Solve for x |x+3|+5<10

$| x + 3 | + 5 < 10$

Step 1

Write $| x + 3 | + 5 < 10$ as a piecewise.

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

$x + 3 \geq 0$

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

In the piece where $x + 3$ is non-negative, remove the absolute value.

$x + 3 + 5 < 10$

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

$x + 3 < 0$

Subtract $3$ from both sides of the inequality.

$x < - 3$

In the piece where $x + 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 3) + 5 < 10$

Write as a piecewise.

${\begin{cases} x + 3 + 5 < 10 & x \geq - 3 \\ - (x + 3) + 5 < 10 & x < - 3 \end{cases}$

Add $3$ and $5$ .

${\begin{cases} x + 8 < 10 & x \geq - 3 \\ - (x + 3) + 5 < 10 & x < - 3 \end{cases}$

Simplify $- (x + 3) + 5 < 10$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x + 8 < 10 & x \geq - 3 \\ - x - 1 \cdot 3 + 5 < 10 & x < - 3 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} x + 8 < 10 & x \geq - 3 \\ - x - 3 + 5 < 10 & x < - 3 \end{cases}$

Add $- 3$ and $5$ .

${\begin{cases} x + 8 < 10 & x \geq - 3 \\ - x + 2 < 10 & x < - 3 \end{cases}$

Step 2

Solve $x + 8 < 10$ when $x \geq - 3$ .

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

Subtract $8$ from both sides of the inequality.

$x < 10 - 8$

Subtract $8$ from $10$ .

$x < 2$

Find the intersection of $x < 2$ and $x \geq - 3$ .

$- 3 \leq x < 2$

Step 3

Solve $- x + 2 < 10$ when $x < - 3$ .

Solve $- x + 2 < 10$ for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- x < 10 - 2$

Subtract $2$ from $10$ .

$- x < 8$

Divide each term in $- x < 8$ by $- 1$ and simplify.

Divide each term in $- x < 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{8}{- 1}$

Divide $x$ by $1$ .

$x > \frac{8}{- 1}$

Simplify the right side.

Divide $8$ by $- 1$ .

$x > - 8$

Find the intersection of $x > - 8$ and $x < - 3$ .

$- 8 < x < - 3$

Step 4

Find the union of the solutions.

$- 8 < x < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < 2$

Interval Notation:

$(- 8, 2)$

Step 6

Solve for x |x+3|+5<10

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