Solve the Absolute Value Inequality for a |6a-5|=4a

$| 6 a - 5 | = 4 a$

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

$6 a - 5 = \pm 4 a$

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.

$6 a - 5 = 4 a$

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

Subtract $4 a$ from both sides of the equation.

$6 a - 5 - 4 a = 0$

Subtract $4 a$ from $6 a$ .

$2 a - 5 = 0$

Add $5$ to both sides of the equation.

$2 a = 5$

Divide each term in $2 a = 5$ by $2$ and simplify.

Divide each term in $2 a = 5$ by $2$ .

$\frac{2 a}{2} = \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{5}{2}$

Divide $a$ by $1$ .

$a = \frac{5}{2}$

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

$6 a - 5 = - 4 a$

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

Add $4 a$ to both sides of the equation.

$6 a - 5 + 4 a = 0$

Add $6 a$ and $4 a$ .

$10 a - 5 = 0$

Add $5$ to both sides of the equation.

$10 a = 5$

Divide each term in $10 a = 5$ by $10$ and simplify.

Divide each term in $10 a = 5$ by $10$ .

$\frac{10 a}{10} = \frac{5}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 a}{10} = \frac{5}{10}$

Divide $a$ by $1$ .

$a = \frac{5}{10}$

Simplify the right side.

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5$ .

$a = \frac{5 (1)}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$a = \frac{5 \cdot 1}{5 \cdot 2}$

Cancel the common factor.

$a = \frac{5 \cdot 1}{5 \cdot 2}$

Rewrite the expression.

$a = \frac{1}{2}$

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

$a = \frac{5}{2}, \frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{5}{2}, \frac{1}{2}$

Decimal Form:

$a = 2.5, 0.5$

Mixed Number Form:

$a = 2 \frac{1}{2}, \frac{1}{2}$

Solve the Absolute Value Inequality for a |6a-5|=4a

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