Solve the Inequality for x 3+4|3x+7|>=-89

$3 + 4 | 3 x + 7 | \geq - 89$

Write $3 + 4 | 3 x + 7 | \geq - 89$ as a piecewise.

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

$3 x + 7 \geq 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$3 x \geq - 7$

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

Divide each term in $3 x \geq - 7$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 7}{3}$

Divide $x$ by $1$ .

$x \geq \frac{- 7}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{7}{3}$

In the piece where $3 x + 7$ is non-negative, remove the absolute value.

$3 + 4 (3 x + 7) \geq - 89$

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

$3 x + 7 < 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$3 x < - 7$

Divide each term in $3 x < - 7$ by $3$ and simplify.

Divide each term in $3 x < - 7$ by $3$ .

$\frac{3 x}{3} < \frac{- 7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 7}{3}$

Divide $x$ by $1$ .

$x < \frac{- 7}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{7}{3}$

In the piece where $3 x + 7$ is negative, remove the absolute value and multiply by $- 1$ .

$3 + 4 (- (3 x + 7)) \geq - 89$

Write as a piecewise.

${\begin{cases} 3 + 4 (3 x + 7) \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Simplify $3 + 4 (3 x + 7) \geq - 89$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 + 4 (3 x) + 4 \cdot 7 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Multiply $3$ by $4$ .

${\begin{cases} 3 + 12 x + 4 \cdot 7 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Multiply $4$ by $7$ .

${\begin{cases} 3 + 12 x + 28 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Add $3$ and $28$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Simplify $3 + 4 (- (3 x + 7)) \geq - 89$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x) - 1 \cdot 7) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- 3 x - 1 \cdot 7) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- 3 x - 7) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Apply the distributive property.

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- 3 x) + 4 \cdot - 7 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Multiply $- 3$ by $4$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 - 12 x + 4 \cdot - 7 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Multiply $4$ by $- 7$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 - 12 x - 28 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Subtract $28$ from $3$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ - 12 x - 25 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Solve $12 x + 31 \geq - 89$ when $x \geq - \frac{7}{3}$ .

Solve $12 x + 31 \geq - 89$ for $x$ .

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

Subtract $31$ from both sides of the inequality.

$12 x \geq - 89 - 31$

Subtract $31$ from $- 89$ .

$12 x \geq - 120$

Divide each term in $12 x \geq - 120$ by $12$ and simplify.

Divide each term in $12 x \geq - 120$ by $12$ .

$\frac{12 x}{12} \geq \frac{- 120}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 x}{12} \geq \frac{- 120}{12}$

Divide $x$ by $1$ .

$x \geq \frac{- 120}{12}$

Simplify the right side.

Divide $- 120$ by $12$ .

$x \geq - 10$

Find the intersection of $x \geq - 10$ and $x \geq - \frac{7}{3}$ .

$x \geq - \frac{7}{3}$

Solve $- 12 x - 25 \geq - 89$ when $x < - \frac{7}{3}$ .

Solve $- 12 x - 25 \geq - 89$ for $x$ .

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

Add $25$ to both sides of the inequality.

$- 12 x \geq - 89 + 25$

Add $- 89$ and $25$ .

$- 12 x \geq - 64$

Divide each term in $- 12 x \geq - 64$ by $- 12$ and simplify.

Divide each term in $- 12 x \geq - 64$ by $- 12$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 12 x}{- 12} \leq \frac{- 64}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 x}{- 12} \leq \frac{- 64}{- 12}$

Divide $x$ by $1$ .

$x \leq \frac{- 64}{- 12}$

Simplify the right side.

Cancel the common factor of $- 64$ and $- 12$ .

Factor $- 4$ out of $- 64$ .

$x \leq \frac{- 4 (16)}{- 12}$

Cancel the common factors.

Factor $- 4$ out of $- 12$ .

$x \leq \frac{- 4 \cdot 16}{- 4 \cdot 3}$

Cancel the common factor.

$x \leq \frac{- 4 \cdot 16}{- 4 \cdot 3}$

Rewrite the expression.

$x \leq \frac{16}{3}$

Find the intersection of $x \leq \frac{16}{3}$ and $x < - \frac{7}{3}$ .

$x < - \frac{7}{3}$

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 3+4|3x+7|>=-89

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