Solve for t |2t-3|=3t-2

$| 2 t - 3 | = 3 t - 2$

Step 1

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

$2 t - 3 = \pm (3 t - 2)$

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.

$2 t - 3 = 3 t - 2$

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

Subtract $3 t$ from both sides of the equation.

$2 t - 3 - 3 t = - 2$

Subtract $3 t$ from $2 t$ .

$- t - 3 = - 2$

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

Add $3$ to both sides of the equation.

$- t = - 2 + 3$

Add $- 2$ and $3$ .

$- t = 1$

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

Divide each term in $- t = 1$ by $- 1$ .

$\frac{- t}{- 1} = \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{1}{- 1}$

Divide $t$ by $1$ .

$t = \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$t = - 1$

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

$2 t - 3 = - (3 t - 2)$

Simplify $- (3 t - 2)$ .

Rewrite.

$2 t - 3 = 0 + 0 - (3 t - 2)$

Simplify by adding zeros.

$2 t - 3 = - (3 t - 2)$

Apply the distributive property.

$2 t - 3 = - (3 t) - - 2$

Multiply.

Multiply $3$ by $- 1$ .

$2 t - 3 = - 3 t - - 2$

Multiply $- 1$ by $- 2$ .

$2 t - 3 = - 3 t + 2$

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

Add $3 t$ to both sides of the equation.

$2 t - 3 + 3 t = 2$

Add $2 t$ and $3 t$ .

$5 t - 3 = 2$

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

Add $3$ to both sides of the equation.

$5 t = 2 + 3$

Add $2$ and $3$ .

$5 t = 5$

Divide each term in $5 t = 5$ by $5$ and simplify.

Divide each term in $5 t = 5$ by $5$ .

$\frac{5 t}{5} = \frac{5}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 t}{5} = \frac{5}{5}$

Divide $t$ by $1$ .

$t = \frac{5}{5}$

Simplify the right side.

Divide $5$ by $5$ .

$t = 1$

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

$t = - 1, 1$

Step 3

Exclude the solutions that do not make $| 2 t - 3 | = 3 t - 2$ true.

$t = 1$

Solve for t |2t-3|=3t-2

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