Trigonometry Examples

$| 4 x - 6 | - 2 < - 5$

Step 1

Write $| 4 x - 6 | - 2 < - 5$ as a piecewise.

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Step 1.1

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

$4 x - 6 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $6$ to both sides of the inequality.

$4 x \geq 6$

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

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

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Step 1.2.2.3.1.1

Factor $2$ out of $6$ .

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

Step 1.2.2.3.1.2

Cancel the common factors.

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Step 1.2.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 1.2.2.3.1.2.2

Cancel the common factor.

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

Step 1.2.2.3.1.2.3

Rewrite the expression.

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

Step 1.3

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

$4 x - 6 - 2 < - 5$

Step 1.4

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

$4 x - 6 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $6$ to both sides of the inequality.

$4 x < 6$

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $4$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

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

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Step 1.5.2.3.1.1

Factor $2$ out of $6$ .

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

Step 1.5.2.3.1.2

Cancel the common factors.

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Step 1.5.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 1.5.2.3.1.2.2

Cancel the common factor.

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

Step 1.5.2.3.1.2.3

Rewrite the expression.

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

Step 1.6

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

$- (4 x - 6) - 2 < - 5$

Step 1.7

Write as a piecewise.

${\begin{cases} 4 x - 6 - 2 < - 5 & x \geq \frac{3}{2} \\ - (4 x - 6) - 2 < - 5 & x < \frac{3}{2} \end{cases}$

Step 1.8

Subtract $2$ from $- 6$ .

${\begin{cases} 4 x - 8 < - 5 & x \geq \frac{3}{2} \\ - (4 x - 6) - 2 < - 5 & x < \frac{3}{2} \end{cases}$

Step 1.9

Simplify $- (4 x - 6) - 2 < - 5$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 4 x - 8 < - 5 & x \geq \frac{3}{2} \\ - (4 x) - - 6 - 2 < - 5 & x < \frac{3}{2} \end{cases}$

Step 1.9.1.2

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 8 < - 5 & x \geq \frac{3}{2} \\ - 4 x - - 6 - 2 < - 5 & x < \frac{3}{2} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 6$ .

${\begin{cases} 4 x - 8 < - 5 & x \geq \frac{3}{2} \\ - 4 x + 6 - 2 < - 5 & x < \frac{3}{2} \end{cases}$

Step 1.9.2

Subtract $2$ from $6$ .

${\begin{cases} 4 x - 8 < - 5 & x \geq \frac{3}{2} \\ - 4 x + 4 < - 5 & x < \frac{3}{2} \end{cases}$

Step 2

Solve $4 x - 8 < - 5$ when $x \geq \frac{3}{2}$ .

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Step 2.1

Solve $4 x - 8 < - 5$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Add $8$ to both sides of the inequality.

$4 x < - 5 + 8$

Step 2.1.1.2

Add $- 5$ and $8$ .

$4 x < 3$

Step 2.1.2

Divide each term in $4 x < 3$ by $4$ and simplify.

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Step 2.1.2.1

Divide each term in $4 x < 3$ by $4$ .

$\frac{4 x}{4} < \frac{3}{4}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{3}{4}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{3}{4}$

Step 2.2

Find the intersection of $x < \frac{3}{4}$ and $x \geq \frac{3}{2}$ .

No solution

Step 3

Solve $- 4 x + 4 < - 5$ when $x < \frac{3}{2}$ .

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Step 3.1

Solve $- 4 x + 4 < - 5$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $4$ from both sides of the inequality.

$- 4 x < - 5 - 4$

Step 3.1.1.2

Subtract $4$ from $- 5$ .

$- 4 x < - 9$

Step 3.1.2

Divide each term in $- 4 x < - 9$ by $- 4$ and simplify.

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Step 3.1.2.1

Divide each term in $- 4 x < - 9$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{- 9}{- 4}$

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.

$\frac{- 4 x}{- 4} > \frac{- 9}{- 4}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 9}{- 4}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Dividing two negative values results in a positive value.

$x > \frac{9}{4}$

Step 3.2

Find the intersection of $x > \frac{9}{4}$ and $x < \frac{3}{2}$ .

No solution

Step 4

Find the union of the solutions.

No solution