1$ .

$8 - - (2 x - 1) \geq 6$

Write as a piecewise.

${\begin{cases} 8 - (2 x - 1) \geq 6 & x \geq \frac{1}{2} \\ 8 - - (2 x - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Simplify $8 - (2 x - 1) \geq 6$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 8 - (2 x) - - 1 \geq 6 & x \geq \frac{1}{2} \\ 8 - - (2 x - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 8 - 2 x - - 1 \geq 6 & x \geq \frac{1}{2} \\ 8 - - (2 x - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 8 - 2 x + 1 \geq 6 & x \geq \frac{1}{2} \\ 8 - - (2 x - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Add $8$ and $1$ .

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 - - (2 x - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Simplify $8 - - (2 x - 1) \geq 6$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 - (- (2 x) - - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 - (- 2 x - - 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 - (- 2 x + 1) \geq 6 & x < \frac{1}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 - (- 2 x) - 1 \cdot 1 \geq 6 & x < \frac{1}{2} \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 + 2 x - 1 \cdot 1 \geq 6 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 8 + 2 x - 1 \geq 6 & x < \frac{1}{2} \end{cases}$

Subtract $1$ from $8$ .

${\begin{cases} - 2 x + 9 \geq 6 & x \geq \frac{1}{2} \\ 2 x + 7 \geq 6 & x < \frac{1}{2} \end{cases}$

Solve $- 2 x + 9 \geq 6$ when $x \geq \frac{1}{2}$ .

Solve $- 2 x + 9 \geq 6$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $9$ from both sides of the inequality.

$- 2 x \geq 6 - 9$

Subtract $9$ from $6$ .

$- 2 x \geq - 3$

Divide each term in $- 2 x \geq - 3$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 3}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{3}{2}$

Find the intersection of $x \leq \frac{3}{2}$ and $x \geq \frac{1}{2}$ .

$\frac{1}{2} \leq x \leq \frac{3}{2}$

Solve $2 x + 7 \geq 6$ when $x < \frac{1}{2}$ .

Solve $2 x + 7 \geq 6$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $7$ from both sides of the inequality.

$2 x \geq 6 - 7$

Subtract $7$ from $6$ .

$2 x \geq - 1$

Divide each term in $2 x \geq - 1$ by $2$ and simplify.

Divide each term in $2 x \geq - 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 1}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{1}{2}$

Find the intersection of $x \geq - \frac{1}{2}$ and $x < \frac{1}{2}$ .

$- \frac{1}{2} \leq x < \frac{1}{2}$

Find the union of the solutions.

$- \frac{1}{2} \leq x \leq \frac{3}{2}$

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{2} \leq x \leq \frac{3}{2}$

Interval Notation:

$[- \frac{1}{2}, \frac{3}{2}]$

Solve the Inequality for x 8-|2x-1|>=6

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