Algebra Examples

$5 | x + 9 | \leq 10$

Step 1

Divide each term in $5 | x + 9 | \leq 10$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 | x + 9 | \leq 10$ by $5$ .

$\frac{5 | x + 9 |}{5} \leq \frac{10}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 | x + 9 |}{5} \leq \frac{10}{5}$

Step 1.2.1.2

Divide $| x + 9 |$ by $1$ .

$| x + 9 | \leq \frac{10}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $10$ by $5$ .

$| x + 9 | \leq 2$

Step 2

Write $| x + 9 | \leq 2$ 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.

$x + 9 \geq 0$

Step 2.2

Subtract $9$ from both sides of the inequality.

$x \geq - 9$

Step 2.3

In the piece where $x + 9$ is non-negative, remove the absolute value.

$x + 9 \leq 2$

Step 2.4

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

$x + 9 < 0$

Step 2.5

Subtract $9$ from both sides of the inequality.

$x < - 9$

Step 2.6

In the piece where $x + 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 9) \leq 2$

Step 2.7

Write as a piecewise.

${\begin{cases} x + 9 \leq 2 & x \geq - 9 \\ - (x + 9) \leq 2 & x < - 9 \end{cases}$

Step 2.8

Simplify $- (x + 9) \leq 2$ .

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Step 2.8.1

Apply the distributive property.

${\begin{cases} x + 9 \leq 2 & x \geq - 9 \\ - x - 1 \cdot 9 \leq 2 & x < - 9 \end{cases}$

Step 2.8.2

Multiply $- 1$ by $9$ .

${\begin{cases} x + 9 \leq 2 & x \geq - 9 \\ - x - 9 \leq 2 & x < - 9 \end{cases}$

Step 3

Solve $x + 9 \leq 2$ when $x \geq - 9$ .

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Step 3.1

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

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Step 3.1.1

Subtract $9$ from both sides of the inequality.

$x \leq 2 - 9$

Step 3.1.2

Subtract $9$ from $2$ .

$x \leq - 7$

Step 3.2

Find the intersection of $x \leq - 7$ and $x \geq - 9$ .

$- 9 \leq x \leq - 7$

Step 4

Solve $- x - 9 \leq 2$ when $x < - 9$ .

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Step 4.1

Solve $- x - 9 \leq 2$ 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

Add $9$ to both sides of the inequality.

$- x \leq 2 + 9$

Step 4.1.1.2

Add $2$ and $9$ .

$- x \leq 11$

Step 4.1.2

Divide each term in $- x \leq 11$ by $- 1$ and simplify.

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Step 4.1.2.1

Divide each term in $- x \leq 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{11}{- 1}$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{11}{- 1}$

Step 4.1.2.2.2

Divide $x$ by $1$ .

$x \geq \frac{11}{- 1}$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $11$ by $- 1$ .

$x \geq - 11$

Step 4.2

Find the intersection of $x \geq - 11$ and $x < - 9$ .

$- 11 \leq x < - 9$

Step 5

Find the union of the solutions.

$- 11 \leq x \leq - 7$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$- 11 \leq x \leq - 7$

Interval Notation:

$[- 11, - 7]$

Step 7