Precalculus Examples

$2 | 2 x - 7 | < 14$

Step 1

Divide each term in $2 | 2 x - 7 | < 14$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 | 2 x - 7 | < 14$ by $2$ .

$\frac{2 | 2 x - 7 |}{2} < \frac{14}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 | 2 x - 7 |}{2} < \frac{14}{2}$

Step 1.2.1.2

Divide $| 2 x - 7 |$ by $1$ .

$| 2 x - 7 | < \frac{14}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $14$ by $2$ .

$| 2 x - 7 | < 7$

Step 2

Write $| 2 x - 7 | < 7$ 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.

$2 x - 7 \geq 0$

Step 2.2

Solve the inequality.

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Step 2.2.1

Add $7$ to both sides of the inequality.

$2 x \geq 7$

Step 2.2.2

Divide each term in $2 x \geq 7$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 x \geq 7$ by $2$ .

$\frac{2 x}{2} \geq \frac{7}{2}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{7}{2}$

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{7}{2}$

Step 2.3

In the piece where $2 x - 7$ is non-negative, remove the absolute value.

$2 x - 7 < 7$

Step 2.4

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

$2 x - 7 < 0$

Step 2.5

Solve the inequality.

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Step 2.5.1

Add $7$ to both sides of the inequality.

$2 x < 7$

Step 2.5.2

Divide each term in $2 x < 7$ by $2$ and simplify.

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Step 2.5.2.1

Divide each term in $2 x < 7$ by $2$ .

$\frac{2 x}{2} < \frac{7}{2}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{7}{2}$

Step 2.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{7}{2}$

Step 2.6

In the piece where $2 x - 7$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 7) < 7$

Step 2.7

Write as a piecewise.

${\begin{cases} 2 x - 7 < 7 & x \geq \frac{7}{2} \\ - (2 x - 7) < 7 & x < \frac{7}{2} \end{cases}$

Step 2.8

Simplify $- (2 x - 7) < 7$ .

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Step 2.8.1

Apply the distributive property.

${\begin{cases} 2 x - 7 < 7 & x \geq \frac{7}{2} \\ - (2 x) - - 7 < 7 & x < \frac{7}{2} \end{cases}$

Step 2.8.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 7 < 7 & x \geq \frac{7}{2} \\ - 2 x - - 7 < 7 & x < \frac{7}{2} \end{cases}$

Step 2.8.3

Multiply $- 1$ by $- 7$ .

${\begin{cases} 2 x - 7 < 7 & x \geq \frac{7}{2} \\ - 2 x + 7 < 7 & x < \frac{7}{2} \end{cases}$

Step 3

Solve $2 x - 7 < 7$ when $x \geq \frac{7}{2}$ .

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Step 3.1

Solve $2 x - 7 < 7$ 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

Add $7$ to both sides of the inequality.

$2 x < 7 + 7$

Step 3.1.1.2

Add $7$ and $7$ .

$2 x < 14$

Step 3.1.2

Divide each term in $2 x < 14$ by $2$ and simplify.

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Step 3.1.2.1

Divide each term in $2 x < 14$ by $2$ .

$\frac{2 x}{2} < \frac{14}{2}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{14}{2}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{14}{2}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $14$ by $2$ .

$x < 7$

Step 3.2

Find the intersection of $x < 7$ and $x \geq \frac{7}{2}$ .

$\frac{7}{2} \leq x < 7$

Step 4

Solve $- 2 x + 7 < 7$ when $x < \frac{7}{2}$ .

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Step 4.1

Solve $- 2 x + 7 < 7$ for $x$ .

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Step 4.1.1

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

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Step 4.1.1.1

Subtract $7$ from both sides of the inequality.

$- 2 x < 7 - 7$

Step 4.1.1.2

Subtract $7$ from $7$ .

$- 2 x < 0$

Step 4.1.2

Divide each term in $- 2 x < 0$ by $- 2$ and simplify.

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Step 4.1.2.1

Divide each term in $- 2 x < 0$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{0}{- 2}$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{0}{- 2}$

Step 4.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{0}{- 2}$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $0$ by $- 2$ .

$x > 0$

Step 4.2

Find the intersection of $x > 0$ and $x < \frac{7}{2}$ .

$0 < x < \frac{7}{2}$

Step 5

Find the union of the solutions.

$0 < x < 7$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 7$

Interval Notation:

$(0, 7)$

Step 7