Solve the Inequality for a |-8a-3|>11

$| - 8 a - 3 | > 11$

Write $| - 8 a - 3 | > 11$ as a piecewise.

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

$- 8 a - 3 \geq 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$- 8 a \geq 3$

Divide each term in $- 8 a \geq 3$ by $- 8$ and simplify.

Divide each term in $- 8 a \geq 3$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 a}{- 8} \leq \frac{3}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 a}{- 8} \leq \frac{3}{- 8}$

Divide $a$ by $1$ .

$a \leq \frac{3}{- 8}$

Simplify the right side.

Move the negative in front of the fraction.

$a \leq - \frac{3}{8}$

In the piece where $- 8 a - 3$ is non-negative, remove the absolute value.

$- 8 a - 3 > 11$

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

$- 8 a - 3 < 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$- 8 a < 3$

Divide each term in $- 8 a < 3$ by $- 8$ and simplify.

Divide each term in $- 8 a < 3$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 a}{- 8} > \frac{3}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 a}{- 8} > \frac{3}{- 8}$

Divide $a$ by $1$ .

$a > \frac{3}{- 8}$

Simplify the right side.

Move the negative in front of the fraction.

$a > - \frac{3}{8}$

In the piece where $- 8 a - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 8 a - 3) > 11$

Write as a piecewise.

${\begin{cases} - 8 a - 3 > 11 & a \leq - \frac{3}{8} \\ - (- 8 a - 3) > 11 & a > - \frac{3}{8} \end{cases}$

Simplify $- (- 8 a - 3) > 11$ .

Apply the distributive property.

${\begin{cases} - 8 a - 3 > 11 & a \leq - \frac{3}{8} \\ - (- 8 a) - - 3 > 11 & a > - \frac{3}{8} \end{cases}$

Multiply $- 8$ by $- 1$ .

${\begin{cases} - 8 a - 3 > 11 & a \leq - \frac{3}{8} \\ 8 a - - 3 > 11 & a > - \frac{3}{8} \end{cases}$

Multiply $- 1$ by $- 3$ .

${\begin{cases} - 8 a - 3 > 11 & a \leq - \frac{3}{8} \\ 8 a + 3 > 11 & a > - \frac{3}{8} \end{cases}$

Solve $- 8 a - 3 > 11$ for $a$ .

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

Add $3$ to both sides of the inequality.

$- 8 a > 11 + 3$

Add $11$ and $3$ .

$- 8 a > 14$

Divide each term in $- 8 a > 14$ by $- 8$ and simplify.

Divide each term in $- 8 a > 14$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 a}{- 8} < \frac{14}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 a}{- 8} < \frac{14}{- 8}$

Divide $a$ by $1$ .

$a < \frac{14}{- 8}$

Simplify the right side.

Cancel the common factor of $14$ and $- 8$ .

Factor $2$ out of $14$ .

$a < \frac{2 (7)}{- 8}$

Cancel the common factors.

Factor $2$ out of $- 8$ .

$a < \frac{2 \cdot 7}{2 \cdot - 4}$

Cancel the common factor.

$a < \frac{2 \cdot 7}{2 \cdot - 4}$

Rewrite the expression.

$a < \frac{7}{- 4}$

Move the negative in front of the fraction.

$a < - \frac{7}{4}$

Solve $8 a + 3 > 11$ for $a$ .

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

Subtract $3$ from both sides of the inequality.

$8 a > 11 - 3$

Subtract $3$ from $11$ .

$8 a > 8$

Divide each term in $8 a > 8$ by $8$ and simplify.

Divide each term in $8 a > 8$ by $8$ .

$\frac{8 a}{8} > \frac{8}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 a}{8} > \frac{8}{8}$

Divide $a$ by $1$ .

$a > \frac{8}{8}$

Simplify the right side.

Divide $8$ by $8$ .

$a > 1$

Find the union of the solutions.

$a < - \frac{7}{4}$ or $a > 1$

The result can be shown in multiple forms.

Inequality Form:

$a < - \frac{7}{4} or a > 1$

Interval Notation:

$(- \infty, - \frac{7}{4}) \cup (1, \infty)$

Solve the Inequality for a |-8a-3|>11

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