Solve the Absolute Value Inequality for b |4b-5|=19

$| 4 b - 5 | = 19$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$4 b - 5 = \pm 19$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$4 b - 5 = 19$

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

Add $5$ to both sides of the equation.

$4 b = 19 + 5$

Add $19$ and $5$ .

$4 b = 24$

Divide each term in $4 b = 24$ by $4$ and simplify.

Divide each term in $4 b = 24$ by $4$ .

$\frac{4 b}{4} = \frac{24}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 b}{4} = \frac{24}{4}$

Divide $b$ by $1$ .

$b = \frac{24}{4}$

Simplify the right side.

Divide $24$ by $4$ .

$b = 6$

Next, use the negative value of the $\pm$ to find the second solution.

$4 b - 5 = - 19$

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

Add $5$ to both sides of the equation.

$4 b = - 19 + 5$

Add $- 19$ and $5$ .

$4 b = - 14$

Divide each term in $4 b = - 14$ by $4$ and simplify.

Divide each term in $4 b = - 14$ by $4$ .

$\frac{4 b}{4} = \frac{- 14}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 b}{4} = \frac{- 14}{4}$

Divide $b$ by $1$ .

$b = \frac{- 14}{4}$

Simplify the right side.

Cancel the common factor of $- 14$ and $4$ .

Factor $2$ out of $- 14$ .

$b = \frac{2 (- 7)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$b = \frac{2 \cdot - 7}{2 \cdot 2}$

Cancel the common factor.

$b = \frac{2 \cdot - 7}{2 \cdot 2}$

Rewrite the expression.

$b = \frac{- 7}{2}$

Move the negative in front of the fraction.

$b = - \frac{7}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$b = 6, - \frac{7}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$b = 6, - \frac{7}{2}$

Decimal Form:

$b = 6, - 3.5$

Mixed Number Form:

$b = 6, - 3 \frac{1}{2}$

Solve the Absolute Value Inequality for b |4b-5|=19

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