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Solution - Absolute value equations

Exact form:

t = - \frac{2}{3}, - \frac{4}{15}

t=-\frac{2}{3} , -\frac{4}{15}

Decimal form:

t = - 0.667, - 0.267

t=-0.667 , -0.267

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
$3 | 3 t + 1 | = | 6 t + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 3 t + 1 \| = \| 6 t + 1 \|$
$x = + y$	$3 (3 t + 1) = (6 t + 1)$
$x = - y$	$3 (3 t + 1) = - (6 t + 1)$
$+ x = y$	$3 (3 t + 1) = (6 t + 1)$
$- x = y$	$3 (- (3 t + 1)) = (6 t + 1)$

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:

$\| x \| = \| y \|$	$3 \| 3 t + 1 \| = \| 6 t + 1 \|$
$x = + y$ , $+ x = y$	$3 (3 t + 1) = (6 t + 1)$
$x = - y$ , $- x = y$	$3 (3 t + 1) = - (6 t + 1)$

2. Solve the two equations for t

12 additional steps

$3 \cdot (3 t + 1) = (6 t + 1)$

Expand the parentheses:

$3 \cdot 3 t + 3 \cdot 1 = (6 t + 1)$

Multiply the coefficients:

$9 t + 3 \cdot 1 = (6 t + 1)$

Simplify the arithmetic:

$9 t + 3 = (6 t + 1)$

Subtract from both sides:

$(9 t + 3) - 6 t = (6 t + 1) - 6 t$

Group like terms:

$(9 t - 6 t) + 3 = (6 t + 1) - 6 t$

Simplify the arithmetic:

$3 t + 3 = (6 t + 1) - 6 t$

Group like terms:

$3 t + 3 = (6 t - 6 t) + 1$

Simplify the arithmetic:

$3 t + 3 = 1$

Subtract from both sides:

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

Simplify the arithmetic:

$3 t = 1 - 3$

Simplify the arithmetic:

$3 t = - 2$

Divide both sides by :

$\frac{(3 t)}{3} = \frac{- 2}{3}$

Simplify the fraction:

$t = \frac{- 2}{3}$

13 additional steps

$3 \cdot (3 t + 1) = - (6 t + 1)$

Expand the parentheses:

$3 \cdot 3 t + 3 \cdot 1 = - (6 t + 1)$

Multiply the coefficients:

$9 t + 3 \cdot 1 = - (6 t + 1)$

Simplify the arithmetic:

$9 t + 3 = - (6 t + 1)$

Expand the parentheses:

$9 t + 3 = - 6 t - 1$

Add to both sides:

$(9 t + 3) + 6 t = (- 6 t - 1) + 6 t$

Group like terms:

$(9 t + 6 t) + 3 = (- 6 t - 1) + 6 t$

Simplify the arithmetic:

$15 t + 3 = (- 6 t - 1) + 6 t$

Group like terms:

$15 t + 3 = (- 6 t + 6 t) - 1$

Simplify the arithmetic:

$15 t + 3 = - 1$

Subtract from both sides:

$(15 t + 3) - 3 = - 1 - 3$

Simplify the arithmetic:

$15 t = - 1 - 3$

Simplify the arithmetic:

$15 t = - 4$

Divide both sides by :

$\frac{(15 t)}{15} = \frac{- 4}{15}$

Simplify the fraction:

$t = \frac{- 4}{15}$

3. List the solutions

$t = - \frac{2}{3}, - \frac{4}{15}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 3 t + 1 |$
$y = | 6 t + 1 |$
The equation is true where the two lines cross.

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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

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