Solve the Absolute Value Inequality for x -16-3|2x-5|>=-49

$- 16 - 3 | 2 x - 5 | \geq - 49$

Step 1

Write $- 16 - 3 | 2 x - 5 | \geq - 49$ as a piecewise.

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

$2 x - 5 \geq 0$

Solve the inequality.

Add $5$ to 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 $1$ .

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

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

$- 16 - 3 (2 x - 5) \geq - 49$

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

$2 x - 5 < 0$

Solve the inequality.

Add $5$ to 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 $1$ .

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

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

$- 16 - 3 (- (2 x - 5)) \geq - 49$

Write as a piecewise.

${\begin{cases} - 16 - 3 (2 x - 5) \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- (2 x - 5)) \geq - 49 & x < \frac{5}{2} \end{cases}$

Simplify $- 16 - 3 (2 x - 5) \geq - 49$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 16 - 3 (2 x) - 3 \cdot - 5 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- (2 x - 5)) \geq - 49 & x < \frac{5}{2} \end{cases}$

Multiply $2$ by $- 3$ .

${\begin{cases} - 16 - 6 x - 3 \cdot - 5 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- (2 x - 5)) \geq - 49 & x < \frac{5}{2} \end{cases}$

Multiply $- 3$ by $- 5$ .

${\begin{cases} - 16 - 6 x + 15 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- (2 x - 5)) \geq - 49 & x < \frac{5}{2} \end{cases}$

Add $- 16$ and $15$ .

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- (2 x - 5)) \geq - 49 & x < \frac{5}{2} \end{cases}$

Simplify $- 16 - 3 (- (2 x - 5)) \geq - 49$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- (2 x) - - 5) \geq - 49 & x < \frac{5}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- 2 x - - 5) \geq - 49 & x < \frac{5}{2} \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- 2 x + 5) \geq - 49 & x < \frac{5}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 - 3 (- 2 x) - 3 \cdot 5 \geq - 49 & x < \frac{5}{2} \end{cases}$

Multiply $- 2$ by $- 3$ .

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 + 6 x - 3 \cdot 5 \geq - 49 & x < \frac{5}{2} \end{cases}$

Multiply $- 3$ by $5$ .

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ - 16 + 6 x - 15 \geq - 49 & x < \frac{5}{2} \end{cases}$

Subtract $15$ from $- 16$ .

${\begin{cases} - 6 x - 1 \geq - 49 & x \geq \frac{5}{2} \\ 6 x - 31 \geq - 49 & x < \frac{5}{2} \end{cases}$

Step 2

Solve $- 6 x - 1 \geq - 49$ when $x \geq \frac{5}{2}$ .

Solve $- 6 x - 1 \geq - 49$ for $x$ .

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

Add $1$ to both sides of the inequality.

$- 6 x \geq - 49 + 1$

Add $- 49$ and $1$ .

$- 6 x \geq - 48$

Divide each term in $- 6 x \geq - 48$ by $- 6$ and simplify.

Divide each term in $- 6 x \geq - 48$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \leq \frac{- 48}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} \leq \frac{- 48}{- 6}$

Divide $x$ by $1$ .

$x \leq \frac{- 48}{- 6}$

Simplify the right side.

Divide $- 48$ by $- 6$ .

$x \leq 8$

Find the intersection of $x \leq 8$ and $x \geq \frac{5}{2}$ .

$\frac{5}{2} \leq x \leq 8$

Step 3

Solve $6 x - 31 \geq - 49$ when $x < \frac{5}{2}$ .

Solve $6 x - 31 \geq - 49$ for $x$ .

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

Add $31$ to both sides of the inequality.

$6 x \geq - 49 + 31$

Add $- 49$ and $31$ .

$6 x \geq - 18$

Divide each term in $6 x \geq - 18$ by $6$ and simplify.

Divide each term in $6 x \geq - 18$ by $6$ .

$\frac{6 x}{6} \geq \frac{- 18}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{- 18}{6}$

Divide $x$ by $1$ .

$x \geq \frac{- 18}{6}$

Simplify the right side.

Divide $- 18$ by $6$ .

$x \geq - 3$

Find the intersection of $x \geq - 3$ and $x < \frac{5}{2}$ .

$- 3 \leq x < \frac{5}{2}$

Step 4

Find the union of the solutions.

$- 3 \leq x \leq 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x \leq 8$

Interval Notation:

$[- 3, 8]$

Step 6

Solve the Absolute Value Inequality for x -16-3|2x-5|>=-49

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