Algebra Examples

$| 5 s - 10 | \geq 0$

Step 1

Write $| 5 s - 10 | \geq 0$ 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.

$5 s - 10 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $10$ to both sides of the inequality.

$5 s \geq 10$

Step 1.2.2

Divide each term in $5 s \geq 10$ by $5$ and simplify.

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Step 1.2.2.1

Divide each term in $5 s \geq 10$ by $5$ .

$\frac{5 s}{5} \geq \frac{10}{5}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{5 s}{5} \geq \frac{10}{5}$

Step 1.2.2.2.1.2

Divide $s$ by $1$ .

$s \geq \frac{10}{5}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $10$ by $5$ .

$s \geq 2$

Step 1.3

In the piece where $5 s - 10$ is non-negative, remove the absolute value.

$5 s - 10 \geq 0$

Step 1.4

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

$5 s - 10 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $10$ to both sides of the inequality.

$5 s < 10$

Step 1.5.2

Divide each term in $5 s < 10$ by $5$ and simplify.

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Step 1.5.2.1

Divide each term in $5 s < 10$ by $5$ .

$\frac{5 s}{5} < \frac{10}{5}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $5$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{5 s}{5} < \frac{10}{5}$

Step 1.5.2.2.1.2

Divide $s$ by $1$ .

$s < \frac{10}{5}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $10$ by $5$ .

$s < 2$

Step 1.6

In the piece where $5 s - 10$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 s - 10) \geq 0$

Step 1.7

Write as a piecewise.

${\begin{cases} 5 s - 10 \geq 0 & s \geq 2 \\ - (5 s - 10) \geq 0 & s < 2 \end{cases}$

Step 1.8

Simplify $- (5 s - 10) \geq 0$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} 5 s - 10 \geq 0 & s \geq 2 \\ - (5 s) - - 10 \geq 0 & s < 2 \end{cases}$

Step 1.8.2

Multiply $5$ by $- 1$ .

${\begin{cases} 5 s - 10 \geq 0 & s \geq 2 \\ - 5 s - - 10 \geq 0 & s < 2 \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 10$ .

${\begin{cases} 5 s - 10 \geq 0 & s \geq 2 \\ - 5 s + 10 \geq 0 & s < 2 \end{cases}$

Step 2

Solve $5 s - 10 \geq 0$ for $s$ .

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Step 2.1

Add $10$ to both sides of the inequality.

$5 s \geq 10$

Step 2.2

Divide each term in $5 s \geq 10$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 s \geq 10$ by $5$ .

$\frac{5 s}{5} \geq \frac{10}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 s}{5} \geq \frac{10}{5}$

Step 2.2.2.1.2

Divide $s$ by $1$ .

$s \geq \frac{10}{5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $10$ by $5$ .

$s \geq 2$

Step 3

Solve $- 5 s + 10 \geq 0$ when $s < 2$ .

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Step 3.1

Solve $- 5 s + 10 \geq 0$ for $s$ .

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Step 3.1.1

Subtract $10$ from both sides of the inequality.

$- 5 s \geq - 10$

Step 3.1.2

Divide each term in $- 5 s \geq - 10$ by $- 5$ and simplify.

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Step 3.1.2.1

Divide each term in $- 5 s \geq - 10$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 s}{- 5} \leq \frac{- 10}{- 5}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 5 s}{- 5} \leq \frac{- 10}{- 5}$

Step 3.1.2.2.1.2

Divide $s$ by $1$ .

$s \leq \frac{- 10}{- 5}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 10$ by $- 5$ .

$s \leq 2$

Step 3.2

Find the intersection of $s \leq 2$ and $s < 2$ .

$s < 2$

Step 4

Find the union of the solutions for any value of $s$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 6