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Solution - Absolute value equations

Exact form:

t = 8, 0

t=8 , 0

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
$| t + \frac{2}{1} | = | \frac{3}{2} t - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| t + \frac{2}{1} \| = \| \frac{3}{2} t - 2 \|$
$x = + y$	$(t + \frac{2}{1}) = (\frac{3}{2} t - 2)$
$x = - y$	$(t + \frac{2}{1}) = - (\frac{3}{2} t - 2)$
$+ x = y$	$(t + \frac{2}{1}) = (\frac{3}{2} t - 2)$
$- x = y$	$- (t + \frac{2}{1}) = (\frac{3}{2} t - 2)$

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:

$\| x \| = \| y \|$	$\| t + \frac{2}{1} \| = \| \frac{3}{2} t - 2 \|$
$x = + y$ , $+ x = y$	$(t + \frac{2}{1}) = (\frac{3}{2} t - 2)$
$x = - y$ , $- x = y$	$(t + \frac{2}{1}) = - (\frac{3}{2} t - 2)$

2. Solve the two equations for t

20 additional steps

$t + \frac{2}{1} = (\frac{3}{2} t - 2)$

A variable's value does not change when it is divided by 1, so we can eliminate it:

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

$(1 + \frac{- 3}{2}) t + 2 = (\frac{3}{2} \cdot t - 2) - \frac{3}{2} t$

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{- 1}{2} \cdot t + 2 = (\frac{3}{2} \cdot t - 2) - \frac{3}{2} t$

Group like terms:

$\frac{- 1}{2} \cdot t + 2 = (\frac{3}{2} \cdot t + \frac{- 3}{2} t) - 2$

Combine the fractions:

$\frac{- 1}{2} \cdot t + 2 = \frac{(3 - 3)}{2} t - 2$

Combine the numerators:

$\frac{- 1}{2} \cdot t + 2 = \frac{0}{2} t - 2$

Reduce the zero numerator:

$\frac{- 1}{2} t + 2 = 0 t - 2$

Simplify the arithmetic:

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

Subtract from both sides:

$(\frac{- 1}{2} t + 2) - 2 = - 2 - 2$

Simplify the arithmetic:

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

Simplify the arithmetic:

$\frac{- 1}{2} t = - 4$

Multiply both sides by inverse fraction :

$(\frac{- 1}{2} t) \cdot \frac{2}{- 1} = - 4 \cdot \frac{2}{- 1}$

Group like terms:

$(\frac{- 1}{2} \cdot - 2) t = - 4 \cdot \frac{2}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 2)}{2} t = - 4 \cdot \frac{2}{- 1}$

Simplify the arithmetic:

$1 t = - 4 \cdot \frac{2}{- 1}$

$t = - 4 \cdot \frac{2}{- 1}$

Simplify the arithmetic:

$t = 8$

16 additional steps

$t + \frac{2}{1} = - (\frac{3}{2} t - 2)$

A variable's value does not change when it is divided by 1, so we can eliminate it:

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

$(1 + \frac{3}{2}) t + 2 = (\frac{- 3}{2} \cdot t + 2) + \frac{3}{2} t$

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{5}{2} \cdot t + 2 = (\frac{- 3}{2} \cdot t + 2) + \frac{3}{2} t$

Group like terms:

$\frac{5}{2} \cdot t + 2 = (\frac{- 3}{2} \cdot t + \frac{3}{2} t) + 2$

Combine the fractions:

$\frac{5}{2} \cdot t + 2 = \frac{(- 3 + 3)}{2} t + 2$

Combine the numerators:

$\frac{5}{2} \cdot t + 2 = \frac{0}{2} t + 2$

Reduce the zero numerator:

$\frac{5}{2} t + 2 = 0 t + 2$

Simplify the arithmetic:

$\frac{5}{2} t + 2 = 2$

Subtract from both sides:

$(\frac{5}{2} t + 2) - 2 = 2 - 2$

Simplify the arithmetic:

$\frac{5}{2} t = 2 - 2$

Simplify the arithmetic:

$\frac{5}{2} t = 0$

Divide both sides by the coefficient:

$t = 0$

3. List the solutions

$t = 8, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | t + \frac{2}{1} |$
$y = | \frac{3}{2} t - 2 |$
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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