Solve the Absolute Value Inequality for w (|5+3w|)/-2<=-1

$\frac{| 5 + 3 w |}{- 2} \leq - 1$

Step 1

Multiply both sides by $- 2$ .

$\frac{| 5 + 3 w |}{- 2} \cdot - 2 \geq - - 2$

Step 2

Simplify.

Simplify the left side.

Simplify $\frac{| 5 + 3 w |}{- 2} \cdot - 2$ .

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$\frac{| 5 + 3 w |}{2 (- 1)} \cdot - 2 \geq - - 2$

Factor $2$ out of $- 2$ .

$\frac{| 5 + 3 w |}{2 \cdot - 1} \cdot (2 \cdot - 1) \geq - - 2$

Cancel the common factor.

$\frac{| 5 + 3 w |}{2 \cdot - 1} \cdot (2 \cdot - 1) \geq - - 2$

Rewrite the expression.

$\frac{| 5 + 3 w |}{- 1} \cdot - 1 \geq - - 2$

Combine $\frac{| 5 + 3 w |}{- 1}$ and $- 1$ .

$\frac{| 5 + 3 w | \cdot - 1}{- 1} \geq - - 2$

Simplify the expression.

Move the negative one from the denominator of $\frac{| 5 + 3 w | \cdot - 1}{- 1}$ .

$- 1 \cdot (| 5 + 3 w | \cdot - 1) \geq - - 2$

Rewrite $- 1 \cdot (| 5 + 3 w | \cdot - 1)$ as $- (| 5 + 3 w | \cdot - 1)$ .

$- (| 5 + 3 w | \cdot - 1) \geq - - 2$

Move $- 1$ to the left of $| 5 + 3 w |$ .

$- (- 1 \cdot | 5 + 3 w |) \geq - - 2$

Rewrite $- 1 | 5 + 3 w |$ as $- | 5 + 3 w |$ .

$- - | 5 + 3 w | \geq - - 2$

Multiply $- - | 5 + 3 w |$ .

Multiply $- 1$ by $- 1$ .

$1 | 5 + 3 w | \geq - - 2$

Multiply $| 5 + 3 w |$ by $1$ .

$| 5 + 3 w | \geq - - 2$

Simplify the right side.

Multiply $- 1$ by $- 2$ .

$| 5 + 3 w | \geq 2$

Step 3

Solve for $w$ .

Write $| 5 + 3 w | \geq 2$ as a piecewise.

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

$5 + 3 w \geq 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$3 w \geq - 5$

Divide each term in $3 w \geq - 5$ by $3$ and simplify.

Divide each term in $3 w \geq - 5$ by $3$ .

$\frac{3 w}{3} \geq \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 w}{3} \geq \frac{- 5}{3}$

Divide $w$ by $1$ .

$w \geq \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$w \geq - \frac{5}{3}$

In the piece where $5 + 3 w$ is non-negative, remove the absolute value.

$5 + 3 w \geq 2$

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

$5 + 3 w < 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$3 w < - 5$

Divide each term in $3 w < - 5$ by $3$ and simplify.

Divide each term in $3 w < - 5$ by $3$ .

$\frac{3 w}{3} < \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 w}{3} < \frac{- 5}{3}$

Divide $w$ by $1$ .

$w < \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$w < - \frac{5}{3}$

In the piece where $5 + 3 w$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 + 3 w) \geq 2$

Write as a piecewise.

${\begin{cases} 5 + 3 w \geq 2 & w \geq - \frac{5}{3} \\ - (5 + 3 w) \geq 2 & w < - \frac{5}{3} \end{cases}$

Simplify $- (5 + 3 w) \geq 2$ .

Apply the distributive property.

${\begin{cases} 5 + 3 w \geq 2 & w \geq - \frac{5}{3} \\ - 1 \cdot 5 - (3 w) \geq 2 & w < - \frac{5}{3} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 + 3 w \geq 2 & w \geq - \frac{5}{3} \\ - 5 - (3 w) \geq 2 & w < - \frac{5}{3} \end{cases}$

Multiply $3$ by $- 1$ .

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

Solve $5 + 3 w \geq 2$ for $w$ .

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

Subtract $5$ from both sides of the inequality.

$3 w \geq 2 - 5$

Subtract $5$ from $2$ .

$3 w \geq - 3$

Divide each term in $3 w \geq - 3$ by $3$ and simplify.

Divide each term in $3 w \geq - 3$ by $3$ .

$\frac{3 w}{3} \geq \frac{- 3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 w}{3} \geq \frac{- 3}{3}$

Divide $w$ by $1$ .

$w \geq \frac{- 3}{3}$

Simplify the right side.

Divide $- 3$ by $3$ .

$w \geq - 1$

Solve $- 5 - 3 w \geq 2$ for $w$ .

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

Add $5$ to both sides of the inequality.

$- 3 w \geq 2 + 5$

Add $2$ and $5$ .

$- 3 w \geq 7$

Divide each term in $- 3 w \geq 7$ by $- 3$ and simplify.

Divide each term in $- 3 w \geq 7$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 w}{- 3} \leq \frac{7}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 w}{- 3} \leq \frac{7}{- 3}$

Divide $w$ by $1$ .

$w \leq \frac{7}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$w \leq - \frac{7}{3}$

Find the union of the solutions.

$w \leq - \frac{7}{3}$ or $w \geq - 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$w \leq - \frac{7}{3} or w \geq - 1$

Interval Notation:

$(- \infty, - \frac{7}{3}] \cup [- 1, \infty)$

Step 5

Solve the Absolute Value Inequality for w (|5+3w|)/-2<=-1

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