Algebra Examples

$7 | 4 a - 6 | < 42$

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $7$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

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

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $42$ by $7$ .

$| 4 a - 6 | < 6$

Step 2

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

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Step 2.1

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

$4 a - 6 \geq 0$

Step 2.2

Solve the inequality.

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Step 2.2.1

Add $6$ to both sides of the inequality.

$4 a \geq 6$

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

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

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Step 2.2.2.3.1.1

Factor $2$ out of $6$ .

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

Step 2.2.2.3.1.2

Cancel the common factors.

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Step 2.2.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 2.2.2.3.1.2.2

Cancel the common factor.

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

Step 2.2.2.3.1.2.3

Rewrite the expression.

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

Step 2.3

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

$4 a - 6 < 6$

Step 2.4

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

$4 a - 6 < 0$

Step 2.5

Solve the inequality.

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Step 2.5.1

Add $6$ to both sides of the inequality.

$4 a < 6$

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

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

Step 2.5.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

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

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Step 2.5.2.3.1.1

Factor $2$ out of $6$ .

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

Step 2.5.2.3.1.2

Cancel the common factors.

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Step 2.5.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 2.5.2.3.1.2.2

Cancel the common factor.

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

Step 2.5.2.3.1.2.3

Rewrite the expression.

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

Step 2.6

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

$- (4 a - 6) < 6$

Step 2.7

Write as a piecewise.

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

Step 2.8

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

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Multiply $4$ by $- 1$ .

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

Step 2.8.3

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

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Step 3.1

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

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Step 3.1.1

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

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Step 3.1.1.1

Add $6$ to both sides of the inequality.

$4 a < 6 + 6$

Step 3.1.1.2

Add $6$ and $6$ .

$4 a < 12$

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $4$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

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

Step 3.1.2.2.1.2

Divide $a$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $12$ by $4$ .

$a < 3$

Step 3.2

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

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Step 4.1

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

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Step 4.1.1

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

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Step 4.1.1.1

Subtract $6$ from both sides of the inequality.

$- 4 a < 6 - 6$

Step 4.1.1.2

Subtract $6$ from $6$ .

$- 4 a < 0$

Step 4.1.2

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

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Step 4.1.2.1

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.

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

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

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

Step 4.1.2.2.1.2

Divide $a$ by $1$ .

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

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $0$ by $- 4$ .

$a > 0$

Step 4.2

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