Solve the Inequality for x (4|x+7|)/3<8

$\frac{4 | x + 7 |}{3} < 8$

Multiply both sides by $3$ .

$\frac{4 | x + 7 |}{3} \cdot 3 < 8 \cdot 3$

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{4 | x + 7 |}{3} \cdot 3 < 8 \cdot 3$

Rewrite the expression.

$4 | x + 7 | < 8 \cdot 3$

Simplify the right side.

Multiply $8$ by $3$ .

$4 | x + 7 | < 24$

Solve for $x$ .

Divide each term in $4 | x + 7 | < 24$ by $4$ and simplify.

Divide each term in $4 | x + 7 | < 24$ by $4$ .

$\frac{4 | x + 7 |}{4} < \frac{24}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 | x + 7 |}{4} < \frac{24}{4}$

Divide $| x + 7 |$ by $1$ .

$| x + 7 | < \frac{24}{4}$

Simplify the right side.

Divide $24$ by $4$ .

$| x + 7 | < 6$

Write $| x + 7 | < 6$ as a piecewise.

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

$x + 7 \geq 0$

Subtract $7$ from both sides of the inequality.

$x \geq - 7$

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

$x + 7 < 6$

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

$x + 7 < 0$

Subtract $7$ from both sides of the inequality.

$x < - 7$

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

$- (x + 7) < 6$

Write as a piecewise.

${\begin{cases} x + 7 < 6 & x \geq - 7 \\ - (x + 7) < 6 & x < - 7 \end{cases}$

Simplify $- (x + 7) < 6$ .

Apply the distributive property.

${\begin{cases} x + 7 < 6 & x \geq - 7 \\ - x - 1 \cdot 7 < 6 & x < - 7 \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} x + 7 < 6 & x \geq - 7 \\ - x - 7 < 6 & x < - 7 \end{cases}$

Solve $x + 7 < 6$ when $x \geq - 7$ .

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

Subtract $7$ from both sides of the inequality.

$x < 6 - 7$

Subtract $7$ from $6$ .

$x < - 1$

Find the intersection of $x < - 1$ and $x \geq - 7$ .

$- 7 \leq x < - 1$

Solve $- x - 7 < 6$ when $x < - 7$ .

Solve $- x - 7 < 6$ for $x$ .

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

Add $7$ to both sides of the inequality.

$- x < 6 + 7$

Add $6$ and $7$ .

$- x < 13$

Divide each term in $- x < 13$ by $- 1$ and simplify.

Divide each term in $- x < 13$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{13}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{13}{- 1}$

Divide $x$ by $1$ .

$x > \frac{13}{- 1}$

Simplify the right side.

Divide $13$ by $- 1$ .

$x > - 13$

Find the intersection of $x > - 13$ and $x < - 7$ .

$- 13 < x < - 7$

Find the union of the solutions.

$- 13 < x < - 1$

The result can be shown in multiple forms.

Inequality Form:

$- 13 < x < - 1$

Interval Notation:

$(- 13, - 1)$

Solve the Inequality for x (4|x+7|)/3<8

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