Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

t = \frac{2}{7}, - \frac{4}{25}

t=\frac{2}{7} , -\frac{4}{25}

Decimal form:

t = 0.286, - 0.16

t=0.286 , -0.16

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$3 | 3 t + 1 | = | 16 t + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 3 t + 1 \| = \| 16 t + 1 \|$
$x = + y$	$3 (3 t + 1) = (16 t + 1)$
$x = - y$	$3 (3 t + 1) = - (16 t + 1)$
$+ x = y$	$3 (3 t + 1) = (16 t + 1)$
$- x = y$	$3 (- (3 t + 1)) = (16 t + 1)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$3 \| 3 t + 1 \| = \| 16 t + 1 \|$
$x = + y$ , $+ x = y$	$3 (3 t + 1) = (16 t + 1)$
$x = - y$ , $- x = y$	$3 (3 t + 1) = - (16 t + 1)$

2. Solve the two equations for t

14 additional steps

$3 \cdot (3 t + 1) = (16 t + 1)$

Expand the parentheses:

$3 \cdot 3 t + 3 \cdot 1 = (16 t + 1)$

Multiply the coefficients:

$9 t + 3 \cdot 1 = (16 t + 1)$

Simplify the arithmetic:

$9 t + 3 = (16 t + 1)$

Subtract from both sides:

$(9 t + 3) - 16 t = (16 t + 1) - 16 t$

Group like terms:

$(9 t - 16 t) + 3 = (16 t + 1) - 16 t$

Simplify the arithmetic:

$- 7 t + 3 = (16 t + 1) - 16 t$

Group like terms:

$- 7 t + 3 = (16 t - 16 t) + 1$

Simplify the arithmetic:

$- 7 t + 3 = 1$

Subtract from both sides:

$(- 7 t + 3) - 3 = 1 - 3$

Simplify the arithmetic:

$- 7 t = 1 - 3$

Simplify the arithmetic:

$- 7 t = - 2$

Divide both sides by :

$\frac{(- 7 t)}{- 7} = \frac{- 2}{- 7}$

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

$t = \frac{2}{7}$

13 additional steps

$3 \cdot (3 t + 1) = - (16 t + 1)$

Expand the parentheses:

$3 \cdot 3 t + 3 \cdot 1 = - (16 t + 1)$

Multiply the coefficients:

$9 t + 3 \cdot 1 = - (16 t + 1)$

Simplify the arithmetic:

$9 t + 3 = - (16 t + 1)$

Expand the parentheses:

$9 t + 3 = - 16 t - 1$

Add to both sides:

$(9 t + 3) + 16 t = (- 16 t - 1) + 16 t$

Group like terms:

$(9 t + 16 t) + 3 = (- 16 t - 1) + 16 t$

Simplify the arithmetic:

$25 t + 3 = (- 16 t - 1) + 16 t$

Group like terms:

$25 t + 3 = (- 16 t + 16 t) - 1$

Simplify the arithmetic:

$25 t + 3 = - 1$

Subtract from both sides:

$(25 t + 3) - 3 = - 1 - 3$

Simplify the arithmetic:

$25 t = - 1 - 3$

Simplify the arithmetic:

$25 t = - 4$

Divide both sides by :

$\frac{(25 t)}{25} = \frac{- 4}{25}$

Simplify the fraction:

$t = \frac{- 4}{25}$

3. List the solutions

$t = \frac{2}{7}, - \frac{4}{25}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 3 t + 1 |$
$y = | 16 t + 1 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for t

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for t

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved