Solve the Absolute Value Inequality for x |2x-1|<=5

$| 2 x - 1 | \leq 5$

Step 1

Write $| 2 x - 1 | \leq 5$ 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.

$2 x - 1 \leq 5$

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 $- 1$ .

$- (2 x - 1) \leq 5$

Write as a piecewise.

${\begin{cases} 2 x - 1 \leq 5 & x \geq \frac{1}{2} \\ - (2 x - 1) \leq 5 & x < \frac{1}{2} \end{cases}$

Simplify $- (2 x - 1) \leq 5$ .

Apply the distributive property.

${\begin{cases} 2 x - 1 \leq 5 & x \geq \frac{1}{2} \\ - (2 x) - - 1 \leq 5 & x < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 1 \leq 5 & x \geq \frac{1}{2} \\ - 2 x - - 1 \leq 5 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 2 x - 1 \leq 5 & x \geq \frac{1}{2} \\ - 2 x + 1 \leq 5 & x < \frac{1}{2} \end{cases}$

Step 2

Solve $2 x - 1 \leq 5$ when $x \geq \frac{1}{2}$ .

Solve $2 x - 1 \leq 5$ for $x$ .

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

Add $1$ to both sides of the inequality.

$2 x \leq 5 + 1$

Add $5$ and $1$ .

$2 x \leq 6$

Divide each term in $2 x \leq 6$ by $2$ and simplify.

Divide each term in $2 x \leq 6$ by $2$ .

$\frac{2 x}{2} \leq \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{6}{2}$

Divide $x$ by $1$ .

$x \leq \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x \leq 3$

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

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

Step 3

Solve $- 2 x + 1 \leq 5$ when $x < \frac{1}{2}$ .

Solve $- 2 x + 1 \leq 5$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$- 2 x \leq 5 - 1$

Subtract $1$ from $5$ .

$- 2 x \leq 4$

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

Divide each term in $- 2 x \leq 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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

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

Step 4

Find the union of the solutions.

$- 2 \leq x \leq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x \leq 3$

Interval Notation:

$[- 2, 3]$

Step 6

Solve the Absolute Value Inequality for x |2x-1|<=5

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