Solve the Absolute Value Inequality for a 7|4a-6|<42

$7 | 4 a - 6 | < 42$

Step 1

Divide each term in $7 | 4 a - 6 | < 42$ by $7$ and simplify.

Divide each term in $7 | 4 a - 6 | < 42$ by $7$ .

$\frac{7 | 4 a - 6 |}{7} < \frac{42}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 | 4 a - 6 |}{7} < \frac{42}{7}$

Divide $| 4 a - 6 |$ by $1$ .

$| 4 a - 6 | < \frac{42}{7}$

Simplify the right side.

Divide $42$ by $7$ .

$| 4 a - 6 | < 6$

Step 2

Write $| 4 a - 6 | < 6$ as a piecewise.

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

$4 a - 6 \geq 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$4 a \geq 6$

Divide each term in $4 a \geq 6$ by $4$ and simplify.

Divide each term in $4 a \geq 6$ by $4$ .

$\frac{4 a}{4} \geq \frac{6}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} \geq \frac{6}{4}$

Divide $a$ by $1$ .

$a \geq \frac{6}{4}$

Simplify the right side.

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$a \geq \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$a \geq \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$a \geq \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$a \geq \frac{3}{2}$

In the piece where $4 a - 6$ is non-negative, remove the absolute value.

$4 a - 6 < 6$

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

$4 a - 6 < 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$4 a < 6$

Divide each term in $4 a < 6$ by $4$ and simplify.

Divide each term in $4 a < 6$ by $4$ .

$\frac{4 a}{4} < \frac{6}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} < \frac{6}{4}$

Divide $a$ by $1$ .

$a < \frac{6}{4}$

Simplify the right side.

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$a < \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$a < \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$a < \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$a < \frac{3}{2}$

In the piece where $4 a - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 a - 6) < 6$

Write as a piecewise.

${\begin{cases} 4 a - 6 < 6 & a \geq \frac{3}{2} \\ - (4 a - 6) < 6 & a < \frac{3}{2} \end{cases}$

Simplify $- (4 a - 6) < 6$ .

Apply the distributive property.

${\begin{cases} 4 a - 6 < 6 & a \geq \frac{3}{2} \\ - (4 a) - - 6 < 6 & a < \frac{3}{2} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 a - 6 < 6 & a \geq \frac{3}{2} \\ - 4 a - - 6 < 6 & a < \frac{3}{2} \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 4 a - 6 < 6 & a \geq \frac{3}{2} \\ - 4 a + 6 < 6 & a < \frac{3}{2} \end{cases}$

Step 3

Solve $4 a - 6 < 6$ when $a \geq \frac{3}{2}$ .

Solve $4 a - 6 < 6$ for $a$ .

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

Add $6$ to both sides of the inequality.

$4 a < 6 + 6$

Add $6$ and $6$ .

$4 a < 12$

Divide each term in $4 a < 12$ by $4$ and simplify.

Divide each term in $4 a < 12$ by $4$ .

$\frac{4 a}{4} < \frac{12}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} < \frac{12}{4}$

Divide $a$ by $1$ .

$a < \frac{12}{4}$

Simplify the right side.

Divide $12$ by $4$ .

$a < 3$

Find the intersection of $a < 3$ and $a \geq \frac{3}{2}$ .

$\frac{3}{2} \leq a < 3$

Step 4

Solve $- 4 a + 6 < 6$ when $a < \frac{3}{2}$ .

Solve $- 4 a + 6 < 6$ for $a$ .

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

Subtract $6$ from both sides of the inequality.

$- 4 a < 6 - 6$

Subtract $6$ from $6$ .

$- 4 a < 0$

Divide each term in $- 4 a < 0$ by $- 4$ and simplify.

Divide each term in $- 4 a < 0$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 a}{- 4} > \frac{0}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 a}{- 4} > \frac{0}{- 4}$

Divide $a$ by $1$ .

$a > \frac{0}{- 4}$

Simplify the right side.

Divide $0$ by $- 4$ .

$a > 0$

Find the intersection of $a > 0$ and $a < \frac{3}{2}$ .

$0 < a < \frac{3}{2}$

Step 5

Find the union of the solutions.

$0 < a < 3$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$0 < a < 3$

Interval Notation:

$(0, 3)$

Step 7

Solve the Absolute Value Inequality for a 7|4a-6|<42

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