Solve for x |2x-1|+4<=5

$| 2 x - 1 | + 4 \leq 5$

Step 1

Write $| 2 x - 1 | + 4 \leq 5$ 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.

$2 x - 1 + 4 \leq 5$

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 $- 1$ .

$- (2 x - 1) + 4 \leq 5$

Write as a piecewise.

${\begin{cases} 2 x - 1 + 4 \leq 5 & x \geq \frac{1}{2} \\ - (2 x - 1) + 4 \leq 5 & x < \frac{1}{2} \end{cases}$

Add $- 1$ and $4$ .

${\begin{cases} 2 x + 3 \leq 5 & x \geq \frac{1}{2} \\ - (2 x - 1) + 4 \leq 5 & x < \frac{1}{2} \end{cases}$

Simplify $- (2 x - 1) + 4 \leq 5$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 x + 3 \leq 5 & x \geq \frac{1}{2} \\ - (2 x) - - 1 + 4 \leq 5 & x < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 3 \leq 5 & x \geq \frac{1}{2} \\ - 2 x - - 1 + 4 \leq 5 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 2 x + 3 \leq 5 & x \geq \frac{1}{2} \\ - 2 x + 1 + 4 \leq 5 & x < \frac{1}{2} \end{cases}$

Add $1$ and $4$ .

${\begin{cases} 2 x + 3 \leq 5 & x \geq \frac{1}{2} \\ - 2 x + 5 \leq 5 & x < \frac{1}{2} \end{cases}$

Step 2

Solve $2 x + 3 \leq 5$ when $x \geq \frac{1}{2}$ .

Solve $2 x + 3 \leq 5$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

$2 x \leq 5 - 3$

Subtract $3$ from $5$ .

$2 x \leq 2$

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

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

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

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 $2$ .

$x \leq 1$

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

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

Step 3

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

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

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

Subtract $5$ from both sides of the inequality.

$- 2 x \leq 5 - 5$

Subtract $5$ from $5$ .

$- 2 x \leq 0$

Divide each term in $- 2 x \leq 0$ by $- 2$ and simplify.

Divide each term in $- 2 x \leq 0$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{0}{- 2}$

Divide $x$ by $1$ .

$x \geq \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$x \geq 0$

Find the intersection of $x \geq 0$ and $x < \frac{1}{2}$ .

$0 \leq x < \frac{1}{2}$

Step 4

Find the union of the solutions.

$0 \leq x \leq 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$0 \leq x \leq 1$

Interval Notation:

$[0, 1]$

Step 6

Solve for x |2x-1|+4<=5

Related Questions

Support LCMGCF.com by disabling your adblocker.