Solve for t 3|5t-1|+9=23

$3 | 5 t - 1 | + 9 = 23$

Step 1

Move all terms not containing $| 5 t - 1 |$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$3 | 5 t - 1 | = 23 - 9$

Subtract $9$ from $23$ .

$3 | 5 t - 1 | = 14$

Step 2

Divide each term in $3 | 5 t - 1 | = 14$ by $3$ and simplify.

Divide each term in $3 | 5 t - 1 | = 14$ by $3$ .

$\frac{3 | 5 t - 1 |}{3} = \frac{14}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 | 5 t - 1 |}{3} = \frac{14}{3}$

Divide $| 5 t - 1 |$ by $1$ .

$| 5 t - 1 | = \frac{14}{3}$

Step 3

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

$5 t - 1 = \pm \frac{14}{3}$

Step 4

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.

$5 t - 1 = \frac{14}{3}$

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

Add $1$ to both sides of the equation.

$5 t = \frac{14}{3} + 1$

Write $1$ as a fraction with a common denominator.

$5 t = \frac{14}{3} + \frac{3}{3}$

Combine the numerators over the common denominator.

$5 t = \frac{14 + 3}{3}$

Add $14$ and $3$ .

$5 t = \frac{17}{3}$

Divide each term in $5 t = \frac{17}{3}$ by $5$ and simplify.

Divide each term in $5 t = \frac{17}{3}$ by $5$ .

$\frac{5 t}{5} = \frac{\frac{17}{3}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 t}{5} = \frac{\frac{17}{3}}{5}$

Divide $t$ by $1$ .

$t = \frac{\frac{17}{3}}{5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{17}{3} \cdot \frac{1}{5}$

Multiply $\frac{17}{3} \cdot \frac{1}{5}$ .

Multiply $\frac{17}{3}$ by $\frac{1}{5}$ .

$t = \frac{17}{3 \cdot 5}$

Multiply $3$ by $5$ .

$t = \frac{17}{15}$

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

$5 t - 1 = - \frac{14}{3}$

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

Add $1$ to both sides of the equation.

$5 t = - \frac{14}{3} + 1$

Write $1$ as a fraction with a common denominator.

$5 t = - \frac{14}{3} + \frac{3}{3}$

Combine the numerators over the common denominator.

$5 t = \frac{- 14 + 3}{3}$

Add $- 14$ and $3$ .

$5 t = \frac{- 11}{3}$

Move the negative in front of the fraction.

$5 t = - \frac{11}{3}$

Divide each term in $5 t = - \frac{11}{3}$ by $5$ and simplify.

Divide each term in $5 t = - \frac{11}{3}$ by $5$ .

$\frac{5 t}{5} = \frac{- \frac{11}{3}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 t}{5} = \frac{- \frac{11}{3}}{5}$

Divide $t$ by $1$ .

$t = \frac{- \frac{11}{3}}{5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$t = - \frac{11}{3} \cdot \frac{1}{5}$

Multiply $- \frac{11}{3} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{5}$ by $\frac{11}{3}$ .

$t = - \frac{11}{5 \cdot 3}$

Multiply $5$ by $3$ .

$t = - \frac{11}{15}$

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

$t = \frac{17}{15}, - \frac{11}{15}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{17}{15}, - \frac{11}{15}$

Decimal Form:

$t = 1.1\overline{3}, - 0.7\overline{3}$

Mixed Number Form:

$t = 1 \frac{2}{15}, - \frac{11}{15}$

Solve for t 3|5t-1|+9=23

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