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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

t = 0.667, - 2

t=0.667 , -2

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

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

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:

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

2. Solve the two equations for t

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 t - 2 = 2$

Add to both sides:

$(6 t - 2) + 2 = 2 + 2$

Simplify the arithmetic:

$6 t = 2 + 2$

Simplify the arithmetic:

$6 t = 4$

Divide both sides by :

$\frac{(6 t)}{6} = \frac{4}{6}$

Simplify the fraction:

$t = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

5 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 = - 2$

3. List the solutions

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

4. Graph

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