Convert to Set Notation |4x-9|>=6

$| 4 x - 9 | \geq 6$

Step 1

Solve $x \leq \frac{3}{4} or x \geq \frac{15}{4}$ .

Write $| 4 x - 9 | \geq 6$ as a piecewise.

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

$4 x - 9 \geq 0$

Solve the inequality.

Add $9$ to both sides of the inequality.

$4 x \geq 9$

Divide each term in $4 x \geq 9$ by $4$ and simplify.

Divide each term in $4 x \geq 9$ by $4$ .

$\frac{4 x}{4} \geq \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{9}{4}$

Divide $x$ by $1$ .

$x \geq \frac{9}{4}$

In the piece where $4 x - 9$ is non-negative, remove the absolute value.

$4 x - 9 \geq 6$

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

$4 x - 9 < 0$

Solve the inequality.

Add $9$ to both sides of the inequality.

$4 x < 9$

Divide each term in $4 x < 9$ by $4$ and simplify.

Divide each term in $4 x < 9$ by $4$ .

$\frac{4 x}{4} < \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{9}{4}$

Divide $x$ by $1$ .

$x < \frac{9}{4}$

In the piece where $4 x - 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x - 9) \geq 6$

Write as a piecewise.

${\begin{cases} 4 x - 9 \geq 6 & x \geq \frac{9}{4} \\ - (4 x - 9) \geq 6 & x < \frac{9}{4} \end{cases}$

Simplify $- (4 x - 9) \geq 6$ .

Apply the distributive property.

${\begin{cases} 4 x - 9 \geq 6 & x \geq \frac{9}{4} \\ - (4 x) - - 9 \geq 6 & x < \frac{9}{4} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 9 \geq 6 & x \geq \frac{9}{4} \\ - 4 x - - 9 \geq 6 & x < \frac{9}{4} \end{cases}$

Multiply $- 1$ by $- 9$ .

${\begin{cases} 4 x - 9 \geq 6 & x \geq \frac{9}{4} \\ - 4 x + 9 \geq 6 & x < \frac{9}{4} \end{cases}$

Solve $4 x - 9 \geq 6$ for $x$ .

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

Add $9$ to both sides of the inequality.

$4 x \geq 6 + 9$

Add $6$ and $9$ .

$4 x \geq 15$

Divide each term in $4 x \geq 15$ by $4$ and simplify.

Divide each term in $4 x \geq 15$ by $4$ .

$\frac{4 x}{4} \geq \frac{15}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{15}{4}$

Divide $x$ by $1$ .

$x \geq \frac{15}{4}$

Solve $- 4 x + 9 \geq 6$ for $x$ .

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

Subtract $9$ from both sides of the inequality.

$- 4 x \geq 6 - 9$

Subtract $9$ from $6$ .

$- 4 x \geq - 3$

Divide each term in $- 4 x \geq - 3$ by $- 4$ and simplify.

Divide each term in $- 4 x \geq - 3$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 3}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 3}{- 4}$

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{3}{4}$

Find the union of the solutions.

$x \leq \frac{3}{4}$ or $x \geq \frac{15}{4}$

Step 2

Use the inequality $x \leq \frac{3}{4} or x \geq \frac{15}{4}$ to build the set notation.

${x | x \leq \frac{3}{4} or x \geq \frac{15}{4}}$

Step 3

Convert to Set Notation |4x-9|>=6

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