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Solution - Absolute value equations

Exact form:

x = - \frac{9}{11}, \frac{1}{13}

x=-\frac{9}{11} , \frac{1}{13}

Decimal form:

x = - 0.818, 0.077

x=-0.818 , 0.077

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
$\frac{1}{4} | x - 5 | = | 3 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{1}{4} \| x - 5 \| = \| 3 x + 1 \|$
$x = + y$	$\frac{1}{4} (x - 5) = (3 x + 1)$
$x = - y$	$\frac{1}{4} (x - 5) = - (3 x + 1)$
$+ x = y$	$\frac{1}{4} (x - 5) = (3 x + 1)$
$- x = y$	$\frac{1}{4} (- (x - 5)) = (3 x + 1)$

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:

$\| x \| = \| y \|$	$\frac{1}{4} \| x - 5 \| = \| 3 x + 1 \|$
$x = + y$ , $+ x = y$	$\frac{1}{4} (x - 5) = (3 x + 1)$
$x = - y$ , $- x = y$	$\frac{1}{4} (x - 5) = - (3 x + 1)$

2. Solve the two equations for x

26 additional steps

$\frac{1}{4} \cdot (x - 5) = (3 x + 1)$

Multiply the fraction(s):

$\frac{(1 \cdot (x - 5))}{4} = (3 x + 1)$

Break up the fraction:

$\frac{x}{4} + \frac{- 5}{4} = (3 x + 1)$

Subtract from both sides:

$(\frac{x}{4} + \frac{- 5}{4}) - 3 x = (3 x + 1) - 3 x$

Group like terms:

$(\frac{x}{4} - 3 x) + \frac{- 5}{4} = (3 x + 1) - 3 x$

Group the coefficients:

$(\frac{1}{4} - 3) x + \frac{- 5}{4} = (3 x + 1) - 3 x$

Convert the integer into a fraction:

$(\frac{1}{4} + \frac{- 12}{4}) x + \frac{- 5}{4} = (3 x + 1) - 3 x$

Combine the fractions:

$\frac{(1 - 12)}{4} x + \frac{- 5}{4} = (3 x + 1) - 3 x$

Combine the numerators:

$\frac{- 11}{4} x + \frac{- 5}{4} = (3 x + 1) - 3 x$

Group like terms:

$\frac{- 11}{4} x + \frac{- 5}{4} = (3 x - 3 x) + 1$

Simplify the arithmetic:

$\frac{- 11}{4} x + \frac{- 5}{4} = 1$

Add to both sides:

$(\frac{- 11}{4} x + \frac{- 5}{4}) + \frac{5}{4} = 1 + \frac{5}{4}$

Combine the fractions:

$\frac{- 11}{4} x + \frac{(- 5 + 5)}{4} = 1 + \frac{5}{4}$

Combine the numerators:

$\frac{- 11}{4} x + \frac{0}{4} = 1 + \frac{5}{4}$

Reduce the zero numerator:

$\frac{- 11}{4} x + 0 = 1 + \frac{5}{4}$

Simplify the arithmetic:

$\frac{- 11}{4} x = 1 + \frac{5}{4}$

Convert the integer into a fraction:

$\frac{- 11}{4} x = \frac{4}{4} + \frac{5}{4}$

Combine the fractions:

$\frac{- 11}{4} x = \frac{(4 + 5)}{4}$

Combine the numerators:

$\frac{- 11}{4} x = \frac{9}{4}$

Multiply both sides by inverse fraction :

$(\frac{- 11}{4} x) \cdot \frac{4}{- 11} = (\frac{9}{4}) \cdot \frac{4}{- 11}$

Move the negative sign from the denominator to the numerator:

$\frac{- 11}{4} x \cdot \frac{- 4}{11} = (\frac{9}{4}) \cdot \frac{4}{- 11}$

Group like terms:

$(\frac{- 11}{4} \cdot \frac{- 4}{11}) x = (\frac{9}{4}) \cdot \frac{4}{- 11}$

Multiply the coefficients:

$\frac{(- 11 \cdot - 4)}{(4 \cdot 11)} x = (\frac{9}{4}) \cdot \frac{4}{- 11}$

Simplify the arithmetic:

$1 x = (\frac{9}{4}) \cdot \frac{4}{- 11}$

$x = (\frac{9}{4}) \cdot \frac{4}{- 11}$

Move the negative sign from the denominator to the numerator:

$x = \frac{9}{4} \cdot \frac{- 4}{11}$

Multiply the fraction(s):

$x = \frac{(9 \cdot - 4)}{(4 \cdot 11)}$

Simplify the arithmetic:

$x = \frac{- 9}{11}$

24 additional steps

$\frac{1}{4} \cdot (x - 5) = - (3 x + 1)$

Multiply the fraction(s):

$\frac{(1 \cdot (x - 5))}{4} = - (3 x + 1)$

Break up the fraction:

$\frac{x}{4} + \frac{- 5}{4} = - (3 x + 1)$

Expand the parentheses:

$\frac{x}{4} + \frac{- 5}{4} = - 3 x - 1$

Add to both sides:

$(\frac{x}{4} + \frac{- 5}{4}) + 3 x = (- 3 x - 1) + 3 x$

Group like terms:

$(\frac{x}{4} + 3 x) + \frac{- 5}{4} = (- 3 x - 1) + 3 x$

Group the coefficients:

$(\frac{1}{4} + 3) x + \frac{- 5}{4} = (- 3 x - 1) + 3 x$

Convert the integer into a fraction:

$(\frac{1}{4} + \frac{12}{4}) x + \frac{- 5}{4} = (- 3 x - 1) + 3 x$

Combine the fractions:

$\frac{(1 + 12)}{4} x + \frac{- 5}{4} = (- 3 x - 1) + 3 x$

Combine the numerators:

$\frac{13}{4} x + \frac{- 5}{4} = (- 3 x - 1) + 3 x$

Group like terms:

$\frac{13}{4} x + \frac{- 5}{4} = (- 3 x + 3 x) - 1$

Simplify the arithmetic:

$\frac{13}{4} x + \frac{- 5}{4} = - 1$

Add to both sides:

$(\frac{13}{4} x + \frac{- 5}{4}) + \frac{5}{4} = - 1 + \frac{5}{4}$

Combine the fractions:

$\frac{13}{4} x + \frac{(- 5 + 5)}{4} = - 1 + \frac{5}{4}$

Combine the numerators:

$\frac{13}{4} x + \frac{0}{4} = - 1 + \frac{5}{4}$

Reduce the zero numerator:

$\frac{13}{4} x + 0 = - 1 + \frac{5}{4}$

Simplify the arithmetic:

$\frac{13}{4} x = - 1 + \frac{5}{4}$

Convert the integer into a fraction:

$\frac{13}{4} x = \frac{- 4}{4} + \frac{5}{4}$

Combine the fractions:

$\frac{13}{4} x = \frac{(- 4 + 5)}{4}$

Combine the numerators:

$\frac{13}{4} x = \frac{1}{4}$

Multiply both sides by inverse fraction :

$(\frac{13}{4} x) \cdot \frac{4}{13} = (\frac{1}{4}) \cdot \frac{4}{13}$

Group like terms:

$(\frac{13}{4} \cdot \frac{4}{13}) x = (\frac{1}{4}) \cdot \frac{4}{13}$

Multiply the coefficients:

$\frac{(13 \cdot 4)}{(4 \cdot 13)} x = (\frac{1}{4}) \cdot \frac{4}{13}$

Simplify the fraction:

$x = (\frac{1}{4}) \cdot \frac{4}{13}$

Multiply the fraction(s):

$x = \frac{(1 \cdot 4)}{(4 \cdot 13)}$

Simplify the arithmetic:

$x = \frac{1}{13}$

3. List the solutions

$x = - \frac{9}{11}, \frac{1}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = \frac{1}{4} | x - 5 |$
$y = | 3 x + 1 |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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