Solve the Inequality for x |3-2x|>=1

$| 3 - 2 x | \geq 1$

Write $| 3 - 2 x | \geq 1$ as a piecewise.

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

$3 - 2 x \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 2 x \geq - 3$

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.

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

In the piece where $3 - 2 x$ is non-negative, remove the absolute value.

$3 - 2 x \geq 1$

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

$3 - 2 x < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 2 x < - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

In the piece where $3 - 2 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 - 2 x) \geq 1$

Write as a piecewise.

${\begin{cases} 3 - 2 x \geq 1 & x \leq \frac{3}{2} \\ - (3 - 2 x) \geq 1 & x > \frac{3}{2} \end{cases}$

Simplify $- (3 - 2 x) \geq 1$ .

Apply the distributive property.

${\begin{cases} 3 - 2 x \geq 1 & x \leq \frac{3}{2} \\ - 1 \cdot 3 - (- 2 x) \geq 1 & x > \frac{3}{2} \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 - 2 x \geq 1 & x \leq \frac{3}{2} \\ - 3 - (- 2 x) \geq 1 & x > \frac{3}{2} \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} 3 - 2 x \geq 1 & x \leq \frac{3}{2} \\ - 3 + 2 x \geq 1 & x > \frac{3}{2} \end{cases}$

Solve $3 - 2 x \geq 1$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

$- 2 x \geq 1 - 3$

Subtract $3$ from $1$ .

$- 2 x \geq - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x \leq 1$

Solve $- 3 + 2 x \geq 1$ for $x$ .

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

Add $3$ to both sides of the inequality.

$2 x \geq 1 + 3$

Add $1$ and $3$ .

$2 x \geq 4$

Divide each term in $2 x \geq 4$ by $2$ and simplify.

Divide each term in $2 x \geq 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{4}{2}$

Divide $x$ by $1$ .

$x \geq \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x \geq 2$

Find the union of the solutions.

$x \leq 1$ or $x \geq 2$

The result can be shown in multiple forms.

Inequality Form:

$x \leq 1 or x \geq 2$

Interval Notation:

$(- \infty, 1] \cup [2, \infty)$

Solve the Inequality for x |3-2x|>=1

Related Questions

Support LCMGCF.com by disabling your adblocker.