Solve the Absolute Value Inequality for a |3-9a|<=60

$| 3 - 9 a | \leq 60$

Step 1

Write $| 3 - 9 a | \leq 60$ as a piecewise.

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

$3 - 9 a \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 9 a \geq - 3$

Divide each term in $- 9 a \geq - 3$ by $- 9$ and simplify.

Divide each term in $- 9 a \geq - 3$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 a}{- 9} \leq \frac{- 3}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 a}{- 9} \leq \frac{- 3}{- 9}$

Divide $a$ by $1$ .

$a \leq \frac{- 3}{- 9}$

Simplify the right side.

Cancel the common factor of $- 3$ and $- 9$ .

Factor $- 3$ out of $- 3$ .

$a \leq \frac{- 3 (1)}{- 9}$

Cancel the common factors.

Factor $- 3$ out of $- 9$ .

$a \leq \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Cancel the common factor.

$a \leq \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Rewrite the expression.

$a \leq \frac{1}{3}$

In the piece where $3 - 9 a$ is non-negative, remove the absolute value.

$3 - 9 a \leq 60$

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

$3 - 9 a < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 9 a < - 3$

Divide each term in $- 9 a < - 3$ by $- 9$ and simplify.

Divide each term in $- 9 a < - 3$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 a}{- 9} > \frac{- 3}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 a}{- 9} > \frac{- 3}{- 9}$

Divide $a$ by $1$ .

$a > \frac{- 3}{- 9}$

Simplify the right side.

Cancel the common factor of $- 3$ and $- 9$ .

Factor $- 3$ out of $- 3$ .

$a > \frac{- 3 (1)}{- 9}$

Cancel the common factors.

Factor $- 3$ out of $- 9$ .

$a > \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Cancel the common factor.

$a > \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Rewrite the expression.

$a > \frac{1}{3}$

In the piece where $3 - 9 a$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 - 9 a) \leq 60$

Write as a piecewise.

${\begin{cases} 3 - 9 a \leq 60 & a \leq \frac{1}{3} \\ - (3 - 9 a) \leq 60 & a > \frac{1}{3} \end{cases}$

Simplify $- (3 - 9 a) \leq 60$ .

Apply the distributive property.

${\begin{cases} 3 - 9 a \leq 60 & a \leq \frac{1}{3} \\ - 1 \cdot 3 - (- 9 a) \leq 60 & a > \frac{1}{3} \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 - 9 a \leq 60 & a \leq \frac{1}{3} \\ - 3 - (- 9 a) \leq 60 & a > \frac{1}{3} \end{cases}$

Multiply $- 9$ by $- 1$ .

${\begin{cases} 3 - 9 a \leq 60 & a \leq \frac{1}{3} \\ - 3 + 9 a \leq 60 & a > \frac{1}{3} \end{cases}$

Step 2

Solve $3 - 9 a \leq 60$ when $a \leq \frac{1}{3}$ .

Solve $3 - 9 a \leq 60$ for $a$ .

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

Subtract $3$ from both sides of the inequality.

$- 9 a \leq 60 - 3$

Subtract $3$ from $60$ .

$- 9 a \leq 57$

Divide each term in $- 9 a \leq 57$ by $- 9$ and simplify.

Divide each term in $- 9 a \leq 57$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 a}{- 9} \geq \frac{57}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 a}{- 9} \geq \frac{57}{- 9}$

Divide $a$ by $1$ .

$a \geq \frac{57}{- 9}$

Simplify the right side.

Cancel the common factor of $57$ and $- 9$ .

Factor $3$ out of $57$ .

$a \geq \frac{3 (19)}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

$a \geq \frac{3 \cdot 19}{3 \cdot - 3}$

Cancel the common factor.

$a \geq \frac{3 \cdot 19}{3 \cdot - 3}$

Rewrite the expression.

$a \geq \frac{19}{- 3}$

Move the negative in front of the fraction.

$a \geq - \frac{19}{3}$

Find the intersection of $a \geq - \frac{19}{3}$ and $a \leq \frac{1}{3}$ .

$- \frac{19}{3} \leq a \leq \frac{1}{3}$

Step 3

Solve $- 3 + 9 a \leq 60$ when $a > \frac{1}{3}$ .

Solve $- 3 + 9 a \leq 60$ for $a$ .

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

Add $3$ to both sides of the inequality.

$9 a \leq 60 + 3$

Add $60$ and $3$ .

$9 a \leq 63$

Divide each term in $9 a \leq 63$ by $9$ and simplify.

Divide each term in $9 a \leq 63$ by $9$ .

$\frac{9 a}{9} \leq \frac{63}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 a}{9} \leq \frac{63}{9}$

Divide $a$ by $1$ .

$a \leq \frac{63}{9}$

Simplify the right side.

Divide $63$ by $9$ .

$a \leq 7$

Find the intersection of $a \leq 7$ and $a > \frac{1}{3}$ .

$\frac{1}{3} < a \leq 7$

Step 4

Find the union of the solutions.

$- \frac{19}{3} \leq a \leq 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{19}{3} \leq a \leq 7$

Interval Notation:

$[- \frac{19}{3}, 7]$

Step 6

Solve the Absolute Value Inequality for a |3-9a|<=60

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