Solve the Inequality for x |2x+1|+7>2

$| 2 x + 1 | + 7 > 2$

Write $| 2 x + 1 | + 7 > 2$ 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.

Subtract $1$ from 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 $1$ .

$x \geq \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{1}{2}$

In the piece where $2 x + 1$ is non-negative, remove the absolute value.

$2 x + 1 + 7 > 2$

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.

Subtract $1$ from 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 $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

$- (2 x + 1) + 7 > 2$

Write as a piecewise.

${\begin{cases} 2 x + 1 + 7 > 2 & x \geq - \frac{1}{2} \\ - (2 x + 1) + 7 > 2 & x < - \frac{1}{2} \end{cases}$

Add $1$ and $7$ .

${\begin{cases} 2 x + 8 > 2 & x \geq - \frac{1}{2} \\ - (2 x + 1) + 7 > 2 & x < - \frac{1}{2} \end{cases}$

Simplify $- (2 x + 1) + 7 > 2$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 x + 8 > 2 & x \geq - \frac{1}{2} \\ - (2 x) - 1 \cdot 1 + 7 > 2 & x < - \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 8 > 2 & x \geq - \frac{1}{2} \\ - 2 x - 1 \cdot 1 + 7 > 2 & x < - \frac{1}{2} \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} 2 x + 8 > 2 & x \geq - \frac{1}{2} \\ - 2 x - 1 + 7 > 2 & x < - \frac{1}{2} \end{cases}$

Add $- 1$ and $7$ .

${\begin{cases} 2 x + 8 > 2 & x \geq - \frac{1}{2} \\ - 2 x + 6 > 2 & x < - \frac{1}{2} \end{cases}$

Solve $2 x + 8 > 2$ when $x \geq - \frac{1}{2}$ .

Solve $2 x + 8 > 2$ for $x$ .

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

Subtract $8$ from both sides of the inequality.

$2 x > 2 - 8$

Subtract $8$ from $2$ .

$2 x > - 6$

Divide each term in $2 x > - 6$ by $2$ and simplify.

Divide each term in $2 x > - 6$ by $2$ .

$\frac{2 x}{2} > \frac{- 6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 6}{2}$

Divide $x$ by $1$ .

$x > \frac{- 6}{2}$

Simplify the right side.

Divide $- 6$ by $2$ .

$x > - 3$

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

$x \geq - \frac{1}{2}$

Solve $- 2 x + 6 > 2$ when $x < - \frac{1}{2}$ .

Solve $- 2 x + 6 > 2$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 2 x > 2 - 6$

Subtract $6$ from $2$ .

$- 2 x > - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $- 2$ .

$x < 2$

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

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

Find the union of the solutions.

All real numbers

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve the Inequality for x |2x+1|+7>2

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