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Solution - Absolute value equations

Exact form:

x = - \frac{5}{6}, \frac{5}{2}

x=-\frac{5}{6} , \frac{5}{2}

Mixed number form:

x = - \frac{5}{6}, 2 \frac{1}{2}

x=-\frac{5}{6} , 2\frac{1}{2}

Decimal form:

x = - 0.833, 2.5

x=-0.833 , 2.5

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
$5 | 2 x - 3 | = 4 | x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| 2 x - 3 \| = 4 \| x - 5 \|$
$x = + y$	$5 (2 x - 3) = 4 (x - 5)$
$x = - y$	$5 (2 x - 3) = 4 (- (x - 5))$
$+ x = y$	$5 (2 x - 3) = 4 (x - 5)$
$- x = y$	$5 (- (2 x - 3)) = 4 (x - 5)$

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:

$\| x \| = \| y \|$	$5 \| 2 x - 3 \| = 4 \| x - 5 \|$
$x = + y$ , $+ x = y$	$5 (2 x - 3) = 4 (x - 5)$
$x = - y$ , $- x = y$	$5 (2 x - 3) = 4 (- (x - 5))$

2. Solve the two equations for x

14 additional steps

$5 \cdot (2 x - 3) = 4 \cdot (x - 5)$

Expand the parentheses:

$5 \cdot 2 x + 5 \cdot - 3 = 4 \cdot (x - 5)$

Multiply the coefficients:

$10 x + 5 \cdot - 3 = 4 \cdot (x - 5)$

Simplify the arithmetic:

$10 x - 15 = 4 \cdot (x - 5)$

Expand the parentheses:

$10 x - 15 = 4 x + 4 \cdot - 5$

Simplify the arithmetic:

$10 x - 15 = 4 x - 20$

Subtract from both sides:

$(10 x - 15) - 4 x = (4 x - 20) - 4 x$

Group like terms:

$(10 x - 4 x) - 15 = (4 x - 20) - 4 x$

Simplify the arithmetic:

$6 x - 15 = (4 x - 20) - 4 x$

Group like terms:

$6 x - 15 = (4 x - 4 x) - 20$

Simplify the arithmetic:

$6 x - 15 = - 20$

Add to both sides:

$(6 x - 15) + 15 = - 20 + 15$

Simplify the arithmetic:

$6 x = - 20 + 15$

Simplify the arithmetic:

$6 x = - 5$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 5}{6}$

Simplify the fraction:

$x = \frac{- 5}{6}$

19 additional steps

$5 \cdot (2 x - 3) = 4 \cdot (- (x - 5))$

Expand the parentheses:

$5 \cdot 2 x + 5 \cdot - 3 = 4 \cdot (- (x - 5))$

Multiply the coefficients:

$10 x + 5 \cdot - 3 = 4 \cdot (- (x - 5))$

Simplify the arithmetic:

$10 x - 15 = 4 \cdot (- (x - 5))$

Expand the parentheses:

$10 x - 15 = 4 \cdot (- x + 5)$

$10 x - 15 = 4 \cdot - x + 4 \cdot 5$

Group like terms:

$10 x - 15 = (4 \cdot - 1) x + 4 \cdot 5$

Multiply the coefficients:

$10 x - 15 = - 4 x + 4 \cdot 5$

Simplify the arithmetic:

$10 x - 15 = - 4 x + 20$

Add to both sides:

$(10 x - 15) + 4 x = (- 4 x + 20) + 4 x$

Group like terms:

$(10 x + 4 x) - 15 = (- 4 x + 20) + 4 x$

Simplify the arithmetic:

$14 x - 15 = (- 4 x + 20) + 4 x$

Group like terms:

$14 x - 15 = (- 4 x + 4 x) + 20$

Simplify the arithmetic:

$14 x - 15 = 20$

Add to both sides:

$(14 x - 15) + 15 = 20 + 15$

Simplify the arithmetic:

$14 x = 20 + 15$

Simplify the arithmetic:

$14 x = 35$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{35}{14}$

Simplify the fraction:

$x = \frac{35}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 7)}{(2 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{2}$

3. List the solutions

$x = - \frac{5}{6}, \frac{5}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | 2 x - 3 |$
$y = 4 | x - 5 |$
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 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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