Solve for x 4|5+2x|+1>=13

$4 | 5 + 2 x | + 1 \geq 13$

Step 1

Write $4 | 5 + 2 x | + 1 \geq 13$ as a piecewise.

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

$5 + 2 x \geq 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$2 x \geq - 5$

Divide each term in $2 x \geq - 5$ by $2$ and simplify.

Divide each term in $2 x \geq - 5$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 5}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{5}{2}$

In the piece where $5 + 2 x$ is non-negative, remove the absolute value.

$4 (5 + 2 x) + 1 \geq 13$

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

$5 + 2 x < 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$2 x < - 5$

Divide each term in $2 x < - 5$ by $2$ and simplify.

Divide each term in $2 x < - 5$ by $2$ .

$\frac{2 x}{2} < \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 5}{2}$

Divide $x$ by $1$ .

$x < \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{5}{2}$

In the piece where $5 + 2 x$ is negative, remove the absolute value and multiply by $- 1$ .

$4 (- (5 + 2 x)) + 1 \geq 13$

Write as a piecewise.

${\begin{cases} 4 (5 + 2 x) + 1 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- (5 + 2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Simplify $4 (5 + 2 x) + 1 \geq 13$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 \cdot 5 + 4 (2 x) + 1 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- (5 + 2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Multiply $4$ by $5$ .

${\begin{cases} 20 + 4 (2 x) + 1 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- (5 + 2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Multiply $2$ by $4$ .

${\begin{cases} 20 + 8 x + 1 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- (5 + 2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Add $20$ and $1$ .

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- (5 + 2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Simplify $4 (- (5 + 2 x)) + 1 \geq 13$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- 1 \cdot 5 - (2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- 5 - (2 x)) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ 4 (- 5 - 2 x) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ 4 \cdot - 5 + 4 (- 2 x) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Multiply $4$ by $- 5$ .

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ - 20 + 4 (- 2 x) + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Multiply $- 2$ by $4$ .

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ - 20 - 8 x + 1 \geq 13 & x < - \frac{5}{2} \end{cases}$

Add $- 20$ and $1$ .

${\begin{cases} 8 x + 21 \geq 13 & x \geq - \frac{5}{2} \\ - 8 x - 19 \geq 13 & x < - \frac{5}{2} \end{cases}$

Step 2

Solve $8 x + 21 \geq 13$ for $x$ .

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

Subtract $21$ from both sides of the inequality.

$8 x \geq 13 - 21$

Subtract $21$ from $13$ .

$8 x \geq - 8$

Divide each term in $8 x \geq - 8$ by $8$ and simplify.

Divide each term in $8 x \geq - 8$ by $8$ .

$\frac{8 x}{8} \geq \frac{- 8}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{- 8}{8}$

Divide $x$ by $1$ .

$x \geq \frac{- 8}{8}$

Simplify the right side.

Divide $- 8$ by $8$ .

$x \geq - 1$

Step 3

Solve $- 8 x - 19 \geq 13$ for $x$ .

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

Add $19$ to both sides of the inequality.

$- 8 x \geq 13 + 19$

Add $13$ and $19$ .

$- 8 x \geq 32$

Divide each term in $- 8 x \geq 32$ by $- 8$ and simplify.

Divide each term in $- 8 x \geq 32$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{32}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{32}{- 8}$

Divide $x$ by $1$ .

$x \leq \frac{32}{- 8}$

Simplify the right side.

Divide $32$ by $- 8$ .

$x \leq - 4$

Step 4

Find the union of the solutions.

$x \leq - 4$ or $x \geq - 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 4 or x \geq - 1$

Interval Notation:

$(- \infty, - 4] \cup [- 1, \infty)$

Step 6

Solve for x 4|5+2x|+1>=13

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