Solve the Inequality for b 3b+2<5b-6<=2b+9

$3 b + 2 < 5 b - 6 \leq 2 b + 9$

Step 1

Solve $3 b + 2 < 5 b - 6$ for $b$ .

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

Subtract $5 b$ from both sides of the inequality.

$3 b + 2 - 5 b < - 6$

Subtract $5 b$ from $3 b$ .

$- 2 b + 2 < - 6$

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

Subtract $2$ from both sides of the inequality.

$- 2 b < - 6 - 2$

Subtract $2$ from $- 6$ .

$- 2 b < - 8$

Divide each term in $- 2 b < - 8$ by $- 2$ and simplify.

Divide each term in $- 2 b < - 8$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 b}{- 2} > \frac{- 8}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} > \frac{- 8}{- 2}$

Divide $b$ by $1$ .

$b > \frac{- 8}{- 2}$

Simplify the right side.

Divide $- 8$ by $- 2$ .

$b > 4$

Step 2

Solve $5 b - 6 \leq 2 b + 9$ for $b$ .

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

Subtract $2 b$ from both sides of the inequality.

$5 b - 6 - 2 b \leq 9$

Subtract $2 b$ from $5 b$ .

$3 b - 6 \leq 9$

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

Add $6$ to both sides of the inequality.

$3 b \leq 9 + 6$

Add $9$ and $6$ .

$3 b \leq 15$

Divide each term in $3 b \leq 15$ by $3$ and simplify.

Divide each term in $3 b \leq 15$ by $3$ .

$\frac{3 b}{3} \leq \frac{15}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 b}{3} \leq \frac{15}{3}$

Divide $b$ by $1$ .

$b \leq \frac{15}{3}$

Simplify the right side.

Divide $15$ by $3$ .

$b \leq 5$

Step 3

Find the intersection of $b > 4$ and $b \leq 5$ .

$4 < b \leq 5$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$4 < b \leq 5$

Interval Notation:

$(4, 5]$

Step 5

Solve the Inequality for b 3b+2<5b-6<=2b+9

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