Algebra Examples

$| y | \geq x$

Step 1

Write $| y | \geq x$ 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.

$y \geq 0$

Step 1.2

In the piece where $y$ is non-negative, remove the absolute value.

$y \geq x$

Step 1.3

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

$y < 0$

Step 1.4

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

$- y \geq x$

Step 1.5

Write as a piecewise.

${\begin{cases} y \geq x & y \geq 0 \\ - y \geq x & y < 0 \end{cases}$

Step 2

Find the intersection of $y \geq x$ and $y \geq 0$ .

$y \geq x$ and $y \geq 0$

Step 3

Solve $- y \geq x$ when $y < 0$ .

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Step 3.1

Divide each term in $- y \geq x$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- y \geq x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \leq \frac{x}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \leq \frac{x}{- 1}$

Step 3.1.2.2

Divide $y$ by $1$ .

$y \leq \frac{x}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y \leq - 1 \cdot x$

Step 3.1.3.2

Rewrite $- 1 \cdot x$ as $- x$ .

$y \leq - x$

Step 3.2

Find the intersection of $y \leq - x$ and $y < 0$ .

$y < N o (Min i m u m)$

Step 4

Find the union of the solutions.

$y \geq N o Max i m u m$ or $y < N o Min i m u m$

Step 5