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Solution - Absolute value equations

Exact form:

t = \frac{1}{2}, - \frac{1}{6}

t=\frac{1}{2} , -\frac{1}{6}

Decimal form:

t = 0.5, - 0.167

t=0.5 , -0.167

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
$| 8 t - 2 | = | - 2 t + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 t - 2 \| = \| - 2 t + 3 \|$
$x = + y$	$(8 t - 2) = (- 2 t + 3)$
$x = - y$	$(8 t - 2) = - (- 2 t + 3)$
$+ x = y$	$(8 t - 2) = (- 2 t + 3)$
$- x = y$	$- (8 t - 2) = (- 2 t + 3)$

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:

$\| x \| = \| y \|$	$\| 8 t - 2 \| = \| - 2 t + 3 \|$
$x = + y$ , $+ x = y$	$(8 t - 2) = (- 2 t + 3)$
$x = - y$ , $- x = y$	$(8 t - 2) = - (- 2 t + 3)$

2. Solve the two equations for t

11 additional steps

$(8 t - 2) = (- 2 t + 3)$

Add to both sides:

$(8 t - 2) + 2 t = (- 2 t + 3) + 2 t$

Group like terms:

$(8 t + 2 t) - 2 = (- 2 t + 3) + 2 t$

Simplify the arithmetic:

$10 t - 2 = (- 2 t + 3) + 2 t$

Group like terms:

$10 t - 2 = (- 2 t + 2 t) + 3$

Simplify the arithmetic:

$10 t - 2 = 3$

Add to both sides:

$(10 t - 2) + 2 = 3 + 2$

Simplify the arithmetic:

$10 t = 3 + 2$

Simplify the arithmetic:

$10 t = 5$

Divide both sides by :

$\frac{(10 t)}{10} = \frac{5}{10}$

Simplify the fraction:

$t = \frac{5}{10}$

Find the greatest common factor of the numerator and denominator:

$t = \frac{(1 \cdot 5)}{(2 \cdot 5)}$

Factor out and cancel the greatest common factor:

$t = \frac{1}{2}$

10 additional steps

$(8 t - 2) = - (- 2 t + 3)$

Expand the parentheses:

$(8 t - 2) = 2 t - 3$

Subtract from both sides:

$(8 t - 2) - 2 t = (2 t - 3) - 2 t$

Group like terms:

$(8 t - 2 t) - 2 = (2 t - 3) - 2 t$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 t - 2 = - 3$

Add to both sides:

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

Simplify the arithmetic:

$6 t = - 3 + 2$

Simplify the arithmetic:

$6 t = - 1$

Divide both sides by :

$\frac{(6 t)}{6} = \frac{- 1}{6}$

Simplify the fraction:

$t = \frac{- 1}{6}$

3. List the solutions

$t = \frac{1}{2}, - \frac{1}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 t - 2 |$
$y = | - 2 t + 3 |$
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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