Solve the Inequality for z |2z-9|<=1

$| 2 z - 9 | \leq 1$

Step 1

Write $| 2 z - 9 | \leq 1$ as a piecewise.

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

$2 z - 9 \geq 0$

Solve the inequality.

Add $9$ to both sides of the inequality.

$2 z \geq 9$

Divide each term in $2 z \geq 9$ by $2$ and simplify.

Divide each term in $2 z \geq 9$ by $2$ .

$\frac{2 z}{2} \geq \frac{9}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} \geq \frac{9}{2}$

Divide $z$ by $1$ .

$z \geq \frac{9}{2}$

In the piece where $2 z - 9$ is non-negative, remove the absolute value.

$2 z - 9 \leq 1$

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

$2 z - 9 < 0$

Solve the inequality.

Add $9$ to both sides of the inequality.

$2 z < 9$

Divide each term in $2 z < 9$ by $2$ and simplify.

Divide each term in $2 z < 9$ by $2$ .

$\frac{2 z}{2} < \frac{9}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} < \frac{9}{2}$

Divide $z$ by $1$ .

$z < \frac{9}{2}$

In the piece where $2 z - 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 z - 9) \leq 1$

Write as a piecewise.

${\begin{cases} 2 z - 9 \leq 1 & z \geq \frac{9}{2} \\ - (2 z - 9) \leq 1 & z < \frac{9}{2} \end{cases}$

Simplify $- (2 z - 9) \leq 1$ .

Apply the distributive property.

${\begin{cases} 2 z - 9 \leq 1 & z \geq \frac{9}{2} \\ - (2 z) - - 9 \leq 1 & z < \frac{9}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 z - 9 \leq 1 & z \geq \frac{9}{2} \\ - 2 z - - 9 \leq 1 & z < \frac{9}{2} \end{cases}$

Multiply $- 1$ by $- 9$ .

${\begin{cases} 2 z - 9 \leq 1 & z \geq \frac{9}{2} \\ - 2 z + 9 \leq 1 & z < \frac{9}{2} \end{cases}$

Step 2

Solve $2 z - 9 \leq 1$ when $z \geq \frac{9}{2}$ .

Solve $2 z - 9 \leq 1$ for $z$ .

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

Add $9$ to both sides of the inequality.

$2 z \leq 1 + 9$

Add $1$ and $9$ .

$2 z \leq 10$

Divide each term in $2 z \leq 10$ by $2$ and simplify.

Divide each term in $2 z \leq 10$ by $2$ .

$\frac{2 z}{2} \leq \frac{10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} \leq \frac{10}{2}$

Divide $z$ by $1$ .

$z \leq \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

$z \leq 5$

Find the intersection of $z \leq 5$ and $z \geq \frac{9}{2}$ .

$\frac{9}{2} \leq z \leq 5$

Step 3

Solve $- 2 z + 9 \leq 1$ when $z < \frac{9}{2}$ .

Solve $- 2 z + 9 \leq 1$ for $z$ .

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

Subtract $9$ from both sides of the inequality.

$- 2 z \leq 1 - 9$

Subtract $9$ from $1$ .

$- 2 z \leq - 8$

Divide each term in $- 2 z \leq - 8$ by $- 2$ and simplify.

Divide each term in $- 2 z \leq - 8$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 z}{- 2} \geq \frac{- 8}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 z}{- 2} \geq \frac{- 8}{- 2}$

Divide $z$ by $1$ .

$z \geq \frac{- 8}{- 2}$

Simplify the right side.

Divide $- 8$ by $- 2$ .

$z \geq 4$

Find the intersection of $z \geq 4$ and $z < \frac{9}{2}$ .

$4 \leq z < \frac{9}{2}$

Step 4

Find the union of the solutions.

$4 \leq z \leq 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$4 \leq z \leq 5$

Interval Notation:

$[4, 5]$

Step 6

Solve the Inequality for z |2z-9|<=1

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