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Solution - Absolute value equations

Exact form:

t = - \frac{75}{8}, \frac{225}{23}

t=-\frac{75}{8} , \frac{225}{23}

Mixed number form:

t = - 9 \frac{3}{8}, 9 \frac{18}{23}

t=-9\frac{3}{8} , 9\frac{18}{23}

Decimal form:

t = - 9.375, 9.783

t=-9.375 , 9.783

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
$| - 5 t + 100 | = | - \frac{31}{3} t + 50 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 5 t + 100 \| = \| - \frac{31}{3} t + 50 \|$
$x = + y$	$(- 5 t + 100) = (- \frac{31}{3} t + 50)$
$x = - y$	$(- 5 t + 100) = - (- \frac{31}{3} t + 50)$
$+ x = y$	$(- 5 t + 100) = (- \frac{31}{3} t + 50)$
$- x = y$	$- (- 5 t + 100) = (- \frac{31}{3} t + 50)$

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:

$\| x \| = \| y \|$	$\| - 5 t + 100 \| = \| - \frac{31}{3} t + 50 \|$
$x = + y$ , $+ x = y$	$(- 5 t + 100) = (- \frac{31}{3} t + 50)$
$x = - y$ , $- x = y$	$(- 5 t + 100) = - (- \frac{31}{3} t + 50)$

2. Solve the two equations for t

19 additional steps

$(- 5 t + 100) = (\frac{- 31}{3} t + 50)$

Add to both sides:

$(- 5 t + 100) + \frac{31}{3} \cdot t = (\frac{- 31}{3} t + 50) + \frac{31}{3} t$

Group like terms:

$(- 5 t + \frac{31}{3} \cdot t) + 100 = (\frac{- 31}{3} \cdot t + 50) + \frac{31}{3} t$

Group the coefficients:

$(- 5 + \frac{31}{3}) t + 100 = (\frac{- 31}{3} \cdot t + 50) + \frac{31}{3} t$

Convert the integer into a fraction:

$(\frac{- 15}{3} + \frac{31}{3}) t + 100 = (\frac{- 31}{3} \cdot t + 50) + \frac{31}{3} t$

Combine the fractions:

$\frac{(- 15 + 31)}{3} \cdot t + 100 = (\frac{- 31}{3} \cdot t + 50) + \frac{31}{3} t$

Combine the numerators:

$\frac{16}{3} \cdot t + 100 = (\frac{- 31}{3} \cdot t + 50) + \frac{31}{3} t$

Group like terms:

$\frac{16}{3} \cdot t + 100 = (\frac{- 31}{3} \cdot t + \frac{31}{3} t) + 50$

Combine the fractions:

$\frac{16}{3} \cdot t + 100 = \frac{(- 31 + 31)}{3} t + 50$

Combine the numerators:

$\frac{16}{3} \cdot t + 100 = \frac{0}{3} t + 50$

Reduce the zero numerator:

$\frac{16}{3} t + 100 = 0 t + 50$

Simplify the arithmetic:

$\frac{16}{3} t + 100 = 50$

Subtract from both sides:

$(\frac{16}{3} t + 100) - 100 = 50 - 100$

Simplify the arithmetic:

$\frac{16}{3} t = 50 - 100$

Simplify the arithmetic:

$\frac{16}{3} t = - 50$

Multiply both sides by inverse fraction :

$(\frac{16}{3} t) \cdot \frac{3}{16} = - 50 \cdot \frac{3}{16}$

Group like terms:

$(\frac{16}{3} \cdot \frac{3}{16}) t = - 50 \cdot \frac{3}{16}$

Multiply the coefficients:

$\frac{(16 \cdot 3)}{(3 \cdot 16)} t = - 50 \cdot \frac{3}{16}$

Simplify the fraction:

$t = - 50 \cdot \frac{3}{16}$

Multiply the fraction(s):

$t = \frac{(- 50 \cdot 3)}{16}$

Simplify the arithmetic:

$t = \frac{- 75}{8}$

23 additional steps

$(- 5 t + 100) = - (\frac{- 31}{3} t + 50)$

Expand the parentheses:

$(- 5 t + 100) = \frac{31}{3} t - 50$

Subtract from both sides:

$(- 5 t + 100) - \frac{31}{3} \cdot t = (\frac{31}{3} t - 50) - \frac{31}{3} t$

Group like terms:

$(- 5 t + \frac{- 31}{3} \cdot t) + 100 = (\frac{31}{3} \cdot t - 50) - \frac{31}{3} t$

Group the coefficients:

$(- 5 + \frac{- 31}{3}) t + 100 = (\frac{31}{3} \cdot t - 50) - \frac{31}{3} t$

Convert the integer into a fraction:

$(\frac{- 15}{3} + \frac{- 31}{3}) t + 100 = (\frac{31}{3} \cdot t - 50) - \frac{31}{3} t$

Combine the fractions:

$\frac{(- 15 - 31)}{3} \cdot t + 100 = (\frac{31}{3} \cdot t - 50) - \frac{31}{3} t$

Combine the numerators:

$\frac{- 46}{3} \cdot t + 100 = (\frac{31}{3} \cdot t - 50) - \frac{31}{3} t$

Group like terms:

$\frac{- 46}{3} \cdot t + 100 = (\frac{31}{3} \cdot t + \frac{- 31}{3} t) - 50$

Combine the fractions:

$\frac{- 46}{3} \cdot t + 100 = \frac{(31 - 31)}{3} t - 50$

Combine the numerators:

$\frac{- 46}{3} \cdot t + 100 = \frac{0}{3} t - 50$

Reduce the zero numerator:

$\frac{- 46}{3} t + 100 = 0 t - 50$

Simplify the arithmetic:

$\frac{- 46}{3} t + 100 = - 50$

Subtract from both sides:

$(\frac{- 46}{3} t + 100) - 100 = - 50 - 100$

Simplify the arithmetic:

$\frac{- 46}{3} t = - 50 - 100$

Simplify the arithmetic:

$\frac{- 46}{3} t = - 150$

Multiply both sides by inverse fraction :

$(\frac{- 46}{3} t) \cdot \frac{3}{- 46} = - 150 \cdot \frac{3}{- 46}$

Move the negative sign from the denominator to the numerator:

$\frac{- 46}{3} t \cdot \frac{- 3}{46} = - 150 \cdot \frac{3}{- 46}$

Group like terms:

$(\frac{- 46}{3} \cdot \frac{- 3}{46}) t = - 150 \cdot \frac{3}{- 46}$

Multiply the coefficients:

$\frac{(- 46 \cdot - 3)}{(3 \cdot 46)} t = - 150 \cdot \frac{3}{- 46}$

Simplify the arithmetic:

$1 t = - 150 \cdot \frac{3}{- 46}$

$t = - 150 \cdot \frac{3}{- 46}$

Move the negative sign from the denominator to the numerator:

$t = - 150 \cdot \frac{- 3}{46}$

Multiply the fraction(s):

$t = \frac{(- 150 \cdot - 3)}{46}$

Simplify the arithmetic:

$t = \frac{225}{23}$

3. List the solutions

$t = - \frac{75}{8}, \frac{225}{23}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 5 t + 100 |$
$y = | - \frac{31}{3} t + 50 |$
The equation is true where the two lines cross.

How did we do?

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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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