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Solution - Absolute value equations

Exact form:

t = 2, 6

t=2 , 6

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
$| - 3 t + 6 | = 3 | t - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 t + 6 \| = 3 \| t - 2 \|$
$x = + y$	$(- 3 t + 6) = 3 (t - 2)$
$x = - y$	$(- 3 t + 6) = 3 (- (t - 2))$
$+ x = y$	$(- 3 t + 6) = 3 (t - 2)$
$- x = y$	$- (- 3 t + 6) = 3 (t - 2)$

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:

$\| x \| = \| y \|$	$\| - 3 t + 6 \| = 3 \| t - 2 \|$
$x = + y$ , $+ x = y$	$(- 3 t + 6) = 3 (t - 2)$
$x = - y$ , $- x = y$	$(- 3 t + 6) = 3 (- (t - 2))$

2. Solve the two equations for t

15 additional steps

$(- 3 t + 6) = 3 \cdot (t - 2)$

Expand the parentheses:

$(- 3 t + 6) = 3 t + 3 \cdot - 2$

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 6 t + 6 = - 6$

Subtract from both sides:

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

Simplify the arithmetic:

$- 6 t = - 6 - 6$

Simplify the arithmetic:

$- 6 t = - 12$

Divide both sides by :

$\frac{(- 6 t)}{- 6} = \frac{- 12}{- 6}$

Cancel out the negatives:

$\frac{6 t}{6} = \frac{- 12}{- 6}$

Simplify the fraction:

$t = \frac{- 12}{- 6}$

Cancel out the negatives:

$t = \frac{12}{6}$

Find the greatest common factor of the numerator and denominator:

$t = \frac{(2 \cdot 6)}{(1 \cdot 6)}$

Factor out and cancel the greatest common factor:

$t = 2$

9 additional steps

$(- 3 t + 6) = 3 \cdot (- (t - 2))$

Expand the parentheses:

$(- 3 t + 6) = 3 \cdot (- t + 2)$

$(- 3 t + 6) = 3 \cdot - t + 3 \cdot 2$

Group like terms:

$(- 3 t + 6) = (3 \cdot - 1) t + 3 \cdot 2$

Multiply the coefficients:

$(- 3 t + 6) = - 3 t + 3 \cdot 2$

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 = 6$

3. List the solutions

$t = 2, 6$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 t + 6 |$
$y = 3 | t - 2 |$
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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