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Solution - Absolute value equations

Exact form:

x = \frac{3}{7}, \frac{5}{9}

x=\frac{3}{7} , \frac{5}{9}

Decimal form:

x = 0.429, 0.556

x=0.429 , 0.556

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
$| 2 x - 1 | = \frac{1}{4} | x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \frac{1}{4} \| x - 1 \|$
$x = + y$	$(2 x - 1) = \frac{1}{4} (x - 1)$
$x = - y$	$(2 x - 1) = \frac{1}{4} (- (x - 1))$
$+ x = y$	$(2 x - 1) = \frac{1}{4} (x - 1)$
$- x = y$	$- (2 x - 1) = \frac{1}{4} (x - 1)$

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:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \frac{1}{4} \| x - 1 \|$
$x = + y$ , $+ x = y$	$(2 x - 1) = \frac{1}{4} (x - 1)$
$x = - y$ , $- x = y$	$(2 x - 1) = \frac{1}{4} (- (x - 1))$

2. Solve the two equations for x

23 additional steps

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

Multiply the fraction(s):

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

Break up the fraction:

$(2 x - 1) = \frac{x}{4} + \frac{- 1}{4}$

Subtract from both sides:

$(2 x - 1) - \frac{x}{4} = (\frac{x}{4} + \frac{- 1}{4}) - \frac{x}{4}$

Group like terms:

$(2 x + \frac{- 1}{4} x) - 1 = (\frac{x}{4} + \frac{- 1}{4}) - \frac{x}{4}$

Group the coefficients:

$(2 + \frac{- 1}{4}) x - 1 = (\frac{x}{4} + \frac{- 1}{4}) - \frac{x}{4}$

Convert the integer into a fraction:

$(\frac{8}{4} + \frac{- 1}{4}) x - 1 = (\frac{x}{4} + \frac{- 1}{4}) - \frac{x}{4}$

Combine the fractions:

$\frac{(8 - 1)}{4} x - 1 = (\frac{x}{4} + \frac{- 1}{4}) - \frac{x}{4}$

Combine the numerators:

$\frac{7}{4} x - 1 = (\frac{x}{4} + \frac{- 1}{4}) - \frac{x}{4}$

Group like terms:

$\frac{7}{4} \cdot x - 1 = (\frac{x}{4} + \frac{- 1}{4} x) + \frac{- 1}{4}$

Combine the fractions:

$\frac{7}{4} \cdot x - 1 = \frac{(1 - 1)}{4} x + \frac{- 1}{4}$

Combine the numerators:

$\frac{7}{4} \cdot x - 1 = \frac{0}{4} x + \frac{- 1}{4}$

Reduce the zero numerator:

$\frac{7}{4} x - 1 = 0 x + \frac{- 1}{4}$

Simplify the arithmetic:

$\frac{7}{4} x - 1 = \frac{- 1}{4}$

Add to both sides:

$(\frac{7}{4} x - 1) + 1 = (\frac{- 1}{4}) + 1$

Simplify the arithmetic:

$\frac{7}{4} x = (\frac{- 1}{4}) + 1$

Convert the integer into a fraction:

$\frac{7}{4} x = \frac{- 1}{4} + \frac{4}{4}$

Combine the fractions:

$\frac{7}{4} x = \frac{(- 1 + 4)}{4}$

Combine the numerators:

$\frac{7}{4} x = \frac{3}{4}$

Multiply both sides by inverse fraction :

$(\frac{7}{4} x) \cdot \frac{4}{7} = (\frac{3}{4}) \cdot \frac{4}{7}$

Group like terms:

$(\frac{7}{4} \cdot \frac{4}{7}) x = (\frac{3}{4}) \cdot \frac{4}{7}$

Multiply the coefficients:

$\frac{(7 \cdot 4)}{(4 \cdot 7)} x = (\frac{3}{4}) \cdot \frac{4}{7}$

Simplify the fraction:

$x = (\frac{3}{4}) \cdot \frac{4}{7}$

Multiply the fraction(s):

$x = \frac{(3 \cdot 4)}{(4 \cdot 7)}$

Simplify the arithmetic:

$x = \frac{3}{7}$

24 additional steps

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

Multiply the fraction(s):

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

Expand the parentheses:

$(2 x - 1) = \frac{(- x + 1)}{4}$

Break up the fraction:

$(2 x - 1) = \frac{- x}{4} + \frac{1}{4}$

Add to both sides:

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

Group like terms:

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

Group the coefficients:

$(2 + \frac{1}{4}) x - 1 = (\frac{- x}{4} + \frac{1}{4}) + \frac{1}{4} x$

Convert the integer into a fraction:

$(\frac{8}{4} + \frac{1}{4}) x - 1 = (\frac{- x}{4} + \frac{1}{4}) + \frac{1}{4} x$

Combine the fractions:

$\frac{(8 + 1)}{4} \cdot x - 1 = (\frac{- x}{4} + \frac{1}{4}) + \frac{1}{4} x$

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{9}{4} \cdot x - 1 = \frac{0}{4} x + \frac{1}{4}$

Reduce the zero numerator:

$\frac{9}{4} x - 1 = 0 x + \frac{1}{4}$

Simplify the arithmetic:

$\frac{9}{4} x - 1 = \frac{1}{4}$

Add to both sides:

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

Simplify the arithmetic:

$\frac{9}{4} x = (\frac{1}{4}) + 1$

Convert the integer into a fraction:

$\frac{9}{4} x = \frac{1}{4} + \frac{4}{4}$

Combine the fractions:

$\frac{9}{4} x = \frac{(1 + 4)}{4}$

Combine the numerators:

$\frac{9}{4} x = \frac{5}{4}$

Multiply both sides by inverse fraction :

$(\frac{9}{4} x) \cdot \frac{4}{9} = (\frac{5}{4}) \cdot \frac{4}{9}$

Group like terms:

$(\frac{9}{4} \cdot \frac{4}{9}) x = (\frac{5}{4}) \cdot \frac{4}{9}$

Multiply the coefficients:

$\frac{(9 \cdot 4)}{(4 \cdot 9)} x = (\frac{5}{4}) \cdot \frac{4}{9}$

Simplify the fraction:

$x = (\frac{5}{4}) \cdot \frac{4}{9}$

Multiply the fraction(s):

$x = \frac{(5 \cdot 4)}{(4 \cdot 9)}$

Simplify the arithmetic:

$x = \frac{5}{9}$

3. List the solutions

$x = \frac{3}{7}, \frac{5}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 1 |$
$y = \frac{1}{4} | x - 1 |$
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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