Solve the Inequality for x 3|x-7|<=27

$3 | x - 7 | \leq 27$

Step 1

Divide each term in $3 | x - 7 | \leq 27$ by $3$ and simplify.

Divide each term in $3 | x - 7 | \leq 27$ by $3$ .

$\frac{3 | x - 7 |}{3} \leq \frac{27}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 | x - 7 |}{3} \leq \frac{27}{3}$

Divide $| x - 7 |$ by $1$ .

$| x - 7 | \leq \frac{27}{3}$

Simplify the right side.

Divide $27$ by $3$ .

$| x - 7 | \leq 9$

Step 2

Write $| x - 7 | \leq 9$ as a piecewise.

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

$x - 7 \geq 0$

Add $7$ to both sides of the inequality.

$x \geq 7$

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

$x - 7 \leq 9$

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

$x - 7 < 0$

Add $7$ to both sides of the inequality.

$x < 7$

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

$- (x - 7) \leq 9$

Write as a piecewise.

${\begin{cases} x - 7 \leq 9 & x \geq 7 \\ - (x - 7) \leq 9 & x < 7 \end{cases}$

Simplify $- (x - 7) \leq 9$ .

Apply the distributive property.

${\begin{cases} x - 7 \leq 9 & x \geq 7 \\ - x - - 7 \leq 9 & x < 7 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} x - 7 \leq 9 & x \geq 7 \\ - x + 7 \leq 9 & x < 7 \end{cases}$

Step 3

Solve $x - 7 \leq 9$ when $x \geq 7$ .

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

Add $7$ to both sides of the inequality.

$x \leq 9 + 7$

Add $9$ and $7$ .

$x \leq 16$

Find the intersection of $x \leq 16$ and $x \geq 7$ .

$7 \leq x \leq 16$

Step 4

Solve $- x + 7 \leq 9$ when $x < 7$ .

Solve $- x + 7 \leq 9$ for $x$ .

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

Subtract $7$ from both sides of the inequality.

$- x \leq 9 - 7$

Subtract $7$ from $9$ .

$- x \leq 2$

Divide each term in $- x \leq 2$ by $- 1$ and simplify.

Divide each term in $- x \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{2}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$x \geq - 2$

Find the intersection of $x \geq - 2$ and $x < 7$ .

$- 2 \leq x < 7$

Step 5

Find the union of the solutions.

$- 2 \leq x \leq 16$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x \leq 16$

Interval Notation:

$[- 2, 16]$

Step 7

Solve the Inequality for x 3|x-7|<=27

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