Solve the Absolute Value Inequality for b |2b-9|=|b-6|

$| 2 b - 9 | = | b - 6 |$

Step 1

Rewrite the equation as $| b - 6 | = | 2 b - 9 |$ .

$| b - 6 | = | 2 b - 9 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$b - 6 = 2 b - 9$

$b - 6 = - (2 b - 9)$

$- (b - 6) = 2 b - 9$

$- (b - 6) = - (2 b - 9)$

Step 3

After simplifying, there are only two unique equations to be solved.

$b - 6 = 2 b - 9$

$b - 6 = - (2 b - 9)$

Step 4

Solve $b - 6 = 2 b - 9$ for $b$ .

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

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

$b - 6 - 2 b = - 9$

Subtract $2 b$ from $b$ .

$- b - 6 = - 9$

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

Add $6$ to both sides of the equation.

$- b = - 9 + 6$

Add $- 9$ and $6$ .

$- b = - 3$

Divide each term in $- b = - 3$ by $- 1$ and simplify.

Divide each term in $- b = - 3$ by $- 1$ .

$\frac{- b}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{b}{1} = \frac{- 3}{- 1}$

Divide $b$ by $1$ .

$b = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$b = 3$

Step 5

Solve $b - 6 = - (2 b - 9)$ for $b$ .

Simplify $- (2 b - 9)$ .

Rewrite.

$b - 6 = 0 + 0 - (2 b - 9)$

Simplify by adding zeros.

$b - 6 = - (2 b - 9)$

Apply the distributive property.

$b - 6 = - (2 b) - - 9$

Multiply.

Multiply $2$ by $- 1$ .

$b - 6 = - 2 b - - 9$

Multiply $- 1$ by $- 9$ .

$b - 6 = - 2 b + 9$

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

Add $2 b$ to both sides of the equation.

$b - 6 + 2 b = 9$

Add $b$ and $2 b$ .

$3 b - 6 = 9$

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

Add $6$ to both sides of the equation.

$3 b = 9 + 6$

Add $9$ and $6$ .

$3 b = 15$

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

Divide each term in $3 b = 15$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $b$ by $1$ .

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

Simplify the right side.

Divide $15$ by $3$ .

$b = 5$

Step 6

List all of the solutions.

$b = 3, 5$

Solve the Absolute Value Inequality for b |2b-9|=|b-6|

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