Algebra Examples

$| x + 1 | \leq 4$

Step 1

Write $| x + 1 | \leq 4$ 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.

$x + 1 \geq 0$

Step 1.2

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

Step 1.3

In the piece where $x + 1$ is non-negative, remove the absolute value.

$x + 1 \leq 4$

Step 1.4

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

$x + 1 < 0$

Step 1.5

Subtract $1$ from both sides of the inequality.

$x < - 1$

Step 1.6

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

$- (x + 1) \leq 4$

Step 1.7

Write as a piecewise.

${\begin{cases} x + 1 \leq 4 & x \geq - 1 \\ - (x + 1) \leq 4 & x < - 1 \end{cases}$

Step 1.8

Simplify $- (x + 1) \leq 4$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} x + 1 \leq 4 & x \geq - 1 \\ - x - 1 \cdot 1 \leq 4 & x < - 1 \end{cases}$

Step 1.8.2

Multiply $- 1$ by $1$ .

${\begin{cases} x + 1 \leq 4 & x \geq - 1 \\ - x - 1 \leq 4 & x < - 1 \end{cases}$

Step 2

Solve $x + 1 \leq 4$ when $x \geq - 1$ .

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Step 2.1

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

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Step 2.1.1

Subtract $1$ from both sides of the inequality.

$x \leq 4 - 1$

Step 2.1.2

Subtract $1$ from $4$ .

$x \leq 3$

Step 2.2

Find the intersection of $x \leq 3$ and $x \geq - 1$ .

$- 1 \leq x \leq 3$

Step 3

Solve $- x - 1 \leq 4$ when $x < - 1$ .

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Step 3.1

Solve $- x - 1 \leq 4$ 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 $1$ to both sides of the inequality.

$- x \leq 4 + 1$

Step 3.1.1.2

Add $4$ and $1$ .

$- x \leq 5$

Step 3.1.2

Divide each term in $- x \leq 5$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- x \leq 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{5}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{5}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x \geq \frac{5}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $5$ by $- 1$ .

$x \geq - 5$

Step 3.2

Find the intersection of $x \geq - 5$ and $x < - 1$ .

$- 5 \leq x < - 1$

Step 4

Find the union of the solutions.

$- 5 \leq x \leq 3$

Step 5

Convert the inequality to interval notation.

$[- 5, 3]$

Step 6