Solve the Inequality for x x+1<-2x+11<3x+5

$x + 1 < - 2 x + 11 < 3 x + 5$

Step 1

Solve $x + 1 < - 2 x + 11$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Add $2 x$ to both sides of the inequality.

$x + 1 + 2 x < 11$

Add $x$ and $2 x$ .

$3 x + 1 < 11$

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

Subtract $1$ from both sides of the inequality.

$3 x < 11 - 1$

Subtract $1$ from $11$ .

$3 x < 10$

Divide each term in $3 x < 10$ by $3$ and simplify.

Divide each term in $3 x < 10$ by $3$ .

$\frac{3 x}{3} < \frac{10}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{10}{3}$

Divide $x$ by $1$ .

$x < \frac{10}{3}$

Step 2

Solve $- 2 x + 11 < 3 x + 5$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Subtract $3 x$ from both sides of the inequality.

$- 2 x + 11 - 3 x < 5$

Subtract $3 x$ from $- 2 x$ .

$- 5 x + 11 < 5$

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

Subtract $11$ from both sides of the inequality.

$- 5 x < 5 - 11$

Subtract $11$ from $5$ .

$- 5 x < - 6$

Divide each term in $- 5 x < - 6$ by $- 5$ and simplify.

Divide each term in $- 5 x < - 6$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{- 6}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{- 6}{- 5}$

Divide $x$ by $1$ .

$x > \frac{- 6}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{6}{5}$

Step 3

Find the intersection of $x < \frac{10}{3}$ and $x > \frac{6}{5}$ .

$\frac{6}{5} < x < \frac{10}{3}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$\frac{6}{5} < x < \frac{10}{3}$

Interval Notation:

$(\frac{6}{5}, \frac{10}{3})$

Step 5

Solve the Inequality for x x+1<-2x+11<3x+5

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