Solve the Inequality for x 4|x+5|+1<=33

$4 | x + 5 | + 1 \leq 33$

Step 1

Write $4 | x + 5 | + 1 \leq 33$ 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.

$4 (x + 5) + 1 \leq 33$

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 $- 1$ .

$4 (- (x + 5)) + 1 \leq 33$

Write as a piecewise.

${\begin{cases} 4 (x + 5) + 1 \leq 33 & x \geq - 5 \\ 4 (- (x + 5)) + 1 \leq 33 & x < - 5 \end{cases}$

Simplify $4 (x + 5) + 1 \leq 33$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 x + 4 \cdot 5 + 1 \leq 33 & x \geq - 5 \\ 4 (- (x + 5)) + 1 \leq 33 & x < - 5 \end{cases}$

Multiply $4$ by $5$ .

${\begin{cases} 4 x + 20 + 1 \leq 33 & x \geq - 5 \\ 4 (- (x + 5)) + 1 \leq 33 & x < - 5 \end{cases}$

Add $20$ and $1$ .

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ 4 (- (x + 5)) + 1 \leq 33 & x < - 5 \end{cases}$

Simplify $4 (- (x + 5)) + 1 \leq 33$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ 4 (- x - 1 \cdot 5) + 1 \leq 33 & x < - 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ 4 (- x - 5) + 1 \leq 33 & x < - 5 \end{cases}$

Apply the distributive property.

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ 4 (- x) + 4 \cdot - 5 + 1 \leq 33 & x < - 5 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ - 4 x + 4 \cdot - 5 + 1 \leq 33 & x < - 5 \end{cases}$

Multiply $4$ by $- 5$ .

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ - 4 x - 20 + 1 \leq 33 & x < - 5 \end{cases}$

Add $- 20$ and $1$ .

${\begin{cases} 4 x + 21 \leq 33 & x \geq - 5 \\ - 4 x - 19 \leq 33 & x < - 5 \end{cases}$

Step 2

Solve $4 x + 21 \leq 33$ when $x \geq - 5$ .

Solve $4 x + 21 \leq 33$ for $x$ .

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

Subtract $21$ from both sides of the inequality.

$4 x \leq 33 - 21$

Subtract $21$ from $33$ .

$4 x \leq 12$

Divide each term in $4 x \leq 12$ by $4$ and simplify.

Divide each term in $4 x \leq 12$ by $4$ .

$\frac{4 x}{4} \leq \frac{12}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \leq \frac{12}{4}$

Divide $x$ by $1$ .

$x \leq \frac{12}{4}$

Simplify the right side.

Divide $12$ by $4$ .

$x \leq 3$

Find the intersection of $x \leq 3$ and $x \geq - 5$ .

$- 5 \leq x \leq 3$

Step 3

Solve $- 4 x - 19 \leq 33$ when $x < - 5$ .

Solve $- 4 x - 19 \leq 33$ for $x$ .

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

Add $19$ to both sides of the inequality.

$- 4 x \leq 33 + 19$

Add $33$ and $19$ .

$- 4 x \leq 52$

Divide each term in $- 4 x \leq 52$ by $- 4$ and simplify.

Divide each term in $- 4 x \leq 52$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \geq \frac{52}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \geq \frac{52}{- 4}$

Divide $x$ by $1$ .

$x \geq \frac{52}{- 4}$

Simplify the right side.

Divide $52$ by $- 4$ .

$x \geq - 13$

Find the intersection of $x \geq - 13$ and $x < - 5$ .

$- 13 \leq x < - 5$

Step 4

Find the union of the solutions.

$- 13 \leq x \leq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 13 \leq x \leq 3$

Interval Notation:

$[- 13, 3]$

Step 6

Solve the Inequality for x 4|x+5|+1<=33

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