Solve the Absolute Value Inequality for x |x+7|<=4

$| x + 7 | \leq 4$

Write $| x + 7 | \leq 4$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x + 7 \geq 0$

Subtract $7$ from both sides of the inequality.

$x \geq - 7$

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

$x + 7 \leq 4$

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

$x + 7 < 0$

Subtract $7$ from both sides of the inequality.

$x < - 7$

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

$- (x + 7) \leq 4$

Write as a piecewise.

${\begin{cases} x + 7 \leq 4 & x \geq - 7 \\ - (x + 7) \leq 4 & x < - 7 \end{cases}$

Simplify $- (x + 7) \leq 4$ .

Apply the distributive property.

${\begin{cases} x + 7 \leq 4 & x \geq - 7 \\ - x - 1 \cdot 7 \leq 4 & x < - 7 \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} x + 7 \leq 4 & x \geq - 7 \\ - x - 7 \leq 4 & x < - 7 \end{cases}$

Solve $x + 7 \leq 4$ when $x \geq - 7$ .

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

Subtract $7$ from both sides of the inequality.

$x \leq 4 - 7$

Subtract $7$ from $4$ .

$x \leq - 3$

Find the intersection of $x \leq - 3$ and $x \geq - 7$ .

$- 7 \leq x \leq - 3$

Solve $- x - 7 \leq 4$ when $x < - 7$ .

Solve $- x - 7 \leq 4$ for $x$ .

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

Add $7$ to both sides of the inequality.

$- x \leq 4 + 7$

Add $4$ and $7$ .

$- x \leq 11$

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $11$ by $- 1$ .

$x \geq - 11$

Find the intersection of $x \geq - 11$ and $x < - 7$ .

$- 11 \leq x < - 7$

Find the union of the solutions.

$- 11 \leq x \leq - 3$

The result can be shown in multiple forms.

Inequality Form:

$- 11 \leq x \leq - 3$

Interval Notation:

$[- 11, - 3]$

Solve the Absolute Value Inequality for x |x+7|<=4

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