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Solution - Absolute value equations

Exact form:

x = \frac{11}{9}, \frac{7}{5}

x=\frac{11}{9} , \frac{7}{5}

Mixed number form:

x = 1 \frac{2}{9}, 1 \frac{2}{5}

x=1\frac{2}{9} , 1\frac{2}{5}

Decimal form:

x = 1.222, 1.4

x=1.222 , 1.4

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

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$12 x + 4 \cdot - 4 = (3 x - 5)$

Simplify the arithmetic:

$12 x - 16 = (3 x - 5)$

Subtract from both sides:

$(12 x - 16) - 3 x = (3 x - 5) - 3 x$

Group like terms:

$(12 x - 3 x) - 16 = (3 x - 5) - 3 x$

Simplify the arithmetic:

$9 x - 16 = (3 x - 5) - 3 x$

Group like terms:

$9 x - 16 = (3 x - 3 x) - 5$

Simplify the arithmetic:

$9 x - 16 = - 5$

Add to both sides:

$(9 x - 16) + 16 = - 5 + 16$

Simplify the arithmetic:

$9 x = - 5 + 16$

Simplify the arithmetic:

$9 x = 11$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{11}{9}$

Simplify the fraction:

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

15 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$12 x + 4 \cdot - 4 = - (3 x - 5)$

Simplify the arithmetic:

$12 x - 16 = - (3 x - 5)$

Expand the parentheses:

$12 x - 16 = - 3 x + 5$

Add to both sides:

$(12 x - 16) + 3 x = (- 3 x + 5) + 3 x$

Group like terms:

$(12 x + 3 x) - 16 = (- 3 x + 5) + 3 x$

Simplify the arithmetic:

$15 x - 16 = (- 3 x + 5) + 3 x$

Group like terms:

$15 x - 16 = (- 3 x + 3 x) + 5$

Simplify the arithmetic:

$15 x - 16 = 5$

Add to both sides:

$(15 x - 16) + 16 = 5 + 16$

Simplify the arithmetic:

$15 x = 5 + 16$

Simplify the arithmetic:

$15 x = 21$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{21}{15}$

Simplify the fraction:

$x = \frac{21}{15}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 3)}{(5 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{7}{5}$

3. List the solutions

$x = \frac{11}{9}, \frac{7}{5}$
(2 solution(s))

4. Graph

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