Solve the Inequality for x 5|x+5|+2>=42

$5 | x + 5 | + 2 \geq 42$

Step 1

Write $5 | x + 5 | + 2 \geq 42$ as a piecewise.

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

$x + 5 \geq 0$

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

In the piece where $x + 5$ is non-negative, remove the absolute value.

$5 (x + 5) + 2 \geq 42$

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

$x + 5 < 0$

Subtract $5$ from both sides of the inequality.

$x < - 5$

In the piece where $x + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$5 (- (x + 5)) + 2 \geq 42$

Write as a piecewise.

${\begin{cases} 5 (x + 5) + 2 \geq 42 & x \geq - 5 \\ 5 (- (x + 5)) + 2 \geq 42 & x < - 5 \end{cases}$

Simplify $5 (x + 5) + 2 \geq 42$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x + 5 \cdot 5 + 2 \geq 42 & x \geq - 5 \\ 5 (- (x + 5)) + 2 \geq 42 & x < - 5 \end{cases}$

Multiply $5$ by $5$ .

${\begin{cases} 5 x + 25 + 2 \geq 42 & x \geq - 5 \\ 5 (- (x + 5)) + 2 \geq 42 & x < - 5 \end{cases}$

Add $25$ and $2$ .

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ 5 (- (x + 5)) + 2 \geq 42 & x < - 5 \end{cases}$

Simplify $5 (- (x + 5)) + 2 \geq 42$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ 5 (- x - 1 \cdot 5) + 2 \geq 42 & x < - 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ 5 (- x - 5) + 2 \geq 42 & x < - 5 \end{cases}$

Apply the distributive property.

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ 5 (- x) + 5 \cdot - 5 + 2 \geq 42 & x < - 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ - 5 x + 5 \cdot - 5 + 2 \geq 42 & x < - 5 \end{cases}$

Multiply $5$ by $- 5$ .

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ - 5 x - 25 + 2 \geq 42 & x < - 5 \end{cases}$

Add $- 25$ and $2$ .

${\begin{cases} 5 x + 27 \geq 42 & x \geq - 5 \\ - 5 x - 23 \geq 42 & x < - 5 \end{cases}$

Step 2

Solve $5 x + 27 \geq 42$ for $x$ .

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

Subtract $27$ from both sides of the inequality.

$5 x \geq 42 - 27$

Subtract $27$ from $42$ .

$5 x \geq 15$

Divide each term in $5 x \geq 15$ by $5$ and simplify.

Divide each term in $5 x \geq 15$ by $5$ .

$\frac{5 x}{5} \geq \frac{15}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{15}{5}$

Divide $x$ by $1$ .

$x \geq \frac{15}{5}$

Simplify the right side.

Divide $15$ by $5$ .

$x \geq 3$

Step 3

Solve $- 5 x - 23 \geq 42$ for $x$ .

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

Add $23$ to both sides of the inequality.

$- 5 x \geq 42 + 23$

Add $42$ and $23$ .

$- 5 x \geq 65$

Divide each term in $- 5 x \geq 65$ by $- 5$ and simplify.

Divide each term in $- 5 x \geq 65$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \leq \frac{65}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} \leq \frac{65}{- 5}$

Divide $x$ by $1$ .

$x \leq \frac{65}{- 5}$

Simplify the right side.

Divide $65$ by $- 5$ .

$x \leq - 13$

Step 4

Find the union of the solutions.

$x \leq - 13$ or $x \geq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 13 or x \geq 3$

Interval Notation:

$(- \infty, - 13] \cup [3, \infty)$

Step 6

Solve the Inequality for x 5|x+5|+2>=42

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